# Finitely generated simple sharply 2-transitive groups

Simon André and Vincent Guirardel

###### Abstract

We construct the first examples of infinite sharply 2-transitive groups which
are finitely generated. Moreover, we construct such a group that has Kazhdan
property (T), is simple, has exactly four conjugacy classes, and we show that
this number is as small as possible.

##  1 Introduction

An action of a group $G$ on a set $X$ containing at least two elements is said
to be _sharply $2$-transitive_ if, for any two couples $(x_{1},x_{2})$ and
$(y_{1},y_{2})$ of distinct elements of $X$, there exists a unique element of
$G$ mapping $(x_{1},x_{2})$ to $(y_{1},y_{2})$. A group $G$ is called _sharply
$2$-transitive_ if there exists a set $X$ on which $G$ acts sharply
$2$-transitively. For instance, if $K$ is any skew field, the natural action
of the affine group $K^{*}\ltimes K$ on $K$ is sharply 2-transitive. This
generalizes to a near-field $K$ where only one of the two distributive rules
is required (see for instance [DM96, §7.6]). We say that $G$ is _nearly
affine_ if $G\simeq K^{*}\ltimes K$ for some near-field $K$ (this notion is
also referred to as _split_ in the literature).

Zassenhaus proved that every finite sharply $2$-transitive group $G$ is nearly
affine, and classified all finite near-fields in [Zas35a, Zas35b]. In the same
direction, Tits proved in [Tit52, Tit56] that if $G$ is a locally compact
connected group having a continuous sharply 2-transitive action on a locally
compact set, then $G$ is nearly affine. In fact, $G\simeq K^{*}\ltimes K$
where $K$ is either ${\mathbb{R}}$, ${\mathbb{C}}$, ${\mathbb{H}}$ or a near-
field obtained by twisting the multiplication on ${\mathbb{H}}$. More
recently, Glasner and Gulko [GG14] and Glauberman, Mann and Segev [GMS15]
proved that if $G$ is any linear group having a sharply 2-transitive action of
characteristic distinct from 2 (see Subsection 2.1 for a definition), then
again, $G$ is nearly affine.

Until recently, it was an open problem whether every sharply 2-transitive
group $G$ is nearly affine. The first counterexamples were constructed in
[RST17] and [RT19] by Rips, Segev and Tent. Then, infinite simple sharply
2-transitive groups were constructed in [AT21].

But none of the examples constructed in [RST17, RT19, AT21] is finitely
generated, so one may ask whether Zassenhaus classification extends assuming
finite generation: if $G$ is a finitely generated sharply 2-transitive group,
does $G$ have to be nearly affine?

In fact, there is an even more basic question: does there exist an infinite,
sharply 2-transitive group $G$ which is finitely generated? Indeed, no group
of the form $G=K^{*}\ltimes K$ for a skew field $K$ is known to be finitely
generated as the existence of an infinite skew field $K$ whose multiplicative
group is finitely generated is a famous open problem (see for instance [AMH00,
AKBPR19]).

In the present paper, we answer the two questions above: we construct the
first examples of infinite sharply 2-transitive groups which are finitely
generated, and our examples are not nearly affine. These are also the first
examples of such infinite groups having Kazhdan property (T).

######  Theorem 1.1.

There exists an infinite, finitely generated, sharply 2-transitive group $G$.
Moreover, one can construct such a group $G$ which has property (T), is
2-generated and simple (and in particular, not nearly affine).

######  Remark 1.2.

The sharply 2-transitive groups constructed in this paper have characteristic
0, meaning that involutions have a fixed point and that the products of
distinct involutions have infinite order (see Subsection 2.1 for more
details).

In fact, it is well known (see [Ker74, Chapter II §6]) that every sharply
2-transitive group is the group of affine transformations of a _near-domain_ :
this is a structure $(K,+,\cdot)$ similar to a near-field but in which $(K,+)$
is not required to be a commutative group but only a loop with a twisted
associativity condition. In this context, the group $G$ of affine
transformations $x\mapsto ax+b$ of $K$ acts sharply 2-transitively on $K$, but
it is not a semi-direct product in general. Actually, $G$ is a semi-direct
product if and only if $K$ is a near-field, in which case $G$ is nearly affine
(see [Ker74, Theorem 7.1]). Thus, examples of exotic sharply 2-transitive
groups yield exotic near-domains, and Theorem 1.1 can be reformulated in terms
of near-domains as follows.

######  Corollary 1.3.

There exists an infinite near-domain $K$ which is not a near-field and whose
associated group of affine transformations is finitely generated.

We also consider the problem of minimizing the number of conjugacy classes.
First, any sharply 2-transitive group has to have at least three conjugacy
classes (except for ${\mathbb{Z}}/2{\mathbb{Z}}$), and at least four conjugacy
classes in characteristic distinct from $2$ (except for the affine group on
${\mathbb{F}}_{3}$), see Proposition 2.3. Actually, except for the affine
groups on ${\mathbb{F}}_{2}$, ${\mathbb{F}}_{3}$ and ${\mathbb{F}}_{4}$, all
_finite_ sharply 2-transitive groups have at least five conjugacy classes.

On the other hand, Cameron [Cam00] mentions that Cohn’s construction [Coh71,
Theorem 6.3] can be used to construct an infinite skew field $K$ of
characterisic $p$ such that $K^{*}$ has exactly $p$ conjugacy classes. For
$p=2$, the corresponding affine group $K^{*}\ltimes K$ would have exactly
three conjugacy classes. In characteristic 0, all the previously known sharply
2-transitive groups have infinitely many conjugacy classes.

Also note that in the realm of abstract groups, the first infinite, finitely
generated groups with only finitely many conjugacy classes were constructed by
Ivanov, and a construction can be found in [Ols91] (see Theorem 41.2). This
construction yields for each sufficiently large prime number $p$ a group of
exponent $p$ with exactly $p$ conjugacy classes. Then, Osin constructed in
[Osi10] the first infinite finitely generated groups with exactly $n$
conjugacy classes for any natural number $n\geq 2$.

Our construction, relying on Osin’s small cancellation techniques [Osi10],
provides examples of groups which are sharply 2-transitive in characteristic
0, are finitely generated, and have the smallest possible number of conjugacy
classes.

######  Theorem 1.4.

There exists an infinite simple sharply 2-transitive group which is finitely
generated, has property (T), and has exactly four conjugacy classes.

The proof of our results combines the strategy used in [RT19] and [AT21] with
small cancellation theory over relatively hyperbolic groups, as developed
notably by Osin in [Osi10] to construct infinite finitely generated groups
with exactly two conjugacy classes.

#### Acknowledgements.

The first named author would like to thank Katrin Tent for stimulating
discussions, and acknowledges support from the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy EXC
2044–390685587, Mathematics Münster: Dynamics-Geometry-Structure and from CRC
1442 Geometry: Deformations and Rigidity. The second named author benefits
from the support of the French government “Investissements d’Avenir” program
ANR-11-LABX-0020-01 and from the ANR project ANR-22-CE40-0004.

##  2 Conjugacy classes in a sharply 2-transitive group

###  2.1 Characteristic of a sharply 2-transitive group

If $G$ acts sharply 2-transitively on a set $X$ (with $|X|\geq 2$), then $G$
has involutions (take any element exchanging two points $x,y\in X$), and all
involutions are conjugate (any involution swaps a pair of points, and two
involutions can be conjugate to swap the same pair of points).

By analogy with the affine group $K^{*}\ltimes K$, one says that
$G\curvearrowright X$ is of characteristic $2$ if involutions have no fixed
point.

When the action is not of characteristic $2$, every involution has a unique
fixed point, and distinct involutions have distinct fixed points (see for
instance [Ten16, Corollary 3.2]). It follows that there is a natural
equivariant bijection between $X$ and the set ${\mathcal{I}}_{G}$ of
involutions of $G$, on which $G$ acts by conjugation. In particular, a group
has a sharply 2-transitive action of characteristic distinct from 2 on a set
$X$ if and only if it acts sharply 2-transitively on its set of involutions.
It also follows that all pairs of distinct involutions are conjugate. By
analogy with the affine group in characteristic $\neq 2$, we say that an
element of $G$ is a _translation_ if it is the product of two distinct
involutions. Thus, in characteristic $\neq 2$, all translations are conjugate,
and one says that $G\curvearrowright X$ has characteristic 0 if translations
have infinite order (i.e. if two distinct involutions generate an infinite
dihedral group) and characteristic $p$ if translations have order $p$ (i.e.
any two distinct involutions generate a dihedral group of order $2p$), and $p$
has to be some prime number (otherwise the subgroup generated by a translation
$t$ of order $p$ would contain a translation $t^{\prime}$ of order
$p^{\prime}<p$, thus not conjugate to $t$). Note that if $K$ is a field, then
the characteristic of $K$ in the usual sense coincides with the characteristic
of $K^{*}\ltimes K$ as defined above. The sharply 2-transitive groups we will
construct in this paper will have characteristic 0.

###  2.2 A lower bound on the number of conjugacy classes

A sharply 2-transitive group with exactly two conjugacy classes is necessarily
abelian since all its non-trivial elements have order 2. But the only abelian
sharply 2-transitive group is $\mathbb{Z}/2\mathbb{Z}$ (because the stabilizer
of a point in such a group is malnormal), and hence every sharply 2-transitive
group of order $>2$ has at least three conjugacy classes.

The only finite sharply 2-transitive group with exactly three conjugacy
classes is the symmetric group $S_{3}$, but there are infinite examples: Cohn
constructed a skew field $K$ of characteristic 2 whose multiplicative group
$K^{*}$ has exactly two conjugacy classes (see Corollary 2 of Theorem 6.3 in
[Coh71]), and the corresponding affine group $K^{*}\ltimes K$ has exactly
three conjugacy classes.

The goal of this section is Proposition 2.3 below showing that any sharply
2-transitive group $G\curvearrowright X$ of characteristic $\neq 2$ with
$|X|\geq 4$ has at least 4 conjugacy classes. Moreover, in characteristic 2,
the only possibility to have at most 3 conjugacy classes comes from near-
fields of characteristic 2. In Section 4, we will construct infinite sharply
2-transitive groups of characteristic 0 with exactly 4 conjugacy classes.

######  Lemma 2.1 ([Ten16, Corollary 3.2]).

Let $G$ be a group acting sharply 2-transitively on a set $X$, with $|X|\geq
2$. Then two distinct involutions cannot have a common fixed point in $X$.

###### Proof.

The lemma is obvious if the characteristic is $2$, i.e. if involutions don’t
have fixed points. Otherwise, this is [Ten16, Corollary 3.2]. ∎

######  Lemma 2.2.

Let $G$ be a group acting sharply 2-transitively on a set $X$, with $|X|\geq
2$. The product of two distinct involutions cannot fix a point.

###### Proof.

Assume towards a contradiction that a product $uv$ of two distinct involutions
fixes a point $x\in X$. If $v$ fixes $x$, so does $u$, which contradicts the
previous lemma. Thus the involution $v$ exchanges the point $x$ and the point
$y=v\cdot x\neq x$. Since $uv\cdot x=x$, $u$ exchanges $x$ and $y$. By
$2$-sharpness, $u=v$. ∎

######  Proposition 2.3.

Let $G\curvearrowright X$ be a sharply 2-transitive group with $|X|\geq 4$.

  * •

If $G\curvearrowright X$ is not of characteristic 2, then $G$ has at least 4
conjugacy classes.

  * •

If $G\curvearrowright X$ is of characteristic 2, then $G$ has at least 3
conjugacy classes, and at least 4 conjugacy classes unless $G\simeq
K^{*}\ltimes K$ for some near-field $K$ such that all non-trivial elements of
$K^{*}$ are conjugate.

###### Proof.

We already saw at the beginning of this section that if $|X|\geq 3$, $G$ has
at least 3 conjugacy classes.

Fix a point $x\in X$. By Lemmas 2.1 and 2.2, $G_{x}$ contains at most one
involution and no translation (i.e. a product of two distinct involutions).
Moreover, $|X\setminus\\{x\\}|\geq 3$ and $G_{x}$ acts transitively on
$X\setminus\\{x\\}$. Thus, $G_{x}$ contains a non-trivial element $g_{x}$
which is neither an involution nor a translation.

Assume $G\curvearrowright X$ has characteristic $p\neq 2$, and consider two
distinct involutions $u,v$. Then the translation $uv$ is an element of order
$p$ (or infinite order if $p=0$), hence is not an involution. This shows that
the elements $1,u,uv,g_{x}$ are in four distinct conjugacy classes.

In characteristic 2, the argument above says that if some translation $uv$ in
not an involution then $G$ has at least four conjugacy classes. We may thus
assume that the product of any two distinct involutions is an involution, i.e.
that the set ${\mathcal{I}}_{G}\cup\\{1\\}$ is a (normal) subgroup of $G$. By
[Ker74, Theorem 7.1], $G=K^{*}\ltimes K$ for some near-field $K$ (where
$K={\mathcal{I}}_{G}\cup\\{1\\}$ as an additive group). If $G$ has at most 3
conjugacy classes, then $K^{*}$ has at most 2 conjugacy classes. ∎

##  3 The classes of groups ${\mathcal{C}}$ and ${\mathcal{C}}^{\prime}$

In this section that closely follows [AT21, RT19], we introduce two classes of
groups ${\mathcal{C}}$ and ${\mathcal{C}}^{\prime}$ that are stable under
various types of HNN extensions, amalgams and increasing union. Any sharply
2-transitive group of characteristic 0 is in the class ${\mathcal{C}}$, but
the class ${\mathcal{C}}^{\prime}$ is more restrictive. We show that these
classes are stable under

###  3.1 The class ${\mathcal{C}}$

In [AT21] (building on [RT19]), a class of groups $\mathcal{C}$ is introduced
that contains all sharply 2-transitive groups of characteristic 0 and which is
preserved under various HNN extensions and amalgamated products. We
reformulate this definition below (see Definition 3.3).

We denote by $D_{\mathbb{Z}}={\mathbb{Z}}/2{\mathbb{Z}}\ltimes{\mathbb{Z}}$
the infinite dihedral group and by
$D_{\mathbb{Q}}={\mathbb{Z}}/2{\mathbb{Z}}\ltimes{\mathbb{Q}}$ the group of
isometries of ${\mathbb{Q}}$.

A subgroup $H\subset G$ is called _malnormal_ if the intersection $H\cap
gHg^{-1}$ is trivial for every $g\in G\setminus H$, and _quasi-malnormal_ if
$H\cap gHg^{-1}$ has order at most 2 for $g\in G\setminus H$.

Recall that we denote by ${\mathcal{I}}_{G}$ the set of involutions of $G$,
and we denote by ${\mathcal{I}}_{G}^{(2)}=\\{(u,v)\in{\mathcal{I}}_{G}^{2}\ |\
u\neq v\\}$ the set of ordered pairs of involutions, on which $G$ acts by
conjugation.

######  Definition 3.1.

Let $G$ be a group, and $(u,v)\in{\mathcal{I}}_{G}^{(2)}$ be a pair of
distinct involutions.

  * •

We say that $(u,v)$ is of type $D_{\mathbb{Z}}$ if $\langle u,v\rangle$ is
contained in a (necessarily unique) group $D_{u,v}$ isomorphic to
$D_{\mathbb{Z}}$ with $D_{u,v}$ quasi-malnormal in $G$. If $\langle
u,v\rangle=D_{u,v}$ we say that the pair $(u,v)$ is _maximal_.

  * •

We say that $(u,v)$ is of type $D_{\mathbb{Q}}$ if $\langle u,v\rangle$ is
contained in a (maybe non-unique, maybe not quasi-malnormal) group isomorphic
to $D_{\mathbb{Q}}$.

######  Remark 3.2.

  1. 1.

If $(u,v)$ is of one of the two types, then $\langle u,v\rangle$ is infinite
dihedral.

  2. 2.

If $(u,v)$ is of type $D_{\mathbb{Z}}$ and $(u^{\prime},v^{\prime})$ is of
type $D_{\mathbb{Q}}$, then $\langle u,v\rangle\cap\langle
u^{\prime},v^{\prime}\rangle$ has cardinality at most 2. In particular, the
two types are mutually exclusive.

  3. 3.

Recall that the _commensurator_ ${\mathrm{Comm}}_{G}(H)$ of a subgroup
$H\subset G$ is the subgroup consisting of elements $g\in G$ such that the
intersection of $H$ and $gHg^{-1}$ has finite index in both of them. The pair
$(u,v)$ is of type $D_{\mathbb{Z}}$ if and only if the commensurator
${\mathrm{Comm}}_{G}(\langle u,v\rangle)$ is an infinite dihedral group. In
this case, $D_{u,v}={\mathrm{Comm}}_{G}(\langle u,v\rangle)$ is the unique
maximal infinite dihedral group containing $\langle u,v\rangle$.

######  Definition 3.3.

Let $G$ be a group. We say that $G$ belongs to the class $\mathcal{C}$ if

  1. 1.

$G$ acts transitively on ${\mathcal{I}}_{G}$, and freely on
${\mathcal{I}}_{G}^{(2)}$,

  2. 2.

any pair $(u,v)\in{\mathcal{I}}_{G}^{(2)}$ of distinct involutions is of type
$D_{\mathbb{Z}}$ or $D_{\mathbb{Q}}$,

  3. 3.

the set of pairs $(u,v)\in{\mathcal{I}}_{G}^{(2)}$ of type $D_{\mathbb{Q}}$ is
non-empty and $G$ acts transitively on it.

######  Remark 3.4.

If a group $G$ belongs to ${\mathcal{C}}$, and if all pairs of involutions of
$G$ are of type $D_{\mathbb{Q}}$, then $G$ acts sharply 2-transitively on
${\mathcal{I}}_{G}$.

Conversely, any sharply 2-transitive group $G$ of characteristic 0 belongs to
the class ${\mathcal{C}}$, with all pairs of involutions of type
$D_{\mathbb{Q}}$. Indeed, $G$ acts sharply 2-transitively on its set of
involutions (as recalled in Subsection 2.1). It then suffices to show that any
pair of involutions $(u,v)\in{\mathcal{I}}_{G}^{(2)}$ is of type
$D_{\mathbb{Q}}$. Indeed, given $k\geq 1$, $(u,v)$ is thus conjugate to
$(u,(vu)^{k-1}v)$, so consider an inner automorphism
$\sigma_{k}={\mathrm{ad}}_{a_{k}}$ sending $(u,v)$ to $(u,(vu)^{k-1}v)$. Note
that $\sigma_{k}(\langle u,v\rangle)$ is the unique subgroup of index $k$ of
$\langle u,v\rangle$. Moreover, for all $k,l\in{\mathbb{N}}\setminus\\{0\\}$,
$\sigma_{k}\circ\sigma_{l}$ agrees with $\sigma_{kl}$ in restriction to
$\langle u,v\rangle$. Since the pair $(u,v)$ has trivial centralizer, it
follows that $a_{k}a_{l}=a_{kl}$ so $\sigma_{k}$ and $\sigma_{l}$ commute.
Define $D_{1}=\langle u,v\rangle$ and
$D_{n}=\sigma_{n!}^{-1}(D_{1})=\sigma_{n}^{-1}(D_{n-1})$. Since $\langle
u,v\rangle$ contains $\sigma_{n}(\langle u,v\rangle)$ with index $n$,
$D_{n}=\sigma_{n!}^{-1}(\langle u,v\rangle)$ contains
$D_{n-1}=\sigma_{n!}^{-1}\sigma_{n}(\langle u,v\rangle)$ with index $n$. Thus,
$D_{1}\subset D_{2}\subset\dots\subset D_{n}$ with $[D_{n}:D_{n-1}]=n$. It
easily follows that $\cup_{n\in\mathbb{N}}D_{n}$ is an increasing union of
infinite dihedral groups, each one having index $n$ in the next one, and whose
union is isomorphic to $D_{\mathbb{Q}}$.

######  Remark 3.5.

Assertion 1 implies that the centralizer of each involution is malnormal.
Indeed, if $g\notin\mathrm{Cent}_{G}(i)$, any element
$a\in\mathrm{Cent}_{G}(i)^{g}\cap\mathrm{Cent}_{G}(i)$ fixes two distinct
involutions hence is trivial. Assertion 2 implies that any two distinct
involutions generate an infinite dihedral group, and thus do not commute.

######  Remark 3.6.

Definition 3.3 is equivalent to the definition of the class ${\mathcal{C}}$
that appears in [AT21]. Since we do not use the equivalence of the two
definitions in this paper, we leave the verification of the equivalence to the
interested reader.

###  3.2 Three types of elements

The following terminology will be convenient.

######  Definition 3.7.

Consider a group $G$ and an element $h\in G$ of infinite order. We say that

  1. 1.

$h$ is a _translation_ if it is the product of two distinct involutions,

     * •

it is a _translation of type $D_{\mathbb{Q}}$_ if it is the product of two
distinct involutions $h=uv$, with $\langle u,v\rangle$ of type
$D_{\mathbb{Q}}$,

     * •

it is a _translation of type $D_{\mathbb{Z}}$_ if it is the product of two
distinct involutions $h=uv$, with $(u,v)$ of type $D_{\mathbb{Z}}$;

  2. 2.

$h$ is a _homothety_ if $h$ centralizes some involution;

  3. 3.

$h$ is _isolated_ if $h$ is contained in a malnormal cyclic group
$\langle\hat{h}\rangle$. We say that $h$ is _maximal isolated_ if $\langle
h\rangle$ is malnormal.

######  Remark 3.8.

In any group $G$, the two types of translations are mutually exclusive (by the
second point of Remark 3.2). Moreover, it is clear that an isolated element
cannot be a translation or a homothety.

######  Remark 3.9.

The set of translations together with the trivial element does _not_ form a
subgroup of $G$ in general. This is the case for $G=K^{*}\ltimes K$ when $K$
is a skew field, and in fact, if and only if $G=K^{*}\ltimes K$ for some near-
field $K$, where $K={\mathcal{I}}_{G}^{2}$ as an additive group (see [Ker74,
Theorem 7.1]).

######  Lemma 3.10.

In a group belonging to ${\mathcal{C}}$, the three possibilities for $h$
(translation, homothety, isolated) are mutually exclusive.

###### Proof.

In view of the previous remark, we just have to prove that a homothety $h$
cannot be a translation. Let $i$ be an involution centralized by $h$. Arguing
by contradiction, assume that $h=uv$ for two involutions $u\neq v$. We will
use several times that two distinct involutions cannot commute in a group of
class ${\mathcal{C}}$. We note that $u,v\neq i$: indeed, if $u=i$ for
instance, then $v$ has to commute with $i$ so $v=i$ and $h=1$ a contradiction.
The involution $j=uiu^{-1}$ is therefore distinct from $i$. The conjugation by
$u$ exchanges $i$ and $j$ and so does $v$ since $uv$ commutes with $i$. By
$2$-sharpness, $u=v$ a contradiction. ∎

######  Remark 3.11.

If $G$ is in the class ${\mathcal{C}}$ then all involutions are conjugate, and
all translations of type $D_{\mathbb{Q}}$ are also conjugate (because $G$ acts
transitively on pairs of involutions of type $D_{\mathbb{Q}}$).

######  Definition 3.12.

We say that a group $G$ satisfies the _3-type condition_ if its elements are
of order $1,2$, or $\infty$, and each element $h\in G$ of infinite order is a
translation, a homothety or is isolated.

###  3.3 The class ${\mathcal{C}}^{\prime}$

We will use the following subclass ${\mathcal{C}}^{\prime}$ of
${\mathcal{C}}$.

######  Definition 3.13 (The class ${\mathcal{C}}^{\prime}$).

We say that a group $G$ belongs to the class ${\mathcal{C}}^{\prime}$ if it
lies in the class ${\mathcal{C}}$ and satisfies the 3-type condition.

For example, the affine group ${\mathbb{Q}}^{*}\ltimes{\mathbb{Q}}$ belongs to
${\mathcal{C}}^{\prime}$ with no isolated element and no
$D_{\mathbb{Z}}$-translation. More generally, if $K$ is a field of
characteristic 0, then $K^{*}\ltimes K$ belongs to ${\mathcal{C}}^{\prime}$ if
and only if $K^{*}$ contains no primitive $n$-th root of unity with $n>2$.

The following lemmas give some restrictions for groups in
${\mathcal{C}}^{\prime}$ that will be useful.

######  Lemma 3.14.

If $G$ is a group in which every finite subgroup of $G$ has order at most 2,
then every infinite virtually cyclic subgroup of $G$ is isomorphic to
${\mathbb{Z}}$, ${\mathbb{Z}}\times({\mathbb{Z}}/2{\mathbb{Z}})$ or
$D_{\mathbb{Z}}$.

###### Proof.

If $E\subset G$ is virtually cyclic, it can be written as an extension of a
finite normal group $N\subset E$, with $E/N$ isomorphic to ${\mathbb{Z}}$ or
$D_{\mathbb{Z}}$. If $N$ is trivial, we are done. Otherwise,
$N\simeq{\mathbb{Z}}/2{\mathbb{Z}}$ and since finite subgroups of $G$ have
order $\leq 2$, $E/N$ has to be torsion-free, so $E/N\simeq{\mathbb{Z}}$ and
$E\simeq{\mathbb{Z}}\times({\mathbb{Z}}/2{\mathbb{Z}})$. ∎

######  Lemma 3.15.

If $G$ belongs to ${\mathcal{C}}^{\prime}$, then every non-trivial finite
subgroup of $G$ has order 2, and every infinite virtually cyclic subgroup of
$G$ is isomorphic to ${\mathbb{Z}}$,
${\mathbb{Z}}\times({\mathbb{Z}}/2{\mathbb{Z}})$ or $D_{\mathbb{Z}}$.

###### Proof.

If $G$ belongs to ${\mathcal{C}}^{\prime}$, the order of elements of finite
order in $G$ is at most 2, so any finite subgroup $F\subset G$ is commutative.
Since distinct involutions of $G$ never commute, $F$ is trivial or isomorphic
to ${\mathbb{Z}}/2{\mathbb{Z}}$. The second part of the lemma then follows
from Lemma 3.14. ∎

######  Lemma 3.16.

Any group $G$ belonging to ${\mathcal{C}}^{\prime}$ contains a subgroup
isomorphic to ${\mathbb{Z}}\times({\mathbb{Z}}/2{\mathbb{Z}})$.

###### Proof.

By Definition 3.3 defining class ${\mathcal{C}}$, $G$ contains a pair of
involutions $(u,v)$ of type $D_{\mathbb{Q}}$, and all such pairs are in the
same $G$-orbit. Moreover, there are infinitely pairs $(u,v^{\prime})$ of type
$D_{\mathbb{Q}}$ (take for $v^{\prime}$ any involution in $\langle
u,v\rangle\setminus\\{u\\}$). Since the centralizer $Z(u)$ of $u$ acts
transitively on all such $v^{\prime}$, it is infinite. Since $u$ is the only
involution in $Z(u)$, it contains an element of infinite order which proves
the lemma. ∎

###  3.4 Increasing unions

Recall from Definition 3.1 that a pair of involutions $(u,v)$ of type
$D_{\mathbb{Z}}$ is maximal if $\langle u,v\rangle$ is quasi-malnormal in $G$.

######  Definition 3.17.

Consider an embedding of groups $G\subset G^{\prime}$, with $G,G^{\prime}$ in
${\mathcal{C}}$.

  * •

We say that the embedding _preserves maximal pairs_ if for each maximal pair
$(u,v)\in{\mathcal{I}}_{G}^{(2)}$ of type $D_{\mathbb{Z}}$, either the pair
$(u,v)$ is still a maximal pair of type $D_{\mathbb{Z}}$ in $G^{\prime}$, or
$(u,v)$ is of type $D_{\mathbb{Q}}$ in $G^{\prime}$.

  * •

We say that the embedding _preserves maximal isolated elements_ if for each
malnormal cyclic group $\langle h\rangle$, either $\langle h\rangle$ is still
malnormal in $G^{\prime}$, or $h$ is a translation or a homothety in
$G^{\prime}$.

We say that the embedding _preserves maximality_ if it satisfies both
conditions.

######  Lemma 3.18.

Consider $G=\cup_{n\in{\mathbb{N}}}G_{n}$ an increasing union of groups in
${\mathcal{C}}^{\prime}$. Assume that the embeddings $G_{n}\subset G_{n+1}$
preserve maximality. Then $G$ is in the class ${\mathcal{C}}^{\prime}$ and for
each $n\in{\mathbb{N}}$, the embedding $G_{n}\subset G$ preserves maximality.

###### Proof.

Let’s first check that $G$ is in the class ${\mathcal{C}}$. Assertion (1) of
Definition 3.3 is clear. If $(u,v)$ is of type $D_{\mathbb{Q}}$ in $G_{n}$,
then it is clearly of type $D_{\mathbb{Q}}$ in $G$. If
$(u,v)\in{\mathcal{I}}_{G_{n_{0}}}^{(2)}$ is of type $D_{\mathbb{Z}}$ in every
$G_{n}$ for all $n\geq n_{0}$, then since embeddings preserve maximality,
$D_{u,v}$ does not depend on the group $G_{n}$. It follows that $D_{u,v}$ is
quasi-malnormal in every $G_{n}$ hence in $G$. This proves Assertion (2) and
that the embedding $G_{n}\subset G$ preserves maximal pairs. Since any pair of
type $D_{{\mathbb{Q}}}$ in $G$ is of type $D_{{\mathbb{Q}}}$ in any $G_{n}$
containing it, Assertion (3) is clear, so $G$ belongs to ${\mathcal{C}}$.

To prove that $G$ is in ${\mathcal{C}}^{\prime}$, the only non-obvious point
to check is that if $h\in G_{n_{0}}$ is isolated in $G_{n}$ for every $n\geq
n_{0}$ then it is still isolated in $G$. The argument is similar to the
previous one using that inclusions preserve maximal isolated elements. ∎

###  3.5 HNN extensions and amalgams

The following proposition unifies several constructions in [AT21] and [RT19].

######  Proposition 3.19.

Let $G$ be a group in the class ${\mathcal{C}}$. Consider an HNN extension
$G_{1}=G*_{C}$ or an amalgam $G_{1}=G*_{C}H$ and denote by
$G_{1}\curvearrowright T$ the corresponding Bass-Serre tree. Assume that the
following properties hold for some $k\geq 0$:

  1. 1.

almost $k$-acylindricity: the stabilizer of any segment of length $>k$ in $T$
has order $\leq 2$;

  2. 2.

in the case of an amalgam, $H$ has at most one involution (necessarily central
in $H$), and this involution lies in the subgroup $C$.

Then $G_{1}$ is in the class ${\mathcal{C}}$.

If moreover $G$ and $H$ satisfy the 3-type condition (see Definition 3.12),
then $G_{1}$ belongs to ${\mathcal{C}}^{\prime}$ and the embedding $G\subset
G_{1}$ preserves maximality.

######  Example 3.20.

The proposition implies that if $G$ lies in ${\mathcal{C}}$ or
${\mathcal{C}}^{\prime}$, then so does $G*{\mathbb{Z}}$.

######  Remark 3.21.

If $H$ has a unique involution, then all elements of infinite order of $H$ are
homotheties, and $H$ satisfies the 3-type condition if and only if its
elements are of order $1,2$ or $\infty$.

###### Proof.

Before starting the proof, given two involutions $u,v$, we say that $\langle
u,v\rangle$ is hyperbolic if it does not fix a point in $T$ (otherwise it is
elliptic). In this case, we denote by $l_{u,v}$ the unique line of $T$
invariant under $\langle u,v\rangle$. By acylindricity, the pointwise
stabilizer of $l_{u,v}$ has order $\leq 2$, we claim that it is trivial.
Indeed, if $g\in G$ is an involution fixing $l_{u,v}$ pointwise, it is unique
and therefore has to commute with $\langle u,v\rangle$. Since $g$ has a common
fixed point with $u$, the pair $(g,u)$ is conjugate in $G$ (it cannot be
conjugate in $H$ because $g\neq u$). Since no pair of involutions of $G$
commute, we get a contradiction which proves the claim.

We now check that $G_{1}$ satisfies Assertion (1) of Definition 3.3. The fact
that $G_{1}$ acts transitively on the set of its involutions is clear: any
involution of $G_{1}$ is elliptic in $T$, so it is conjugate in $G$ or $H$,
hence in $G$ by assumption (2). Then, proving that the action on
${\mathcal{I}}_{G_{1}}^{(2)}$ is free amounts to checking that the centralizer
$Z$ of a pair $(u,v)\in{\mathcal{I}}_{G_{1}}^{(2)}$ is trivial. If $\langle
u,v\rangle$ is hyperbolic, then any element $g\in Z$ fixes $l_{u,v}$ pointwise
so $g=1$ by the initial claim. If $\langle u,v\rangle$ is elliptic, then the
pair $(u,v)$ is conjugate to a pair of involutions of $G$ (it cannot be
conjugate in $H$ by (2)). The set $F\subset T$ of fixed points of $\langle
u,v\rangle$ is a subtree of diameter $\leq k$. In the case of an amalgam, it
is reduced to a point fixed by a conjugate of $G$ by Assumption (2). Since $Z$
preserves the bounded tree $F$, it fixes a point in $F$. Thus $Z,u,v$ are
contained in a common conjugate of $G$, so $Z$ is trivial since $G$ belongs to
${\mathcal{C}}$. This proves that the action of $G_{1}$ on
${\mathcal{I}}_{G_{1}}^{(2)}$ is free.

Then we prove that $G_{1}$ satisfies Assertion (2) of Definition 3.3, namely
that any pair of distinct involutions is of type $D_{{\mathbb{Z}}}$ or
$D_{{\mathbb{Q}}}$. If $\langle u,v\rangle$ is hyperbolic then the pair
$(u,v)$ is of type $D_{\mathbb{Z}}$: the global stabilizer $D_{u,v}$ of
$l_{u,v}$ of is an infinite dihedral group (it acts faithfully on $l_{u,v}$ by
our initial claim), and it is quasi-malnormal: if $gl_{u,v}\neq l_{u,v}$, then
$D_{u,v}\cap D_{u,v}^{g}$ cannot contain any element of infinite order, so
$D_{u,v}\cap D_{u,v}^{g}$ has order at most $2$. If $\langle u,v\rangle$ is
elliptic, then its commensurator $M={\mathrm{Comm}}_{G_{1}}(\langle
u,v\rangle)$ is elliptic because it preserves the union of fixed points of
finite index subgroups of $\langle u,v\rangle$, a subtree of diameter $\leq
k$. Up to conjugacy, we may thus assume that $\langle u,v\rangle\subset
M\subset G$, so $M={\mathrm{Comm}}_{G}(\langle u,v\rangle)$. If $(u,v)$ has
type $D_{\mathbb{Q}}$ in $G$, it also has type $D_{\mathbb{Q}}$ in $G_{1}$. If
it has type $D_{\mathbb{Z}}$ in $G$, then ${\mathrm{Comm}}_{G}(\langle
u,v\rangle)$ is infinite dihedral, and so is ${\mathrm{Comm}}_{G_{1}}(\langle
u,v\rangle)$ so $(u,v)$ has type $D_{\mathbb{Z}}$ in $G_{1}$. Hence Assertion
(2) of Definition 3.3 is proved. Note that we also proved that the embedding
$G\subset G_{1}$ preserves maximal pairs.

Last, since any pair of type $D_{\mathbb{Q}}$ has to be elliptic in $T$, it is
conjugate to a pair in $G$, so $G_{1}$ acts transitively on the set of pairs
of type $D_{\mathbb{Q}}$. This shows that $G_{1}$ satisfies Assertion (3) of
Definition 3.3 and concludes the proof that $G_{1}$ belongs to
${\mathcal{C}}$.

We now assume that $G$ and $H$ satisfy the 3-type condition and prove that
$G_{1}$ satisfies the 3-type condition, hence belongs to the class
${\mathcal{C}}^{\prime}$. Clearly, all elements of $G_{1}$ have order $1,2$ or
$\infty$. Let $h\in G_{1}$ be an element of infinite order.

If $h$ is hyperbolic in $T$, let $l\subset T$ be its axis, and $A$ be the
global stabilizer of $l$. If the pointwise stabilizer of $l$ is non-trivial,
it is isomorphic to ${\mathbb{Z}}/2{\mathbb{Z}}$ by acylindricity and $h$ is a
homothety. If the pointwise stabilizer of $l$ is trivial, then $A$ acts
faithfully on $l$ so $A$ is either cyclic or infinite dihedral. If dihedral,
then $h$ is a translation. If $A$ is cyclic, then it is malnormal because if
$gAg^{-1}\cap A\neq 1$, then $g$ must preserve the line $l$, so $g\in A$. Thus
$h$ is isolated with $\langle\hat{h}\rangle=A$.

Now assume that $h$ is elliptic in $T$. Consider the subtree $F$ of $T$
consisting of points fixed by some power of $h$. $F$ has diameter at most $k$
and is invariant under the group $M={\mathrm{Comm}}_{G_{1}}(\langle h\rangle)$
so $M$ is elliptic. Thus we may assume that $\langle h\rangle\subset M\subset
H$ or $\langle h\rangle\subset M\subset G$.

If $\langle h\rangle\subset M\subset H$ and $H$ centralizes an involution,
then $h$ is a homothety. If $\langle h\rangle\subset M\subset H$ and $H$ is
torsion-free, then $h$ cannot be a translation or a homothety in $H$ so $h$ is
isolated in $H$, so $M\simeq{\mathbb{Z}}$ and $h$ is isolated in $G_{1}$.

If $\langle h\rangle\subset M\subset G$, and if $h$ is a translation or a
homothety in $G$, then this is also the case in $G_{1}$. If $h$ is isolated in
$G$, then $M\simeq{\mathbb{Z}}$ and $h$ is isolated in $G_{1}$. ∎

The following corollary allows to turn a pair of involutions of type
$D_{\mathbb{Z}}$ into type $D_{\mathbb{Q}}$.

######  Corollary 3.22 ([AT21, Prop. 4.1], [RT19, Prop 1.4]).

Let $G$ be a group in the class $\mathcal{C}^{\prime}$. Let
$(u_{0},v_{0})\in{\mathcal{I}}_{G}^{(2)}$ be a pair of type $D_{\mathbb{Q}}$
and $(u,v)\in{\mathcal{I}}_{G}^{(2)}$ be any maximal pair of type
$D_{\mathbb{Z}}$. Then the following HNN extension belongs to
${\mathcal{C}}^{\prime}$:

$G_{1}=\langle G,t\ |\ tut^{-1}=u_{0},\ tvt^{-1}=v_{0}\rangle.$

Moreover, the inclusion $G\subset G_{1}$ preserves maximality.

###### Proof.

In view of Proposition 3.19, we just have to check that the action on the
Bass-Serre tree is almost $2$-acylindrical. This easily follows from the fact
that $\langle u,v\rangle$ is quasi-malnormal in $G$, and that $|\langle
u_{0},v_{0}\rangle\cap\langle u,v\rangle^{g}|\leq 2$ for all $g\in G$. ∎

The following corollary allows to turn an isolated element into a translation.

######  Corollary 3.23 (see also Proposition 4.2 in [AT21]).

Let $G$ be a group in ${\mathcal{C}}^{\prime}$ and let $g,h$ be two elements
of $G$ of infinite order. Suppose that $g$ is a translation or a homothety in
$G$ and that $h$ is a maximal isolated element. Then the following HNN
extension belongs to ${\mathcal{C}}^{\prime}$:

$G_{1}=\langle G,t\ |\ tgt^{-1}=h\rangle.$

Moreover, the inclusion $G\subset G_{1}$ preserves maximality.

###### Proof.

Acylindricity follows from malnormality of $\langle h\rangle$ in $G$, and from
the fact that $\langle h\rangle\cap\langle g\rangle^{a}=\\{1\\}$ for all $a\in
G$. This last fact holds true because otherwise, some power of $h$ would be
conjugate in a group isomorphic to $D_{\mathbb{Z}}$ or to
${\mathbb{Z}}\times({\mathbb{Z}}/2{\mathbb{Z}})$ contradicting the fact that
$\langle h\rangle$ is malnormal in $G$. ∎

The following result allows to make two given homotheties conjugate.

######  Corollary 3.24.

Let $G$ be a group in the class ${\mathcal{C}}^{\prime}$. Let $h_{1},h_{2}\in
G$ be two homotheties centralizing respectively the involutions $u_{1}$ and
$u_{2}$. Then the following HNN extension belongs to ${\mathcal{C}}^{\prime}$:

$G_{1}=\langle G,t\ |\ tu_{2}t^{-1}=u_{1},\ th_{2}t^{-1}=h_{1}\rangle.$

Moreover, the inclusion $G\subset G_{1}$ preserves maximality.

###### Proof.

This HNN extension is not acylindrical. We rewrite it as an acylindrical
amalgam as follows. First, since all involutions of $G$ are conjugate, we may
replace $h_{2}$ and $u_{2}$ by some conjugates and change $t$ accordingly to
ensure that $u_{2}=u_{1}$, and we let $u=u_{1}=u_{2}$. Then $G_{1}=\langle
G,t\ |\ tut^{-1}=u,\ th_{2}t^{-1}=h_{1}\rangle$. Let $Z=\mathrm{Cent}_{G}(u)$
be the centralizer of $u$ in $G$. Then $G_{1}=G*_{Z}H$ where $H$ is the HNN
extension

$H=\langle Z,t\ |\ tut^{-1}=u,\ th_{2}t^{-1}=h_{1}\rangle.$

Since no two involutions commute in $G$, $u$ is the unique involution of $Z$.
Since $H$ is an HNN extension, any involution of $H$ is conjugate to an
element of $Z$ thus conjugate to $u$. On the other hand, we see that $u$ is
central in $H$, so $u$ is the unique involution of $H$. Moreover, all elements
of $H$ are of order $1$, $2$ or $\infty$ because this is the case for
$Z\subset G$. As noted in Remark 3.21, this shows that $H$ satisfies the
3-type condition.

Now $Z$ is malnormal in $G$ (see Remark 3.5) and it follows that the Bass-
Serre tree of the amalgam $G_{1}=G*_{Z}H$ is $2$-acylindrical. Thus
Proposition 3.19 applies. ∎

##  4 An infinite simple sharply 2-transitive group with exactly four
conjugacy classes

In this section, we use the tools of the previous section to construct an
infinite countable simple sharply 2-transitive group of characteristic 0 in
${\mathcal{C}}^{\prime}$ with exactly four conjugacy classes (but still not
finitely generated). As proved in Proposition 2.3, this number of conjugacy
classes is the smallest possible.

######  Theorem 4.1.

Given a countable group $G_{0}$ belonging to ${\mathcal{C}}^{\prime}$, there
exists a countable group $G$ containing $G_{0}$ such that

  * •

$G$ is sharply 2-transitive and belongs to ${\mathcal{C}}^{\prime}$,

  * •

$G$ has exactly four conjugacy classes: the trivial element, the set of
involutions, the set of translations and the set of homotheties.

###### Proof.

Starting from $G_{0}$, we are going to construct inductively an increasing
sequence of groups $(G_{n})_{n\in\mathbb{N}}$ in the class
$\mathcal{C^{\prime}}$ such that the group $G=\cup_{n\in\mathbb{N}}G_{n}$ is
in the class ${\mathcal{C}}^{\prime}$, and such that all its pairs of
involutions are of type $D_{\mathbb{Q}}$. As noticed in Remark 3.4, this
implies that $G$ acts sharply 2-transitively on its set of involutions.

We note that if $g\in G_{n}$ is a homothety (i.e. it centralizes an
involution) in $G_{n}$, then it still is a homothety in $G_{n+1}$. Similarly,
if $g\in G_{n}$ is a translation (i.e. a product of 2 distinct involutions) in
$G_{n}$, it still is a translation in $G_{n+1}$, its type may change from
$D_{\mathbb{Z}}$ to $D_{\mathbb{Q}}$ but not the other way around.

Suppose that the group $G_{n}$ has already been constructed, and let’s
construct $G_{n+1}$ as an increasing union of groups $G^{m}_{n}$, starting
with $G_{n}^{0}=G_{n}$. We fix $(u_{0},v_{0})$ a pair of involutions of type
$D_{\mathbb{Q}}$ in $G_{0}$. Consider an enumeration $g_{1},g_{2},\dots,$ of
the maximal isolated elements of $G_{n}$, an enumeration $h_{1},h_{2},\ldots$
of its homotheties, and an enumeration $(u_{1},v_{1}),(u_{2},v_{2}),\dots$ of
the set maximal pairs of involutions of type $D_{\mathbb{Z}}$. For each
homothety $h_{k}$, we denote by $i_{k}$ the unique involution it centralizes
(uniqueness follows from 2-sharpness of the action on involutions). Starting
with $G_{n}^{0}=G_{n}$, we define inductively an increasing sequence of groups
$G_{n}^{m}$ as follows.

  1. 1.

Define $G_{n}^{m+1/3}=\langle G_{n}^{m},t\ |\ tu_{m}t^{-1}=u_{0},\
tv_{m}t^{-1}=v_{0}\rangle$ as in Corollary 3.22 if $(u_{m},v_{m})$ is maximal
of type $D_{\mathbb{Z}}$ in $G_{n}^{m}$ and $G_{n}^{m+1/3}=G_{n}^{m}$
otherwise.

  2. 2.

Define $G_{n}^{m+2/3}=\langle G_{n}^{m+1/3},t\ |\
tg_{m}t^{-1}=u_{0}v_{0}\rangle$ as in Corollary 3.23 if $g_{m}$ is a maximal
isolated element in $G_{n}^{m+1/3}$, and $G_{n}^{m+2/3}=G_{n}^{m+1/3}$
otherwise.

  3. 3.

Define $G_{n}^{m+1}=\langle G_{n}^{m+2/3},t\ |\ ti_{m}t^{-1}=i_{1},\
th_{m}t^{-1}=h_{1}\rangle$ as in Corollary 3.24;

We finally define $G_{n+1}=\bigcup_{m}G_{n}^{m}$. Corollaries 3.22, 3.23 and
3.24 show that each group $G_{n}^{m}$ is in ${\mathcal{C}}^{\prime}$ and that
the embedding of each group in the next one preserves maximality. Lemma 3.18
then concludes that $G_{n+1}$ belongs to ${\mathcal{C}}^{\prime}$, and that
the embedding $G_{n}\subset G_{n+1}$ preserves maximality. Applying again
Lemma 3.18 to $G=\cup_{n}G_{n}$, we see that $G$ belongs to
${\mathcal{C}}^{\prime}$.

We now check that $G$ acts sharply 2-transitively on its set of involutions.
Since $G$ belongs to ${\mathcal{C}}$, it suffices to check that no pair of
involutions of $G$ is of type $D_{\mathbb{Z}}$ (see Remark 3.4). So consider
$(u,v)$ a pair of involutions of $G$, and let $n\in{\mathbb{N}}$ be such that
$u,v\in G_{n}$. If $(u,v)$ is of type $D_{\mathbb{Q}}$ in $G_{n}$, it is
obviously of type $D_{\mathbb{Q}}$ in $G$. Otherwise, let
$(\tilde{u},\tilde{v})$ be a maximal pair of type $D_{\mathbb{Z}}$ in $G_{n}$
such that $\langle u,v\rangle\subset\langle\tilde{u},\tilde{v}\rangle$. Since
all embeddings preserve maximality, at each step, the pair
$(\tilde{u},\tilde{v})$ remains maximal unless it becomes of type
$D_{\mathbb{Q}}$. Then Step 1 of the construction ensures that
$(\tilde{u},\tilde{v})$ and $(u,v)$ become of type $D_{\mathbb{Q}}$ at some
step. This shows that $(u,v)$ is of type $D_{\mathbb{Q}}$ in $G_{n+1}$, hence
also in $G$. This shows that $G$ acts sharply 2-transitively on its set of
involutions. It also follows that all translations of $G$ are conjugate.

Since $G$ belongs to ${\mathcal{C}}^{\prime}$, there remains to show that $G$
has no isolated element and that all homotheties of $G$ are conjugate.

Assume by contradiction that there exists $g\in G$ a maximal isolated element,
(equivalently, $\langle g\rangle$ is malnormal in $G$). Then $g$ is a maximal
isolated element in all $G^{m}_{n}$, but step 2 ensures that $g$ is conjugate
in $\langle u_{0},v_{0}\rangle$ in $G_{n+1}$, contradicting that $g$ is
isolated in $G$.

Similarly, step 3 ensures that all homotheties of $G_{n}$ become conjugate in
$G_{n+1}$. Since any homothety of $G$ is a homothety in some $G_{n}$, this
shows that all homotheties of $G$ are conjugate.∎

Having only four conjugacy classes, the group $G$ above is not far from being
simple, but may still fail to be so. The following result shows that we can
additionally ensure that $G$ is simple (see also [AT21]).

######  Theorem 4.2.

Given a countable group $G_{0}$ belonging to ${\mathcal{C}}^{\prime}$, there
exists a countable group $G$ containing $G_{0}$ such that

  * •

$G$ is sharply 2-transitive and belongs to ${\mathcal{C}}^{\prime}$,

  * •

$G$ has exactly four conjugacy classes: the trivial element, the set of
involutions, the set of translations and the set of homotheties,

  * •

and $G$ is simple.

Taking for instance $G_{0}={\mathbb{Q}}^{*}\ltimes{\mathbb{Q}}$, we get the
following result.

######  Corollary 4.3.

There exists a countable sharply 2-transitive group in the class
${\mathcal{C}}^{\prime}$ which is simple and has exactly four conjugacy
classes.∎

###### Proof of Theorem 4.2.

We claim that one can construct a group $G^{\prime}_{0}$ containing $G_{0}$
and in the class ${\mathcal{C}}^{\prime}$ such that there are four distinct
involutions $u_{1},u_{2},u_{3},u_{4}\in G^{\prime}_{0}$ such that
$h=u_{1}u_{2}u_{3}$ and $h^{\prime}=u_{1}u_{2}u_{3}u_{4}$ are two homotheties.

If the claim holds, then one can take for $G$ the group constructed by
applying Theorem 4.1 to $G^{\prime}_{0}$. Indeed, $h$ and $h^{\prime}$ are
still homotheties in $G$, and let us check that $G$ is simple. Consider $N$
any non-trivial normal subgroup. If $N$ contains an involution or a
translation, then it contains all of them so contains $h$ or $h^{\prime}$
respectively. This shows that in all cases, $N$ contains all homotheties, so
$N$ also contains the involution $h^{\prime}h^{-1}$, hence all involutions,
all translations, and $N=G$.

We now prove the claim. The group $G_{0}*\langle t\rangle$ belongs to
${\mathcal{C}}^{\prime}$ by Example 3.20, and if $u,v,w,z\in G_{0}$ are 4
distinct involutions then the element $h=uv(twt^{-1})$ is isolated, and
Proposition 3.23 yields a larger group $G_{1}$ in which $h$ is a homothety.
Similarly, the free product $G_{1}*\langle s\rangle$ belongs to
${\mathcal{C}}^{\prime}$, the element $h^{\prime}=h(szs^{-1})\in G_{1}*\langle
s\rangle$ is isolated, and we can embed $G_{1}*\langle s\rangle$ using
Proposition 3.23 into a larger group $G^{\prime}_{0}$ to ensure that
$h^{\prime}$ is a homothety. This proves our initial claim. ∎

##  5 Getting finite generation

###  5.1 Small cancellation over relatively hyperbolic groups

Let $G$ be a group hyperbolic relative to a collection of subgroups
$\mathcal{H}=\\{H_{\lambda}\\}_{\lambda\in\Lambda}$. Recall that an element
$g\in H$ is _parabolic_ if it is conjugate in some $H_{\lambda}$, and
_hyperbolic_ if it is not parabolic and of infinite order (we note that in
[Osi10], hyperbolic elements are synonymous with non-parabolic, and may have
finite order). Given a hyperbolic element $h\in G$, its commensurator
${\mathrm{Comm}}_{G}(\langle h\rangle)=\\{g\in G\ |\ gh^{n}g^{-1}=h^{\pm
n}\text{ for some }n\in\mathbb{N}^{\ast}\\}$ is the unique maximal virtually
cyclic subgroup of $G$ containing $h$. Following [Osi10], we use the following
technical definition.

######  Definition 5.1.

Let $G$ be a relatively hyperbolic group. We say that a subgroup $H\subset G$
is _suitable_ if it contains two hyperbolic elements $h_{1},h_{2}\in H$ (of
infinite order) such that ${\mathrm{Comm}}_{G}(\langle
h_{1}\rangle)\cap{\mathrm{Comm}}_{G}(\\{h_{2}\\})=\\{1\\}$.

We need the following slight refinement of Theorem 2.4 in [Osi10].

######  Theorem 5.2.

Let $G$ be a group hyperbolic relative to a subgroup $P$, and $H\subset G$ be
a suitable subgroup. Let $t_{1},\dots,t_{n}$ be arbitrary elements of $G$.
Then there exists an epimorphism $\eta:G\twoheadrightarrow Q$ such that:

  1. 1.

$Q$ is hyperbolic relative to $\eta(P)$;

  2. 2.

$\eta$ is injective in restriction to $P$;

  3. 3.

$\eta(H)$ is a suitable subgroup of $Q$;

  4. 4.

$\eta(t_{1}),\dots,\eta(t_{n})$ belong to $\eta(H)$;

  5. 5.

for every finite subgroup $F$ of $Q$, there exists a subgroup
$F^{\prime}\subset G$ isomorphic to $F$ such that $\eta(F^{\prime})=F$.

###### Proof.

This is Theorem 2.4 in [Osi10] except for Assertion 5 which is only stated for
finite cyclic groups but the argument actually works for all finite groups.
One can also refer to [Cou13, Proposition 6.12] where this assertion is proved
for small cancellation quotients of hyperbolic groups, or to [DG18, Lemma 4.3]
in the context of Dehn fillings.∎

###  5.2 Construction

The main result of this section is the following Theorem from which we will
deduce Theorems 1.1 and 1.4.

######  Theorem 5.3.

Let $G_{0}$ be a sharply 2-transitive group belonging to the class
$\mathcal{C}^{\prime}$. Then there exists a 2-generated sharply 2-transitive
group $G$ with Kazhdan property (T), such that $G$ contains $G_{0}$ and every
element of $G$ is conjugate to an element of $G_{0}$.

Theorems 1.1 and 1.4 are immediate consequences of the following corollary.

######  Corollary 5.4.

There exists an infinite sharply 2-transitive group $G$ of characteristic 0
with the following properties:

  * •

$G$ is generated by two elements,

  * •

$G$ has exactly four conjugacy classes,

  * •

$G$ is simple,

  * •

$G$ has property (T).

###### Proof of the corollary.

Start with a countable simple sharply 2-transitive group $G_{0}$ of
characteristic 0 with four conjugacy classes and belonging to
${\mathcal{C}}^{\prime}$, whose existence is proved in Theorem 4.2. Let $G$ be
the 2-generated group provided by Theorem 5.3. Then $G$ has at most four
conjugacy classes, and at least four by Proposition 2.3. Simplicity of $G$
follows immediately from the simplicity of $G_{0}$ since all conjugacy classes
of $G$ intersect $G_{0}$. ∎

The following notation will be convenient.

######  Definition 5.5.

Consider a group $G$ with two subgroups $P,H\subset G$. We say that $(G,P,H)$
satisfies $(*)$ if the following hold:

  1. 1.

$G$ is a group in the class ${\mathcal{C}}^{\prime}$,

  2. 2.

$G$ is hyperbolic relative to $P$,

  3. 3.

$H$ is a suitable subgroup of $G$.

The proof of Theorem 5.3 is based on the following result which will be
applied iteratively.

######  Proposition 5.6.

Consider a group $G$ with two subgroups $P,H\subset G$ such that $(G,P,H)$
satisfies $(*)$. Let $E\subset G$ be a finite or virtually cyclic subgroup of
$G$. Then there exists an epimorphism $\eta:G\twoheadrightarrow\bar{G}$ which
is injective on $P$, such that $(\bar{G},\eta(P),\eta(H))$ satisfies $(*)$ and
such that $\eta(E)\subset\eta(H)$ and $\eta(E)$ is conjugate to a subgroup of
$\eta(P)$. Moreover, every involution of $\bar{G}$ is the image of an
involution of $G$.

We first deduce Theorem 5.3 from Proposition 5.6.

###### Proof of Theorem 5.3.

Let $G_{0}$ be a sharply 2-transitive group belonging to
${\mathcal{C}}^{\prime}$. Let $H_{1}$ be a 2-generated torsion-free hyperbolic
group with Kazhdan property (T); the existence of such group is standard: for
instance, take a common torsion-free hyperbolic quotient ([Ols93],[Cha00, Th.
5.19]) of a free group $F_{2}$ and a torsion-free cocompact lattice in
$\mathrm{Sp}(2,1)$. Consider the free product $G_{1}=G_{0}*H_{1}$. It is
hyperbolic relative to $P_{1}:=G_{0}$, $H_{1}$ is a suitable subgroup of
$G_{1}$, and it is in the class ${\mathcal{C}}^{\prime}$ by Proposition 3.19,
so $(G_{1},P_{1},H_{1})$ satisfies $(*)$.

Let $E_{1},\dots,E_{n},\dots$ be an enumeration of all the virtually cyclic
subgroups of $G_{1}$ (including all cyclic groups of order 2 or $\infty$). We
construct inductively a chain of quotients using Proposition 5.6 as follows.
Apply Proposition 5.6 to $(G_{1},P_{1},H_{1})$ and $E_{1}\subset G_{1}$, and
denote by $G_{2}=\bar{G}_{1}$ the obtained quotient and by
$\eta_{1}:G_{1}\twoheadrightarrow G_{2}$ the quotient map. Define
$H_{2}=\eta(H_{1})$ and $P_{2}=\eta(P_{1})\simeq G_{0}$ so that the image of
$E_{1}$ is contained in $H_{2}$ and conjugate in $P_{2}$. Since
$(G_{2},P_{2},H_{2})$ satisfies $(*)$, one can repeat the argument and obtain
a chain of quotients

$G_{1}\xrightarrow[]{\eta_{1}}\mathrel{\mkern-14.0mu}\rightarrow
G_{2}\xrightarrow[]{\eta_{2}}\mathrel{\mkern-14.0mu}\rightarrow\dots\xrightarrow[]{\eta_{n-1}}\mathrel{\mkern-14.0mu}\rightarrow
G_{n}\xrightarrow[]{\eta_{n}}\mathrel{\mkern-14.0mu}\rightarrow\dots$

and subgroups $H_{n}=\eta_{n-1}(H_{n-1})$, $P_{n}=\eta_{n-1}(P_{n-1})\simeq
G_{0}$ of $G_{n}$ such that $(G_{n},P_{n},H_{n})$ satisfies $(*)$ and such
that the image of $E_{1},\dots,E_{n-1}$ in $G_{n}$ are contained in $H_{n}$
and conjugate in $P_{n}$.

Let $G_{\infty}$ be the direct limit of this chain and denote the
corresponding epimorphism by $\eta_{\infty}:G_{1}\rightarrow G_{\infty}$.
Denote $H_{\infty}=\eta_{\infty}(H_{1})$ and
$P_{\infty}=\eta_{\infty}(P_{1})\simeq G_{0}$. Any element $g\in G_{1}$ is
contained in some $E_{j}$, so its image in $G_{j+1}$ lies in $H_{j+1}$, hence
$\eta_{\infty}(g)\in H_{\infty}$. This shows that $\eta_{\infty|H_{1}}$ is
onto, so $G_{\infty}$, as a quotient of $H_{1}$, has property (T) and is
2-generated. Similarly, if $g\in E_{j}$, then its image in $G_{i+1}$ is
conjugate in $P_{j+1}$, which shows that every element of $G_{\infty}$ has a
conjugate in $P_{\infty}$.

We finally check that $G_{\infty}$ acts sharply 2-transitively on its set of
involutions. Let $u,v\in G_{\infty}$ be a pair of involutions, and let us
prove that it is conjugate to a pair of involutions of $P_{\infty}$. There
exist $n$ and preimages $u_{n},v_{n}\in G_{n}$ of $u,v$ such that
$u_{n},v_{n}$ are involutions. Since every involution of $G_{i}$ is the image
of an involution of $G_{i-1}$ under $\eta_{i-1}$, there exist involutions
$u_{1},v_{1}\in G_{1}$ that respectively map to $u,v\in G_{\infty}$. Since the
elementary group $\langle u_{1},v_{1}\rangle=E_{j}$ for some index $j$, its
image in $G_{j+1}$ is conjugate in $P_{j+1}$ hence $\langle u,v\rangle$ is
conjugate in $P_{\infty}$. Since $P_{\infty}\simeq G_{0}$ acts transitively on
its pairs of involutions, so does $G_{\infty}$.

It remains to check that the centralizer of a pair of distinct involutions
$u,v\in G_{\infty}$ is trivial. If $z\in G_{\infty}$ centralizes $u\neq v$,
then there exist $n$ and lifts $u_{n},v_{n},z_{n}\in G_{n}$ of $u,v,z$ such
that $u_{n}\neq v_{n}$ are involutions and $z_{n}$ centralizes $u_{n},v_{n}$.
Since $G_{n}$ belongs to ${\mathcal{C}}$, it acts freely on its pairs of
involutions so $z_{n}=1$ and $z=1$ which concludes the proof. ∎

There remains to prove Proposition 5.6.

###### Proof of Proposition 5.6.

Consider $(G,P,H)$ satisfying $(*)$. In a first step, we are going to embed
$G$ in a group $G_{1}$ so that $E$ is conjugate to a subgroup of $P$ in
$G_{1}$. If $E$ is already parabolic in $G$, we let $G_{1}=G$ so assume
otherwise. This implies that $E$ is infinite because, since $G$ belongs to
${\mathcal{C}}^{\prime}$, every finite subgroup of $G$ has order at most 2
(Lemma 3.15) and all involutions are conjugate (Definition 3.3).

Since $G$ is relatively hyperbolic, the commensurator $\hat{E}$ of $E$ is
virtually cyclic infinite. By Lemma 3.15, $\hat{E}$ is isomorphic to
${\mathbb{Z}}$, $D_{\mathbb{Z}}$ or
${\mathbb{Z}}\times({\mathbb{Z}}/2{\mathbb{Z}})$. Note that if $\langle
h\rangle$ is a maximal infinite cyclic subgroup of $\hat{E}$, then $h$ is
isolated, a translation of type $D_{\mathbb{Z}}$, or a homothety accordingly.

Consider $E^{\prime}$ a subgroup of $P$ isomorphic to $\hat{E}$: if $E$ is
isomorphic to ${\mathbb{Z}}$ or
${\mathbb{Z}}\times({\mathbb{Z}}/2{\mathbb{Z}})$, Lemma 3.16 ensures that
$E^{\prime}$ exists, and if $\hat{E}\simeq=\langle u,v\rangle\simeq
D_{\mathbb{Z}}$, we choose $E^{\prime}=\langle u_{0},v_{0}\rangle$ where
$(u_{0},v_{0})$ is a pair of involutions of type $D_{\mathbb{Q}}$, which
exists by Definition 3.3. Let $\sigma:\hat{E}\rightarrow E^{\prime}$ be an
isomorphism and let $G_{1}$ be the HNN extension $G_{1}=\langle G,t\ |\
txt^{-1}=\sigma(x),\ x\in\hat{E}\rangle$, and let us check that $(G_{1},P,H)$
satisfies $(*)$.

The group $G_{1}$ belongs to ${\mathcal{C}}^{\prime}$:

  * •

if $\hat{E}=\langle h\rangle\simeq{\mathbb{Z}}$, this follows from Corollary
3.23;

  * •

if $\hat{E}\simeq D_{\mathbb{Z}}$, this follows from Corollary 3.22 (where
$u,v\in\hat{E}$ are two involutions that generate $\hat{E}$ and
$E^{\prime}=\langle u_{0},v_{0}\rangle\subset P$ is chosen so that
$(u_{0},v_{0})$ has type $D_{\mathbb{Q}}$);

  * •

if $\hat{E}=\langle h\rangle\times\langle
u\rangle\simeq{\mathbb{Z}}\times({\mathbb{Z}}/2{\mathbb{Z}})$, then this
follows from Corollary 3.24.

The group $G_{1}$ is hyperbolic relative to $P$ by Dahmani’s combination
theorem [Dah03] as stated in [Osi10, Theorem 2.5] where we view $G$ as
hyperbolic relative to $\\{P,\hat{E}\\}$ (see for instance [Osi10, Th 2.1]).

We now check that $H$ is suitable in $G_{1}$. By [Osi10, Lemma 2.3], $H$
contains infinitely many non-commensurable hyperbolic elements
$h_{1},h_{2},\ldots\in H$ of infinite order such that
${\mathrm{Comm}}_{G}(\langle h_{i}\rangle)=\langle h_{i}\rangle$. Up to
discarding at most two elements, we may assume that no power of any $h_{i}$ is
$G$-conjugate in $\hat{E}$ or $\hat{E}^{\prime}$. To check that
${\mathrm{Comm}}_{G_{1}}(\langle h_{i}\rangle)=\langle h_{i}\rangle$, consider
$G_{1}\curvearrowright T$ be the Bass-Serre tree of the HNN extension defining
$G_{1}$, and $x\in T$ a vertex with stabilizer $G$. For all $k\geq 1$, the set
of fixed points of $h_{i}^{k}$ in $T$ is exactly $\\{x\\}$. It follows that
${\mathrm{Comm}}_{G_{1}}(\langle h_{i}\rangle)$ fixes $x$ so
${\mathrm{Comm}}_{G_{1}}(\langle h_{i}\rangle)={\mathrm{Comm}}_{G}(\langle
h_{i}\rangle)=\langle h_{i}\rangle$. This shows that $H$ is suitable in
$G_{1}$.

We thus have embedded $G$ in a group $G_{1}$ such that $(G_{1},P,H)$ satisfies
$(*)$, and such that $E$ is conjugate to a subgroup of $P$ in $G_{1}$.

Let $\\{t_{1},t_{2}\\}$ be a generating set of $E$, and recall that $t$
denotes the stable letter of the HNN extension defining $G_{1}$; in the case
where we defined $G=G_{1}$, we let $t=1$. Let
$\eta_{1}:G_{1}\twoheadrightarrow\bar{G}$ be the quotient of $G_{1}$ given by
Theorem 5.2 applied to the elements $t,t_{1},t_{2}\in G_{1}$. We denote by
$\eta:G\rightarrow\bar{G}$ the restriction of $\eta_{1}$. Since $G_{1}$ is
generated by $G$ and $t$, and since
$\eta_{1}(t)\in\eta_{1}(H)\subset\eta_{1}(G)$, it follows that
$\eta:G\rightarrow\bar{G}$ is onto. Theorem 5.2 says that $\eta$ is injective
in restriction to $P$, that the group $\bar{G}$ is hyperbolic relative to
$\eta(P)$, that $\eta(H)$ is suitable in $\bar{G}$, and that
$\eta(E)=\langle\eta(t_{1}),\eta(t_{2})\rangle$ is contained in $\eta(H)$.
Since $E$ is conjugate in $P$ in $G_{1}$, $\eta(E)$ is conjugate in $\eta(P)$
in $\bar{G}$.

It also says that any involution $v\in\bar{G}$ is the image of some involution
$\tilde{v}$ in $G_{1}$. It follows that $v$ is the image of some involution in
$G$: $\tilde{v}$ is conjugate to some involution $\tilde{v}^{\prime}\in G$,
and since $\eta$ is onto, there exists $g\in G$ such that
$\eta(g\tilde{v}^{\prime}g^{-1})=v$. We note that since all involutions of $G$
are conjugate to each other, this is also the case in $\bar{G}$.

There remains to check that $(\bar{G},\eta(P),\eta(H))$ satisfies $(*)$, and
the only remaining fact to prove is that $\bar{G}$ belongs to
${\mathcal{C}}^{\prime}$. In what follows we denote $\bar{P}=\eta(P)$.

We first characterize finite and virtually cyclic subgroups of $\bar{G}$.
Since $G_{1}$ belongs to ${\mathcal{C}}^{\prime}$, every non-trivial finite
subgroup of $G_{1}$ is isomorphic to ${\mathbb{Z}}/2{\mathbb{Z}}$ (see Lemma
3.15), and the same holds for $\bar{G}$ by Assertion (5) of Theorem 5.2. By
Lemma 3.15, it follows that every infinite virtually cyclic subgroup of
$\bar{G}$ is isomorphic to ${\mathbb{Z}}$, $D_{\mathbb{Z}}$ or
${\mathbb{Z}}\times({\mathbb{Z}}/2{\mathbb{Z}})$.

We now check that $\bar{G}$ belongs to ${\mathcal{C}}$ (Definition 3.3). The
following consequence of relative hyperbolicity will be useful: $\bar{P}$ is
almost malnormal in $\bar{G}$, i.e. if $\bar{P}^{g}\cap\bar{P}$ is infinite,
then $g\in\bar{P}$ (see for instance Lemma 8.3 in [Osi10]).

First, for any pair of involutions $(u,v)\in{\mathcal{I}}_{\bar{G}}^{(2)}$,
the group $\langle u,v\rangle$ is infinite dihedral since finite subgroups of
$\bar{G}$ have order at most 2. We already noticed that $\bar{G}$ acts
transitively on its involutions. Let $(u,v)\in{\mathcal{I}}_{\bar{G}}^{(2)}$
be a pair of involutions, and let us check that its centralizer
$Z=Z_{\bar{G}}(\langle u,v\rangle)$ is trivial and that it is either or type
$D_{\mathbb{Z}}$ or $D_{\mathbb{Q}}$.

If $\langle u,v\rangle$ is not parabolic, then its commensurator
${\mathrm{Comm}}_{\bar{G}}(\langle u,v\rangle)\supset{\bar{G}}(\langle
u,v\rangle)$ is virtually cyclic, necessarily isomorphic to $D_{\mathbb{Z}}$,
so $(u,v)$ is of type $D_{\mathbb{Z}}$ and its centralizer $Z$ is trivial.

If $\langle u,v\rangle$ is parabolic in $\bar{G}$, we may assume that $\langle
u,v\rangle\subset\bar{P}$. Then $Z$ is contained in $\bar{P}$ by almost-
malnormality of $\bar{P}$, so $Z$ is trivial because $\bar{P}$ is isomorphic
to the subgroup $P$ of $G$, and $G$ belongs to ${\mathcal{C}}$. Denote by
$\tilde{u},\tilde{v}\in P$ the preimages of $u,v$ by the isomorphism
$\eta_{|P}:P\xrightarrow{\sim}\bar{P}$.

If $(\tilde{u},\tilde{v})$ is of type $D_{\mathbb{Q}}$ in $G$, then
$\langle\tilde{u},\tilde{v}\rangle$ is contained in a subgroup $\tilde{D}$ of
$G$ isomorphic to $D_{\mathbb{Q}}$. Such a group $\tilde{D}$ has to be
parabolic, hence conjugate in $P$. It follows that $(u,v)$ is of type
$D_{\mathbb{Q}}$ since $\langle u,v\rangle\subset\eta(\tilde{D})\simeq
D_{\mathbb{Q}}$. If $(\tilde{u},\tilde{v})$ is of type $D_{\mathbb{Z}}$ in
$G$, then ${\mathrm{Comm}}_{P}(\langle\tilde{u},\tilde{v}\rangle)\simeq
D_{\mathbb{Z}}$. Using the isomorphism $\eta$,
${\mathrm{Comm}}_{\bar{P}}(\langle u,v\rangle)\simeq D_{\mathbb{Z}}$. By
almost malnormality of $\bar{P}$, we get that
${\mathrm{Comm}}_{\bar{G}}(\langle
u,v\rangle)={\mathrm{Comm}}_{\bar{P}}(\langle u,v\rangle)$ so $(u,v)$ is of
type $D_{\mathbb{Z}}$ in $\bar{G}$.

To prove that $\bar{G}$ belongs to ${\mathcal{C}}$, there remains to prove
that $\bar{G}$ acts transitively on the set of pairs
$(u,v)\in{\mathcal{I}}_{\bar{G}}^{(2)}$ of type $D_{\mathbb{Q}}$. But any such
pair is conjugate in $\bar{P}$ because a group isomorphic to $D_{\mathbb{Q}}$
has to be parabolic, so there remains to check that $\bar{P}$ acts
transitively on its set of pairs $(u,v)\in{\mathcal{I}}_{\bar{P}}^{(2)}$ of
type $D_{\mathbb{Q}}$. Viewing $\bar{P}\simeq P$ as a subgroup of $G$, we know
that any two pairs in ${\mathcal{I}}_{P}^{(2)}$ of type $D_{\mathbb{Q}}$ are
in the same $G$-orbit. Since $P$ is almost malnormal in $G$, they are actually
in the same $P$-orbit. This concludes the proof that $\bar{G}$ is in the class
${\mathcal{C}}$.

To check that $\bar{G}$ belongs to ${\mathcal{C}}^{\prime}$, there remains to
show that any element $h\in\bar{G}$ of infinite order is a translation, a
homothety or is isolated. If $\langle h\rangle$ is not parabolic, this follows
from the classification of elementary subgroups (the three cases occurring
when ${\mathrm{Comm}}_{\bar{G}}(\langle h\rangle)$ is isomorphic to
$D_{\mathbb{Z}}$, ${\mathbb{Z}}\times({\mathbb{Z}}/2{\mathbb{Z}})$ or
${\mathbb{Z}}$ respectively). Thus, we may assume that $h\in\bar{P}$. Let
$\tilde{h}\in P$ be its preimage in $P$ under the isomorphism
$\eta:P\xrightarrow{\sim}\bar{P}$. If $\tilde{h}$ is a translation or a
homothety in $G$, it stays so in $\bar{G}$ because $\eta$ does not kill any
involution. If $\tilde{h}$ is isolated in $G$, it is isolated in $P$ and $h$
is isolated in $\bar{P}$. Its commensurator ${\mathrm{Comm}}_{\bar{G}}(\langle
h\rangle)$ is contained in $\bar{P}$ by almost malnormality. It follows that
$h$ is isolated in $\bar{G}$, which proves that $\bar{G}$ belongs to
${\mathcal{C}}^{\prime}$ and concludes the proof. ∎

## Bibliography

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Simon André
Institut für Mathematische Logik und Grundlagenforschung,
WWU Münster, Einsteinstraße 62 48149 Münster, Germany.
E-mail address<EMAIL_ADDRESS>
Vincent Guirardel
Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France.
E-mail address<EMAIL_ADDRESS>


# Random Paraunitary Projections

Ricardo L. de Queiroz The author is with the Department of Computer Science,
Universidade de Brasilia, Brazil, e-mail<EMAIL_ADDRESS>
###### Abstract

Transforms using random matrices have been found to have many applications. We
are concerned with the projection of a signal onto Gaussian-distributed random
orthogonal bases. We also would like to easily invert the process through
transposes in order to facilitate iterative reconstruction. We derive an
efficient method to implement random unitary matrices of larger sizes through
a set of Givens rotations. Random angles are hierarchically generated on-the-
fly and the inverse merely requires traversing the angles in reverse order.
Hierarchical randomization of angles also enables reduced storage. Using the
random unitary matrices as building blocks we introduce random paraunitary
systems (filter banks). We also highlight an efficient implementation of the
paraunitary system and of its inverse. We also derive an adaptive under-
decimated system, wherein one can control and adapt the amount of projections
the signal undergoes, in effect, varying the sampling compression ratio as we
go along the signal, without segmenting it. It may locally range from very
compressive sampling matrices to (para) unitary random ones. One idea is to
adapt to local sparseness characteristics of non-stationary signals.

##  I Introduction

Random matrices [1] are very popular in multivariate statistics and have been
used in number theory, nuclear physics, quantum information, mechanics, and
wireless telecommunications. More recently, it has also been used in
compressive sensing [2], wherein a signal, assumed sparse in some domain, is
projected onto a random matrix of reduced dimensionality. Easily computing the
inverse or transpose of the projection may be useful for iterative
reconstruction systems such as COSAMP [3]. We are interested in an algorithm
that would easily allow transforming very large vectors using Gaussian-
distributed random orthogonal transforms or filter banks that could be easily
reversed. The transform should be randomly generated on-the-fly rather than
pre-stored.

##  II Random unitary transform

A unitary matrix can be decomposed into plane rotations, known as Givens
rotations [4]. An $M\times M$ unitary matrix can be decomposed into $M(M-1)/2$
rotations, i.e. rotations involving every pair of axis of the transformation.
Let ${\bf R}_{ij}$ be a matrix with elements $\\{r_{ij}\\}$ representing the
rotation of an angle $\theta_{ij}$ along the plane containing the $i$-th and
$j$-th axes of the transform, i.e. ${\bf R}_{ij}$ is like the identity matrix
but replacing the elements $r_{ii}=r_{jj}=\cos(\theta_{ij})$ and
$r_{ji}=-r_{ij}=\sin(\theta_{ij})$. Let S be a reflection matrix, i.e. a
diagonal matrix with $\pm 1$ in its diagonal. Then, the unitary transformation
U can also be represented as

${\bf U}={\bf S}\prod_{i=0}^{M-2}\prod_{j=i+1}^{M-1}{\bf R}_{ij},$ (1)

which is the expression for the Givens factorization [4]. The inverse
transform is easily found by reversing the order of the rotations and
inverting the angles.

Unitary random matrices have been generated by QR factorization of non-unitary
random matrices [6], and by using Householder rotations [7]. We generate
unitary random matrices by randomly generating the rotation angles
$\theta_{ij}$ rather than randomly generating the matrix elements $u_{ij}$
[8]. The random matrix can be computed using (1), so that the resulting matrix
U is perfectly unitary. Nevertheless, the statistical correlation and
probability distribution function (PDF) of the samples of U are not trivially
found from the PDF of $\theta_{ij}$ [8],[9]. If $\theta$ is uniformly
distributed, the PDFs of $\cos(\theta)$ and of $\sin(\theta)$ are biased and
the many operations derived from the successive plane rotations tend to
further concentrate the samples. Under certain restrictions it can be shown
that [9] if each angle is randomly chosen according to the following
ditribution:

$p(\theta_{ij})=\frac{\Gamma\left(\frac{j-i+1}{2}\right)}{\sqrt{\pi}\
\Gamma\left(\frac{j-i}{2}\right)}\cos^{j-i-1}(\theta_{ij}),$ (2)

then the resulting PDF would tend to be Gaussian. Also, of great importance to
us is the autocorrelation of the resulting matrix, which ideally should be an
impulse. The trivial approach is to generate random matrix entries in order to
ensure decorrelation. Even though we use an indirect method, generating random
rotation angles, our simulations show that the above distribution of angles
leads to decorrelated near-perfect Gaussian-distributed entries.

It is a case of interest to project an $M$-tuple x into $N<<M$ bases, i.e. an
orthogonal $N\times M$ Gaussian random transform A, so that ${\bf y}={\bf
A}{\bf x}$. This can be easily accomplished through pruning the unnecessary
rotations. One can obtain A through generating U and discarding $M-N$ rows.
For that, it can be shown that we may also discard the last $M-N-1$ stages in
(1). Its transpose (rather than its inverse transform) $\hat{\bf x}={\bf
A}^{T}{\bf y}$, requires traversing the rotations backwards from $N$ samples
to an approximation of the original $M$-tuple.

### Low-memory implementation

If we implement the transform through in-place rotations, we can completely
avoid storing or calculating the matrix entries. Angles are generated as they
are used in sequence following (1). Each stage (rotation) demands calculating
$\cos(\theta_{ij})$ and $\sin(\theta_{ij})$, 4 multiplications and 2
additions. There is no need for matrix inversion and we only store the input
vector. The inverse transform is accomplished by reversing the order of the
planes and by applying negative rotations. However, the angles would have to
be generated in reverse order which normally would require to buffer all the
random angles. Let the set of $\ell=M(M-1)/2$ angles be
${\bf\Theta}=\\{\theta_{0},\ldots,\theta_{\ell-1}\\}$. We divide the set of
angles into $N_{s}$ subsets ${\bf\Theta}_{i}$ of $\ell/N_{s}$ angles each as
${\bf\Theta}=\\{{\bf\Theta}_{1},{\bf\Theta}_{2},\ldots,{\bf\Theta}_{N_{s}}\\}$.
We first generate a seed $S_{0}$, from which we generate $N_{s}$ seeds
$S_{k}$, $1\leq k\leq N_{s}$. Each of the subsets ${\bf\Theta}_{k}$ is then
randomly generated using seed $S_{k}$. For the inverse transform angles and
seeds, one would need to convey $S_{0}$ to both forward and inverse transform
stages and to buffer only the set of seeds $\\{S_{k}\\}$. At a time, only
$\ell/N_{s}$ angles for one subset need to be generated and visited in reverse
order. Thus, the only storage required would be necessary for 3 vectors:
input/output, seeds, and random subgroup, of sizes, $M$, $N_{s}$ and
$\ell/N_{s}$, respectively. If $N_{s}=M$, then total storage is roughly
$2.5M$. This makes it possible to make random orthogonal projections (and
their inverses) of very large vectors.

As a note, we disregarded the computation and storage to produce the random
angles with PDF given by (2).

##  III Random Paraunitary Systems

An $M$-input, $M$-output ($M\times M$) paraunitary system ${\bf H}(z)$ of
order111We mean the order of the system as the highest polynomial degree of
its entries, and not the McMillan degree of the system [10]. $K$ can be
constructed using a cascade of $K$ simpler order-1 paraunitary systems:

${\bf H}(z)={\bf U}_{0}\prod_{i=1}^{K}{\bf\Lambda}(z){\bf U}_{i},$ (3)

where ${\bf\Lambda}(z)=diag\\{1,\ldots,1,z^{-1},\ldots,z^{-1}\\}$ and ${\bf
U}_{i}$ are unitary matrices.

We can make a random, Gaussian, paraunitary system if we cascade random
unitary matrices, i.e. if we make ${\bf U}_{i}$ random unitary matrices as
explained in the previous Section. To see that, we begin by showing that ${\bf
H}(z)$ has random independent entries $\\{h_{ij}(z)\\}$. Let ${\bf H}(z)={\bf
H}^{\prime}(z){\bf U}_{n}$ such that
$h_{ij}(z)=\sum_{k}h^{\prime}_{ik}(z)u_{n,kj}$ and that the $\\{u_{n,ij}\\}$
have zero mean. Then,
$E\\{h_{ij}(z)h_{k\ell}(z)\\}=\sum_{uv}E\\{h^{\prime}_{uj}(z)h^{\prime}_{v\ell}(z)\\}E\\{u_{0,iu}u_{0,kv}\\}=0$
if $i,j\neq k,\ell$ and if the entries $\\{h^{\prime}_{ij}\\}$ are
independent. Since ${\bf U}_{0}$ has independent entries, so does ${\bf
U}_{0}{\bf\Lambda}(z)$ and, hence, the same applies to ${\bf H}(z)$ for any
order.

Let ${\bf F}(z)=\sum_{i=0}^{n}{\bf F}_{i}z^{-i}$ be a system of order $n$,
obtained through appending an order-1 stage to ${\bf
E}(z)=\sum_{i=0}^{n-1}{\bf E}_{i}z^{-i}$, i.e. ${\bf F}(z)={\bf
E}(z){\bf\Lambda}(z){\bf U}_{n}$. Let ${\bf\Lambda}(z){\bf U}_{n}={\bf
U}^{\prime}_{n}+z^{-1}{\bf U}^{\prime\prime}_{n}$. Then, we get

${\bf F}_{k}={\bf E}_{k}{\bf U}^{\prime}_{n}+{\bf E}_{k-1}{\bf
U}_{n}^{\prime\prime}\ \ \ 0<k<n$ ${\bf F}_{0}={\bf E}_{0}{\bf
U}^{\prime}_{n}\ \ \ {\bf F}_{n}={\bf E}_{n-1}{\bf U}^{\prime\prime}_{n}.$

The entries $\\{f_{k,ij}\\}$ of ${\bf F}(z)$ are sums of products of random
variables. For large $M$, it is expected that the distribution of
$\\{f_{k,ij}\\}$ approaches a Gaussian PDF. As we can make all ${\bf U}_{i}$
to have Gaussian-distributed entries as in Section II, we expect all
$\\{h_{k,ij}\\}$entries in ${\bf H}(z)=\sum_{i=0}^{K}{\bf H}_{i}z^{-i}$ to be
approximately Gaussian distributed, for large $M$.

Let ${\bf A}^{H}$ denote the Hermitian of matrix ${\bf A}$, i.e. its
transposed conjugate. Also, let the subscript ${\bf A}^{(*)}(z)$ denote to
conjugate only the coefficients of its polynomials, i.e. if ${\bf A}(z)$ has
entries $\\{\sum_{n}a_{ijn}z^{n}\\}$, then ${\bf A}^{(*)}(z)$ has entries
$\\{\sum_{n}a^{*}_{ijn}z^{n}\\}$. Then, the inverse of a paraunitary system
${\bf H}(z)$ is simply its para-conjugated version [10]

${\bf H}^{-1}(z)={\bf H}^{T(*)}(1/z)=\sum_{i=0}^{K}{\bf H}_{i}^{H}z^{i}.$

A paraunitary ${\bf H}(z)$ is, thus, unitary on the unit circle.

### Low-memory implementation

The system can be easily implemented using (3), wherein each random unitary
stage ${\bf U}_{i}$ can be implemented using the techniques described in the
previous section. Each $M$-sample vector of the input signal undergoes a chain
of random transforms, for each ${\bf U}_{i}$ interspersed with shuffles
${\bf\Lambda}(z)$. One has to budget memory to store the $KM/2$ internal
states, in between the random unitary transforms.

The inverse transform can be accomplished by traversing the system backwards.
Let $\tilde{\bf\Lambda}(z)=z^{-1}{\bf\Lambda}(1/z)$, such that
$\tilde{\bf\Lambda}(z){\bf\Lambda}(z)=z^{-1}{\bf I}$. Let

${\bf G}(z)={\bf U}^{H}_{K}\prod_{i=1}^{K}\tilde{\bf\Lambda}(z){\bf
U}^{H}_{K-i}.$ (4)

Then, ${\bf G}(z){\bf H}(z)=z^{K}{\bf I}$ and the above inverse has the same
structure as (3). Hence, both the forward and inverse transforms have the same
implementation, with the proper adaptation of the shuffles in between stages
of unitary transforms. The transpose (inverse) of a unitary transform can be
accomplished by going through the plane rotations in reverse order, with
inverse angles.

As a note, the order can be changed on-the-fly by adding or removing stages to
the cascade of unitary transforms.

##  IV Adaptive sampling matrices

The $M\times M$ system ${\bf H}(z)$ can be seen as an $M$-channel filter bank,
where each filter is down-sampled by a factor of $M$, ($\downarrow M$). The
filters have length $L=(K+1)M$. In such, $M$ samples enter the system and $M$
samples leave it at a time, in what is referred as a critically decimated
system. If we increase the decimation, the systems becomes over-decimated, and
not invertible. Let us denote as ${\bf H}_{\downarrow m}(z)$ the
representation of the paraunitary system, but with down-sampling by a factor
of $m$. Then,

${\bf H}_{\downarrow M}(z)={\bf H}(z)=\sum_{i=0}^{K}{\bf H}_{i}z^{-i}$ (5)
${\bf H}_{\downarrow 2M}(z)=\sum_{i=0}^{\lfloor K/2\rfloor}\left[{\bf
H}_{2i},{\bf H}_{2i+1}\right]z^{-i}$ (6)

where $\lfloor\ \rfloor$ is the “floor” rounding operation. For a more general
down-sampling factor,

${\bf H}_{\downarrow qM}(z)=\sum_{i=0}^{\lfloor K/q\rfloor}\left[{\bf
H}_{qi},{\bf H}_{qi+1},\ldots,{\bf H}_{qi+q-1}\right]z^{-i},$ (7)

where ${\bf H}_{n}=\bf{0}$ for $n>K$. In the extreme case, if $q=K+1$, i.e.
$\downarrow L$, then

${\bf H}_{\downarrow L}(z)=\left[{\bf H}_{0},{\bf H}_{1},\ldots,{\bf
H}_{K}\right],$ (8)

i.e. the under-sampled matrix becomes a (scalar) sampling matrix.

There are $qM$ input samples per $M$ output ones, i.e. $q$ is the sampling
compression parameter. The down-sampling and, thus, $q$ can be changed on the
fly. The system can go from $M$ input to $M$ output samples per clock, to an
$L$-to-$M$ one, and all stages in between. In all cases, the bases are random
in nature and are approximately Gaussian distributed as discussed in the
previous Section. Hence, $q$ can be made adaptive, perhaps adapting how
compressive the sensing is, in order to track local sparseness of non-
stationary signals.

### Low-memory implementation

Starting from a $\downarrow M$ system in (3), a $\downarrow qM$ system can be
implemented by picking one out of every set of $q$ output blocks of $M$
samples. If we view (3) as a flow graph, the system can then be more
efficiently implemented by simply pruning out the stages which are not
necessary and will not be used to produce output. Figure 1 illustrates a case
$K=4$ and indicates unitary stages to be pruned in the $\downarrow L$
(sampling matrix) case. It can be shown that maximum pruning yields
computational savings in the order of $K-(2K)^{-1}$. The reverse (transpose)
system can be accomplished through reversing the system, i.e. using (4) and
setting to zero the samples which are not produced using the over-decimated
system.

Figure 1: A $K=4$ paraunitary system lattice and diagram illustrating the
blocks ignored in the $\downarrow L$ case (white filled). Each branch carries
$M/2$ samples.

##  V Conclusions

In this letter, we present a method to construct Gaussian-distributed random
paraunitary systems. The system can be dynamically applied in order to create
an adaptive compressive sensing framework wherein sampling is compressed by a
factor of $q$ which can be changed on-the-fly. The bases are unitary in nature
and the projection can be easily implemented and reversed. We discussed the
efficient implementation of random unitary matrices, which are used as
building blocks to construct the random paraunitary filter banks. An efficient
implementation of such filter banks is also discussed. Applications of such an
adaptive system are being investigated.

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