Suppose $G\leq O(d)$ is compact and let $H\leq\operatorname{O}(p)$ be a
minimal reduction of $G$. Let $\chi_{H}:=\max_{x\in R(H)}|H_{x}|$. Fix
$V\in\mathbb{R}^{n\times k}$. Then, for generic
$z_{1},\dots,z_{n}\in\mathbb{R}^{d}$, the corresponding max filter bank
$\Phi\colon\mathbb{R}^{d}/G\to\mathbb{R}^{n}$ given by
$\Phi([x])=\\{\langle\langle[x],[z_{i}]\rangle\rangle\\}_{i=1}^{n}$ locally
avoids $\operatorname{im}(V)$ at every $x\in R(G)$ provided
$n>\chi_{H}(2c+k-2)$.

The proof of Theorem 92 is almost immediate once we establish the following
key lemma:

######  Lemma 93.

Suppose $G\leq\operatorname{O}(d)$ is compact with trivial principal isotropy
group. Fix $V\in\mathbb{R}^{n\times k}$ and put $\chi:=\max_{x\in
R(G)}|G_{x}|$. For $z_{1},\dots,z_{n}\in\mathbb{R}^{d}$, denote the
corresponding max filter bank by
$\Phi\colon\mathbb{R}^{d}/G\to\mathbb{R}^{n}$. Then, the set

$R_{V}:=\\{\\{z_{i}\\}_{i=1}^{n}\in(\mathbb{R}^{d})^{n}:\Phi\text{ fails to
locally avoid $\operatorname{im}(V)$ at every $x\in R(G)$}\\}$

is semialgebraic and it holds that

$\dim(R_{V})\leq
nd-1-\left(\left\lceil\frac{n}{\chi}\right\rceil-k-2c+1\right).$

###### Proof of Theorem 92.

An isometry $\psi\colon\mathbb{R}^{d}/G\to\mathbb{R}^{p}/H$ maps $[R(G)]$ to
$[R(H)]$ isometrically. This is because by Proposition 1.8 and the discussion
that follows it in [20], $[R(G)]$ and $[R(H)]$ are characterized by having
tangent cones of maximal dimension and tangent cones are isometry invariants.
The result now follows by Corollary 89 and Lemma 93. ∎

The rest of the section is dedicated to proving Lemma 93. We benefit from a
characterization of regular points through the notion of local open Voronoi
cells.

######  Definition 94.

Fix a compact $G\leq\operatorname{O}(d)$ with identity component $G_{0}$. For
$z\in\mathbb{R}^{d}$, we define the local open Voronoi cell $V_{z}^{loc}$
through the following characterization: we say $x\in V_{z}^{loc}$ if there
exists an open neighborhood $U$ of $z$ and an open neighborhood $V$ of $x$
such that

$\forall q\in V,\ \ \Big{|}\arg\sup_{p\in[z]_{0}\cap U}\langle
p,q\rangle\Big{|}=1.$

######  Lemma 95.

Suppose $G\leq\operatorname{O}(d)$ is compact with trivial principal isotropy
group. For $x\in\mathbb{R}^{d}$, the following statements are equivalent:

  1. (a)

$x\in R(G)$.

  2. (b)

$z\in V_{x}^{1}$ implies $x\in V_{z}^{loc}$.

###### Proof.

See Section A.4. ∎

In the following proof of Lemma 93, we will refer to a technical result in the
appendix, namely Lemma 112.

###### Proof of Lemma 93.

We proceed in a similar manner to the proof of Lemma 60. Let $N:=N_{w}$ for
some $w\in P(G)$. Then, $G\cdot N=\mathbb{R}^{d}$ by Proposition 56. A witness
of failure $x\in R(G)\cap\mathbb{S}^{d-1}\cap N$ has at most $c-1$ degrees of
freedom. Having $n-m$ templates in $(Q_{x}^{i})^{c}$ allows for at most
$(n-m)(d-1)$ degrees of freedom so we may assume $m$ templates are in
$Q_{x}^{i}$ and show that those templates have at most
$md-1-\left(m-n+\left\lceil\frac{n}{\chi}\right\rceil-k-c\right)$ degrees of
freedom after truncating $V$ to $V_{m}$. In such case, each template has a
version of it in $V_{x}^{1}$ and with such versions fixed, there are
$m\dim(G)=m(d-c)$ degrees of freedom for the orbits of those versions. As
such, we may assume each template is in $V_{x}^{1}$ and show that those have
at most $mc-1-\left(m-n+\left\lceil\frac{n}{\chi}\right\rceil-k-c\right)$
degrees of freedom. Since
$m-n\leq\left\lceil\frac{m}{\chi}\right\rceil-\left\lceil\frac{n}{\chi}\right\rceil$,
it suffices to bound the degrees of freedom by
$mc-1-\left(\left\lceil\frac{m}{\chi}\right\rceil-k-c\right)$. Stated
precisely, our aim to show that $\dim(D_{V_{m}}^{x})\leq
mc-1-\left(\left\lceil\frac{m}{\chi}\right\rceil-k-c\right)$ where

$D_{V_{m}}^{x}:=\big{\\{}\\{z_{i}\\}_{i=1}^{n}\in(V_{x}^{1})^{m}:\Phi\text{
fails to locally avoid }\operatorname{im}(V_{m})\text{ at }x\big{\\}}.$

The bound is trivial when $\left\lceil\frac{m}{\chi}\right\rceil\leq c+k-1$
that is $m\leq\chi(c+k-1)$. As such, we assume
$m\geq\left\lceil\frac{m}{\chi}\right\rceil\geq c+k$. Let

$E:=\bigcap_{\\{h_{l}\\}_{l=1}^{m}\in(G_{x})^{m}}\bigcap_{\begin{subarray}{c}I\subseteq\\{1,\dots,m\\}\\\
|I|=\left\lceil\frac{m}{\chi}\right\rceil\end{subarray}}\begin{array}[]{@{}c@{}}$
$\\\ \big{\\{}\\{z_{i}\\}_{i=1}^{n}\in(N_{x})^{m}:\text{$\\{\langle
h_{i}z_{i},\cdot\rangle\\}_{i\in I}$ is injective}\\\ \text{ \qquad and
$\operatorname{im}(\\{\langle h_{i}z_{i},\cdot\rangle\\}_{i\in
I})\cap\operatorname{im}(V_{I})=\\{0\\}$}\big{\\}}.\end{array}$

We show that $D_{V_{m}}^{x}\subseteq(N_{x})^{m}\setminus E$, and we bound
$\dim((N_{x})^{m}\setminus E)$. To show
$D_{V_{m}}^{x}\subseteq(N_{x})^{m}\setminus E$ we make use of the technical
Lemma 112 stated and proved in the appendix. There, we denote

$S^{1}(z,x):=\\{w\in[z]:\Psi_{1}([z],[x])=\langle w,x\rangle\\}$

which we will refer to in the next paragraph.

Suppose there exists $\\{z_{i}\\}_{i=1}^{n}\in D_{V_{m}}^{x}\cap E$. Let $Q$
denote the corresponding difference quotient. Then, there exists sequences
$x_{j},y_{j}\to x$ such that $x_{j},y_{j}\in N_{x}$, $[x_{j}]\neq[y_{j}]$ and
$\lim_{j\to\infty}Q(x_{j},y_{j})\in\operatorname{im}(V)$. It suffices to show
that there exists $I\subseteq\\{1,\dots,m\\}$ such that
$|I|\geq\left\lceil\frac{m}{\chi}\right\rceil$ and $u\in N_{x}$ with
$\|u\|\geq 1$ such that after taking subsequences, the following convergence
holds

$\frac{\Psi_{1}([x_{j}],[z_{i}])-\Psi_{1}([y_{j}],[z_{i}])}{d([x_{j}],[y_{j}])}\to\langle
w_{i},u\rangle$ (9)

for $i\in I$ where $w_{i}\in[z_{i}]_{G_{x}}$. This way, we obtain the
contradiction $\operatorname{im}(\\{\langle w_{i},\cdot\rangle\\}_{i\in
I})\cap\operatorname{im}(V_{I})\neq\\{0\\}$. Take subsequences so that the
limits $\lim_{j\to\infty}S^{1}(z_{i},x_{j})\subseteq G_{x}z_{i}$ and
$\lim_{j\to\infty}S^{1}(z_{i},y_{j})\subseteq G_{x}z_{i}$ exist for every $i$.
By pigeonhole principle, there exists $h\in G_{x}$ and
$I\subseteq\\{1,\dots,m\\}$ such that
$|I|\geq\left\lceil\frac{m}{\chi}\right\rceil\geq c+k$ and
$L([z_{i}]):=\lim_{j\to\infty}S^{1}(z_{i},x_{j})\cap\lim_{j\to\infty}S^{1}(z_{i},hy_{j})\neq\varnothing$
for $i\in I$. As such and by $G_{x}$-invariance of $N_{x}$ (Proposition 21),
we adjust the sequence $(y_{j})_{j\in\mathbb{N}}$ into
$(hy_{j})_{n\in\mathbb{N}}$ and we show that Lemma 112 applies to the
sequences $(x_{j})_{j\in\mathbb{N}}$ and $(y_{j})_{j\in\mathbb{N}}$. For $i\in
I$, let $w_{i}\in L([z_{i}])=L([w_{i}])$. By injectivity of $\\{\langle
w_{i},\cdot\rangle\\}_{i\in I}$, we get that $\\{w_{i}\\}_{i\in I}$ is
spanning for $V_{x}^{1}$ and hence (H2) of Lemma 112 holds. Moreover, (H1) of
Lemma 112 holds by Lemma 95. Hence, (9) holds by Lemma 112. We obtain that
$D_{V_{m}}^{x}\subseteq(N_{x})^{m}\setminus E$ as desired.

To bound $\dim((N_{x})^{m}\setminus E)$, we view it as a finite union so that
we may fix arbitrary $\\{h_{l}\\}_{l=1}^{m}\in(G_{x})^{m}$ and
$I\subseteq\\{1,\dots,m\\}$ with $|I|=\left\lceil\frac{m}{\chi}\right\rceil$,
and we count the degrees of freedom of
$\\{z_{i}\\}_{i=1}^{n}\in(V_{x}^{1})^{n}$ such that $\\{\langle
h_{i}z_{i},\cdot\rangle\\}_{i\in I}$ is injective and
$\operatorname{im}(\\{\langle h_{i}z_{i},\cdot\rangle\\}_{i\in
I})\cap\operatorname{im}(V_{I})=\\{0\\}$. In fact, since $N_{x}$ is
$G_{x}$-invariant (Proposition 21), we may as well take
$h_{i}=\operatorname{Id}$ and proceed. The templates $\\{z_{i}\\}_{i\notin I}$
roam free with $(m-|I|)c$ degrees of freedom, and by a similar argument as in
Lemma 60 for $\pi_{2}(E_{L})$, the degrees of freedom for $\\{z_{i}\\}_{i\in
I}$ are bounded by $|I|c-1-(|I|-k-c)$. Hence, the combined bounds yield the
desired bound $\dim((N_{x})^{m}\setminus E)\leq
mc-1-\left(\left\lceil\frac{m}{\chi}\right\rceil-k-c\right)$.

∎

##  12 Bi-Lipschitz Embeddings of Orbifold Quotients

In this section, we go through the classification of groups whose actions have
an orbifold quotient (when restricted to the sphere) to the end of showing
that for those groups and with enough generic templates, max filter banks
admit strong avoidance and/or bi-Lipschitz properties.

###  12.1 Preliminary on Polarity and Infinitesemal Polarity

######  Definition 96.

Consider $G\leq\operatorname{O}(d)$ with closed orbits. We say the action of
$G$ is polar if there exists a subspace $\Sigma\subseteq\mathbb{R}^{d}$,
called a section, such that $\Sigma$ orthogonally intersects every orbit of
$G$, i.e., $G\cdot\Sigma=\mathbb{R}^{d}$ and for every $x\in\Sigma$, the
tangent space of the orbit $[x]$ at $x$ resides in the orthogonal complement
$\Sigma^{\perp}$. We say the action of $G$ is infinitesemally polar at
$x\in\mathbb{R}^{d}$ if the action of $G_{x}$ on the invariant subspace
$N_{x}$ is polar.

The following result shows that orbit spaces of polar actions are orbit spaces
of finite actions. Part (a) is immediate, part (b) is given by Propositions
1.2.1 and 1.3.2 in [17] along with Proposition 6 and the remark that follows
it in [13], while part (c) is given by Section 2.3 in [19].

######  Proposition 97.

Consider a compact $G\leq\operatorname{O}(d)$ with identity component $G_{0}$.
The following statements hold:

  * (a)

For $\Sigma\subseteq\mathbb{R}^{d}$, the action of $G$ is polar with section
$\Sigma$ if and only if the action of $G_{0}$ is polar with section $\Sigma$.

  * (b)

If the action of $G$ is polar with section $\Sigma$, then the inclusion
$\Sigma\xhookrightarrow{}\mathbb{R}^{d}$ induces an isometry of $\Sigma/H$ and
$\mathbb{R}^{d}/G$, where $H$ is the largest subgroup of $G$ under which
$\Sigma$ is invariant. Furthermore, $H$ is finite. If $G=G_{0}$, then $H$ is a
finite reflection group.

  * (c)

If $\mathbb{R}^{d}/G$ is isometric to $\mathbb{R}^{k}/H$ for some
$k\in\mathbb{N}$ and some finite $H\leq\operatorname{O}(k)$, then the action
of $G$ is polar.

Next, infinitesemally polarity at a point is equivalent to the orbit space
being an orbifold near that point.

######  Proposition 98 (Theorem 1.1 in [25]).

Consider a compact $G\leq\operatorname{O}(d)$ with identity component $G_{0}$.
The following statements are equivalent:

  * (a)

The action of $G$ is infinitesemally polar at $x$.

  * (b)

The action of $G_{0}$ is infinitesemally polar at $x$.

  * (c)

The sectional curvature of $[P(G)]$ is uniformly bounded near $[x]$.

  * (d)

The quotient $\mathbb{R}^{d}/G$ is an orbifold near $[x]$.

The following result is a classification of groups whose quotients are
Riemannian orbifolds when restricted to the sphere.

######  Proposition 99 (Theorem 1.4 in [18]).

Suppose $G\leq\operatorname{O}(d)$ is compact with identity component $G_{0}$.
Then, $\mathbb{S}^{d-1}/G$ is a Riemannian orbifold if and only if at least
one of the following cases occur:

  1. (a)

The action of $G_{0}$, or equivalently $G$, is polar.

  2. (b)

$G_{0}=U(1)$ and its action is a sum of non-trivial irreducible
$\mathbb{C}$-representations.

  3. (c)

$G_{0}=SU(2)$ and its action is a sum of non-trivial irreducible and even-
dimensional $\mathbb{C}$-representations.

  4. (d)

$G_{0}$ has cohomogeneity $3$.

  5. (e)

$G_{0}$ has the same orbits as one of the representations listed in Table 1.
Here $S_{k+}^{n}(r)$ is the $\frac{1}{2^{k}}^{\text{th}}$ hemisphere of the
round $n$-sphere of radius $r$ and hence curvature $1/r^{2}$.

$G$ | $\rho_{0}$ | Conditions | Orbit Space
---|---|---|---
$Spin(9)$ | $\mathbb{R}^{16}\oplus\mathbb{R}^{16}$ | $-$ | $S_{++}^{3}(1/2)$
$SU(n)$ | $\mathbb{C}^{n}\oplus\mathbb{C}^{n}$ | $n\geq 3$ | $S_{+}^{3}(1/2)$
$U(n)$ | $\mathbb{C}^{n}\oplus\mathbb{C}^{n}$ | $n\geq 2$ | $S_{+}^{3}(1/2)$
$Sp(n)$ | $\mathbb{H}^{n}\oplus\mathbb{H}^{n}$ | $n\geq 2$ | $S_{+}^{5}(1/2)$
$Sp(n)\times U(1)$ | $\mathbb{C}^{2n}\times\mathbb{C}^{2n}$ | $n\geq 2$ | $S_{++}^{4}(1/2)$
$Sp(n)\times Sp(1)$ | $\mathbb{R}^{4n}\oplus\mathbb{R}^{4n}$ | $n\geq 2$ | $S_{++}^{3}(1/2)$
$T^{2}\times Sp(n)$ | $\mathbb{C}^{2n}\oplus\mathbb{C}^{2n}$ | $n\geq 2$ | $S_{+++}^{3}(1/2)$
Table 1: Quotients with constant curvature 4

###  12.2 Groups with a Polar Action

Groups with polar actions constitute part (a) in Proposition 99. When the
action is polar, we have the following result:

######  Theorem 100.

Suppose $G\leq\operatorname{O}(d)$ is compact with polar action of
cohomogeneity $c$. Fix $V\in\mathbb{R}^{n\times k}$. Then, for any generic
templates $z_{1},\ldots,z_{n}\in\mathbb{R}^{d}$, the corresponding max filter
bank $\Phi$ strongly avoids $\operatorname{im}(V)$ provided $n\geq 2c+k$ if
$G$ is not connected and provided $n\geq c+k$ if $G$ is connected.

In the case of coorbit filter banks, we refer to Section 9 for a reduction.
The proof of the theorem follows immediately from the following lemma

######  Lemma 101.

Suppose $G\leq\operatorname{O}(d)$ is compact with polar action of
cohomogeneity $c$. For $z_{1},\dots,z_{n}\in\mathbb{R}^{d}$, denote by $\Phi$
the corresponding max filter bank. Then, for $V\in\mathbb{R}^{n\times k}$,
consider the semialgebraic set

$D_{V}:=\\{\\{z_{i}\\}_{i=1}^{n}\in(\mathbb{R}^{d})^{n}:\Phi\text{ fails to
strongly avoid }\operatorname{im}(V)\\}.$

Then, $\dim(D_{V})\leq nd-1-(n-k-2c)$ if $G$ is disconnected and
$\dim(D_{V})\leq nd-1-(n-k-c)$ if $G$ is connected.

###### Proof.

Let $\Sigma$ be a section. By Proposition 97, the inclusion
$\Sigma\xhookrightarrow{}\mathbb{R}^{d}$ induces an isometry of $\Sigma/H$ and
$\mathbb{R}^{d}/G$, where $H$ is the largest subgroup of $G$ under which
$\Sigma$ is invariant. Furthermore, $H$ is finite, and if $G$ is connected,
then $H$ is a finite reflection group.

Suppose $\\{z_{i}\\}_{i=1}^{n}\in D_{V}$. Then, there exists sequences
$x_{j},y_{j}\in\mathbb{R}^{d}$ such that
$\lim_{j\to\infty}Q(x_{j},y_{j})\in\operatorname{im}(V)$. We may translate
$(x_{j})_{j\in\mathbb{N}},(y_{j})_{j\in\mathbb{N}}$ and
$\\{z_{i}\\}_{i=1}^{n}$ so as to belong to the section $\Sigma$. Then, by
29(a) and the first paragraph,
$\langle\langle[x_{j}]_{G},[z_{i}]_{G}\rangle\rangle_{G}=\langle\langle[x_{j}]_{H},[z_{i}]_{H}\rangle\rangle_{H}$
and similarly for $y_{j}$. Moreover,
$d([x_{j}]_{G},[y_{j}]_{G})=d([x_{j}]_{H},[y_{j}]_{H})$. Hence, by accounting
for $n(d-c)$ degrees of freedom in the orbits of $\\{z_{i}\\}_{i=1}^{n}$, we
reduce the problem to $G=H$ and $\mathbb{R}^{d}=\Sigma$ where now we need to
show $\dim(D_{V})\leq nc-1-(n-k-2c)$ when $H$ is finite, which follows from
Theorem 69, and $\dim(D_{V})\leq nc-1-(n-k-c)$ when $H$ is a finite reflection
group, which follows from reducing the problem to a Weyl chamber of the action
and arguing with singular value decomposition as in the proof of Lemma 60. ∎

######  Remark 102.

The above proof does not extend to coorbit filter banks as they may not be
invariant under isometry of the orbit space when $|\pi_{0}(G)|>1$.

###  12.3 Groups with an Almost Free Action

######  Definition 103.

For a compact $G\leq\operatorname{O}(d)$, we say $G$ has an almost free action
if $G_{x}$ is finite for every $x\in\mathbb{S}^{d-1}$.

Note that $G$ is almost free if and only if $G_{0}$ is almost free. By Section
3.2 in [18], the only almost free actions of connected groups are given by (b)
and (c) in Proposition 99. In such cases, the principal isotropy group is
trivial by the discussion preceeding Lemma 2.1 in [18]. Then, Lemma 93
applies, and when combined with Lemma 54 and Theorem 74, we obtain the
following result:

######  Theorem 104.

Suppose $G\leq\operatorname{O}(d)$ is compact and its identity component
$G_{0}$ has an almost free action with cohomogeneity $c$. Let
$\chi:=\max_{x\in\mathbb{S}^{d-1}}|G_{x}\cap G_{0}|$. Fix
$V\in\mathbb{R}^{n\times k}$ and
$p_{1},\dots,p_{n}\in\\{1,\dots,|\pi_{0}(G)|\\}$. Then, for generic
$z_{1},\dots,z_{n}\in\mathbb{R}^{d}$, the corresponding coorbit filter bank
$\Phi\colon\mathbb{R}^{d}/G\to\mathbb{R}^{n}$ strongly avoids
$\operatorname{im}(V)$ provided $n>|\pi_{0}(G)|\cdot\chi\cdot(2c+k-1)$.

###  12.4 Groups with Cohomogeniety 3

We now consider the case when a group is connectded and has cohomogeneity $3$,
that is part (d) in Proposition 99. We may assume the action is nonpolar since
that yields case (a). Moreover, we may assume $\dim(G)>1$ since otherwise,
$G=S^{1}$ acts on $\mathbb{R}^{4}\cong_{\mathbb{R}}\mathbb{C}^{2}$ in which
case $F_{G}=\\{0\\}$ implies that the action is almost free and
$F_{G}\neq\\{0\\}$ implies that the action is polar.

######  Theorem 105.

Suppose $G\leq\operatorname{O}(d)$ is compact with cohomogeneity $3$,
$\dim(G)>1$ and nonpolar action. Fix $V\in\mathbb{R}^{n\times k}$. Fix
$p_{1},\dots,p_{n}\in\\{1,\dots,|\pi_{0}(G)|\\}$. Then, for generic templates
$z_{1},\ldots,z_{n}\in\mathbb{R}^{d}$, the corresponding coorbit filter bank
$\Phi$ strongly avoids $\operatorname{im}(V)$ provided
$n>|\pi_{0}(G)|(12+2k-1)$.

###### Proof.

The case of nonconnected groups follows from the case of connected groups by
levaraging Theorem 74 and the bound
$\chi_{\pi_{0}}^{\\{p_{i}\\}_{i=1}^{n}}\leq|\pi_{0}(G)|$. As such, we assume
$G$ is connected. In particular, coorbit filter banks are max filter banks in
this case. When viewing $\mathbb{R}^{4}$ as $\mathbb{C}^{2}$, denote

$H_{0}^{1,1}:=\left\\{\begin{bmatrix}e^{i\theta}&0\\\
0&e^{i\theta}\end{bmatrix}:\theta\in\mathbb{R}\right\\}\text{ and
}H_{0}^{1,2}:=\left\\{\begin{bmatrix}e^{i\theta}&0\\\
0&e^{2i\theta}\end{bmatrix}:\theta\in\mathbb{R}\right\\}.$

By Section 4 of [31], it holds that $X_{1,1}:=\mathbb{S}^{3}/H_{0}^{1,1}$ is
isometric to $S^{2}(1/2)$, the round $2$-sphere of radius $1/2$, and
$X_{1,2}:=\mathbb{S}^{3}/H_{0}^{1,2}$ is isometric to a non-round $2$-sphere
with metric

$ds^{2}=\frac{1}{4}\left[\frac{\sin^{2}\varphi}{2\sin^{2}(\varphi/2)+\cos^{2}(\varphi/2)}d\theta^{2}+d\varphi^{2}\right],\qquad
0\leq\theta\leq 2\pi,\ \ 0\leq\varphi\leq\pi.$

Then, by Section 5 in [31], $\mathbb{R}^{d}/G$ is isometric to
$\mathbb{R}^{4}/H$ where one of the following cases occurs:

  * (i)

$H_{0}=H_{0}^{1,1}$ and $\pi_{0}(H)$ is generated by reflections in
$\operatorname{Isom}(S^{2}(1/2))\cong O(3)$.

  * (ii)

$H_{0}=H_{0}^{1,2}$ and $|\pi_{0}(H)|=2$ with the nontrivial element being a
reflection across the great circle with $\theta=\pi$ in
$\operatorname{Isom}(X_{1,2})\cong O(2)$.

Here, $\pi_{0}(H)$ is viewed as a subgroup of
$\operatorname{Isom}(\mathbb{S}^{3}/H_{0})$ (see Remark 85). In both cases
above, $|S^{1}([x]_{0},[y_{0}])|=1$ for all
$[x]_{0},[y]_{0}\in\mathbb{S}^{3}/H_{0}$. As such,
$\chi^{\\{1\\}_{i=1}^{n}}_{\pi_{0}}(H)=1$ and the result follows by Theorem 74
and Theorem 104 (wherein $\chi\leq 2$). ∎

###  12.5 Groups with Hemispherical Quotients

The following result treats the case of hemispherical quotients. Note that
case (e) in Proposition 99 is only concerned with the identity component
having a hemispherical quotient. In general, when the group is not connected
yet its identity component has a hemispherical quotient and when coorbit
filter banks are considered, one may turn the bound below into
$n>|\pi_{0}(G)|(2c-1)$ using Theorem 74. The result is limited to bi-Lipschitz
bounds, but we believe the proof can be extended to strong avoidance of
subspaces.

######  Theorem 106.

Suppose $G\leq\operatorname{O}(d)$ is compact and $\mathbb{S}^{d-1}/G$ is
isometric to $S^{c-1}_{p+}(1/2)$ where $c$ is the cohomogeneity of $G$ where
$p\geq 1$. Then, for any generic templates
$z_{1},\ldots,z_{n}\in\mathbb{R}^{d}$, the corresponding max filter bank
$\Phi$ is bi-Lipschitz provided $n\geq 2c$.

###### Proof.

The orbit space $\mathbb{R}^{d}/G$ is given by a cone
$(S^{c-1}_{p+}(1/2)\times\mathbb{R}_{\geq
0})/(S^{c-1}_{p+}(1/2)\times\\{0\\})$ such that every $[x]\in\mathbb{R}^{d}/G$
can be expressed as $(u_{x},\|x\|)$ where $u_{x}:=[x/\|x\|]$ and where we take
the convention $\frac{0}{\|0\|}:=0$.
Moreover, $d_{\mathbb{R}^{d}/G}((u_{x},1),(u_{y},1))$ is given by the minimal
chord-length $\min_{g\in G}\|g\frac{x}{\|x\|}-\frac{y}{\|y\|}\|$ while
$d_{S^{c-1}_{p+}(1/2)}(u_{x},u_{y})$ is given by the minimal geodesic arc-
length $\min_{g\in G}d_{\mathbb{S}^{d-1}}(g\frac{x}{\|x\|},\frac{y}{\|y\|})$
where $d_{\mathbb{S}^{d-1}}$ is the arc-length metric on the sphere in
$\mathbb{R}^{d}$. From elementary geometry, the two are related by

$d_{\mathbb{R}^{d}/G}((u_{x},1),(u_{y},1))=2\sin\left(\frac{d_{S^{c-1}_{p+}(1/2)}(u_{x},u_{y})}{2}\right).$
(10)

By the radial Pythagorean Theorem (Lemma 12 in [9]), the square quotient
metric is given by

$d^{2}_{\mathbb{R}^{d}/G}((u_{x},\|x\|),(u_{y},\|y\|))=(\|x\|-\|y\|)^{2}+\|x\|\cdot\|y\|\cdot
d^{2}_{\mathbb{R}^{d}/G}((u_{x},1),(u_{y},1)).$ (11)

Now, consider the continuous homeomorphism $\varphi\colon S^{c-1}_{p+}(1/2)\to
S^{c-1}_{p+}(1)\subseteq\mathbb{R}^{c}$ given by rescaling. We take a
continuous extension of $\varphi$ to $\mathbb{R}^{d}/G$ by defining
$\varphi([x]):=\|x\|^{2}\cdot\varphi(u_{x})$. This is again a homeomorphism.
Then, by the half-angle trigonometric identities and since the rescaling
doubles spherical geodesic distance, we use (10) and compute

$\displaystyle d^{2}_{\mathbb{R}^{d}/G}((u_{x},1),(u_{y},1))$
$\displaystyle=4\sin^{2}\left(\frac{d_{S^{c-1}_{p+}(1/2)}(u_{x},u_{y})}{2}\right)$
$\displaystyle=4\sin^{2}\left(\frac{d_{S^{c-1}_{p+}(1)}(\varphi(u_{x}),\varphi(u_{y}))}{4}\right)$
$\displaystyle=2-2\cos\left(\frac{d_{S^{c-1}_{p+}(1)}(\varphi(u_{x}),\varphi(u_{y}))}{2}\right)$
$\displaystyle=2-2\sqrt{\frac{1+\langle\varphi(u_{x}),\varphi(u_{y})\rangle}{2}}.$

In the last step, we only choose the positive square root of the half-angle
formula for cosine since
$d_{S^{c-1}_{p+}(1)}(\varphi(u_{x}),\varphi(u_{y}))\leq\pi$ for every
$\varphi(u_{x}),\varphi(u_{y})\in S^{c-1}_{p+}(1)$. Hence, combining the above
with 29(a), we obtain

$\langle\langle[x],[y]\rangle\rangle=\|x\|\cdot\|y\|\cdot\sqrt{\frac{1+\langle\varphi(u_{x}),\varphi(u_{y})\rangle}{2}}=\sqrt{\frac{\|\varphi([x])\|\|\varphi([y])\|+\langle\varphi([x]),\varphi([y])\rangle}{2}}.$
(12)

Observe that by rearranging (12) and using 29(a) along with
$\|\varphi([x])\|=\|x\|^{2}$ and $\|\varphi([y])\|=\|y\|^{2}$, we get the
following equality which we shall refer to later in the proof:

$\frac{\|\varphi([x])-\varphi([y])\|^{2}}{d([x],[y])^{2}}=d^{2}([x],[y])+4\langle\langle[x],[y]\rangle\rangle.$
(13)

Now, consider any principal template $z_{i}\in P(G)$. Then,
$-\varphi([z_{i}])\notin\operatorname{im}(\varphi)$ because $p\geq 2$ and the
principal stratum in $\mathbb{S}^{d-1}/G$ is precisely the (manifold) interior
of the hemisphere $S^{c-1}_{p+}(1/2)$. As such, the following quantity is
finite and continuous provided $x\neq 0$, $y\neq 0$ and $z_{i}\in P(G)$

$C(z_{i},x,y):=\Big{[}\sqrt{\|\varphi([x])\|\|\varphi([z_{i}])\|+\langle\varphi([x]),\varphi([z_{i}])\rangle}+\sqrt{\|\varphi([y])\|\|\varphi([z_{i}])\|+\langle\varphi([y]),\varphi([z_{i}])\rangle}\Big{]}^{-1}.$

For $x\neq 0$, it holds that $C(z_{i},x,x)<\infty$ is given by

$C(z_{i},x,x)=\left[2\|z_{i}\|\sqrt{1+\langle\varphi(u_{x}),\varphi(u_{z_{i}})\rangle}\right]^{-1}.$
(14)

Then, for arbitrary $x,y,\in\mathbb{R}^{d}$ and $z_{i}\in P(G)$, we use (12)
and we rationalize to obtain

$\displaystyle\quad\frac{\langle\langle[x],[z_{i}]\rangle\rangle-\langle\langle[y],[z_{i}]\rangle\rangle}{d([x],[y])}=$
(15)
$\displaystyle\qquad\frac{1}{\sqrt{2}}C(z_{i},x,y)\cdot\left[\|z_{i}\|^{2}\frac{(\|x\|+\|y\|)(\|x\|-\|y\|)}{d([x],[y])}+\frac{\|\varphi([x])-\varphi([y])\|}{d([x],[y])}\left\langle\varphi([z_{i}]),\frac{\varphi([x])-\varphi([y])}{\|\varphi([x])-\varphi([y])\|}\right\rangle\right].$

Now, let $n\geq 2c$ and consider a max filter bank $\Phi$ corresponding to
templates $\\{z_{i}\\}_{i=1}^{n}\in P(G)^{n}$. By Theorem 53, we have that
$\Phi$ is injective for generic $\\{z_{i}\\}_{i=1}^{n}\in P(G)^{n}$. As such,
it suffices to show $\Phi$ is locally lower Lipschitz at every
$x\in\mathbb{S}^{d-1}$ for generic $\\{z_{i}\\}_{i=1}^{n}\in P(G)^{n}$.

To this end, suppose by contradiction that there exists a nonempty
$G^{n}$-invariant open set $U\subseteq P(G)^{n}$ such that for
$\\{z_{i}\\}_{i=1}^{n}\in U$, $\Phi$ fails to be locally lower Lipschitz at
every $x\in\mathbb{S}^{d-1}$. In particular, $\varphi^{n}([U])$ is nonempty
and open in $\mathbb{R}^{c}$ where the homeomorphism
$\varphi^{n}\colon(\mathbb{R}^{d}/G)^{n}\to\mathbb{(}R^{c})^{n}$ is given by
the $n$-fold product of $\varphi$. Next, considering the open continuous
mapping $h\colon\mathbb{R}^{c}\setminus\\{0\\}\to\mathbb{S}^{c-1}$ given by
$h(z)=z/\|z\|$ and its $n$-fold product
$h^{n}\colon(\mathbb{R}^{c}\setminus\\{0\\})^{n}\to(\mathbb{S}^{c-1})^{n}$
which is also open and continuous, we obtain that $h^{n}\circ\varphi^{n}([U])$
is nonempty and open in $(\mathbb{S}^{c-1})^{n}$. We shall refer to this fact
at the end of the proof.

Now, consider templates $\\{z_{i}\\}_{i=1}^{n}\in U$. Then, there exists
$[x]\in\mathbb{S}^{d-1}/G$ such that $\Phi$ fails to be locally lower
Lipschitz at $x$, meaning there exists sequences $x_{j},y_{j}\to x$ with
$[x_{j}]\neq[y_{j}]$ and $d([x_{j}],[y_{j}])=\|x_{j}-y_{j}\|$ such that
$\lim_{j\to\infty}Q(x_{j},y_{j})=0$ where $Q$ dentoes the difference quotient
corresponding to $\Phi$.

By (13), (14), (15) and the reverse triangle inequality
$|\|x_{j}\|-\|y_{j}\||\leq\|x_{j}-y_{j}\|$, we obtain that there exists
nonzero $u\in\mathbb{R}^{c}$ and $b\in\mathbb{R}$ such that $\|u\|=1$,
$|b|\leq 1$ and after taking subsequences

$\frac{\langle\langle[x_{j}],[z_{i}]\rangle\rangle-\langle\langle[y_{j}],[z_{i}]\rangle\rangle}{d([x_{j}],[y_{j}])}\to\frac{\|z_{i}\|}{\sqrt{2}\sqrt{1+\langle\varphi(u_{x}),\varphi(u_{z_{i}})\rangle}}\big{[}\langle\varphi(u_{[z_{i}]}),u\rangle+b\big{]}.$

Namely,

$u:=\frac{1}{2}\cdot\lim_{j\to\infty}\frac{\|\varphi([x_{j}])-\varphi([y_{j}])\|}{\|x_{j}-y_{j}\|}\cdot\frac{\varphi([x_{j}])-\varphi([y_{j}])}{\|\varphi([x_{j}])-\varphi([y_{j}])\|},$

and

$b:=\frac{1}{2}\cdot\lim_{j\to\infty}\frac{(\|x_{j}\|+\|y_{j}\|)(\|x_{j}\|-\|y_{j}\|)}{\|x_{j}-y_{j}\|}.$

Since $\lim_{j\to\infty}Q(x_{j},y_{j})=0$, we obtain that
$\langle\varphi(u_{z_{i}}),u\rangle=-b$ for each $i\in\\{1,\dots,n\\}$. It
follows that the linear map
$T_{\Phi}:=\\{\langle\varphi(u_{[z_{i}]}),\cdot\rangle\\}_{i=1}^{n}\in\mathbb{R}^{n\times
c}$ is either non-injective or it satisfies
$\operatorname{im}(T_{\Phi})\cap\operatorname{span}(\mathbf{1})\neq\\{0\\}$
where $\mathbf{1}$ is the vector of all ones. Such conditions may not hold for
$\varphi(u_{[z_{i}]})$ in a nonempty open subset of $(\mathbb{S}^{c-1})^{n}$,
provided $n\geq c+1$. This contradicts the fact that so far,
$\varphi(u_{[z_{i}]})\in V:=h^{n}\circ\varphi^{n}([U])$ has been arbitrary and
$V$ is nonempty and open. The result follows. ∎

##  13 Discussion

While we have found that enough generic templates suffice for coorbit filter
banks to be bi-Lipschitz (more generally, strongly avoidant) in the case of
actions whose spheres have orbifold quotients, the following fundamental
question which is weaker than Problems 14 and 18 remains open for general
compact groups:

######  Problem 107.

Does there exist a compact group with which every coorbit filter bank fails to
admit bi-Lipschitz bounds no matter the size or form the templates take?

We conjecture that the answer is false and one way we believe this can be
tackled is by an inductive step on stabilizers acting on the tangent and
normal spaces of a point (for example, in Section 8, we used an induction step
in the simpler case of finite-index stabilizers). This approach could require
a much larger number of templates than would ever be necessary but it would at
least provide a negative answer to the question posed. The first example of
smallest dimension we can think of where the question is unresolved is the
real irreducible representation of $\operatorname{SO}(3)$ acting on
$\mathbb{R}^{7}$. Investigating this example could aid in resolving the
problem for all compact groups.

For the stronger Problem 18, Lemmas 60 and 54 can be viewed as a step forward
towards a solution. It is left to extend those to similar results of local
avoidance at nonprincipal points. This remains an open problem even for
regular points where we had to pass through the size of potentially
arbitrarily large finite stablizers.

Nonetheless, we have shown that with sufficiently many generic templates,
coorbit filter banks are locally lower Lipschitz at principal and regular
points. A natural next step is to aim for tackling orbifold points by
potentially making use of infinitesemal polarity in Proposition 98. Adapting
the proof of Theorem 2.2.1 in [17] to the context of compact subgroups of
$\operatorname{O}(d)$ could be informative for this task although we leave it
for future work.

On the technical improvement side, we conjecture that Lemma 93 holds with $G$
having nontrivial principal isotropy groups. For that, the arguments in
Section A.4 would need to accomodate for that change. If the conjecture holds,
then one may drop the necessity for minimal reduction in Theorem 92 and the
assumption of trivial principal isotropy group in Lemma 93. Either way, with
our proof methods, we may still need to pass to a characteristic
$\chi_{G}:=\max_{x\in R(G)}|G_{x}/G_{p,x}|$ where $G_{p,x}$ is a principal
stabilizer in $G_{x}$.

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##  Appendix A Differential Geometric Arguments

###  A.1 Cut point symmetry for non-complete Riemannian manifolds

######  Lemma 108.

Let $M$ be a Riemannian manifold and let $[x],[z]\in M$. Suppose that the
following hypotheses hold:

  * (i)

There exists a minimal geodesic $\gamma^{r}$ joining $[x]$ to some $[p]\in M$
such that $[z]$ is in the interior of $\gamma^{r}$ and
$\gamma^{r}|_{[x]\to[z]}$ is the unique minimal geodesic joining $[x]$ to
$[z]$.

  * (ii)

There exists a neighborhood $U$ of $[x]$ such that for every $[p]\in U$, there
exists a minimal geodesic joining $[z]$ to $[p]$.

  * (iii)

If $[x_{i}]\to[x]$ and $\sigma_{i}$ are minimal geodesics joining $[z]$ to
$[x_{i}]$, then $\sigma_{i}\rightarrow\gamma$ where $\gamma$ is the unique
minimal geodesic joining $[z]$ to $[x]$ i.e. the reverse parametrization of
$\gamma^{r}$.

Then, $\gamma|_{[z]\to[x]}$ remains minimizing shortly beyond $[x]$.

###### Proof.

Let $\gamma:[0,d([z],[x])]\to M$ be the unique minimal geodesic joining $[z]$
to $[x]$ as in the first hypothesis. Then, there exists $v\in T_{[z]}M$ such
that $\gamma(t)=\operatorname{exp}_{[z]}(tv)$ where $\exp\colon TM\to M$ is
the Riemannian exponential of $M$. By Proposition 5.19 in [24], the
exponential map is smooth and its domain is open. It follows that
$\operatorname{exp}_{[z]}(tv)$ is defined in a neighborhood of $d([z],[x])v\in
T_{[z]}M$.
We seek to show that $\operatorname{exp}_{[z]}(tv)$ remains minimizing for $t$
shortly beyond $d([z],[x])$. Suppose otherwise for the sake of contradiction.
We closely follow the proof of Proposition 13.2.2 in [14] which treats the
case of complete manifolds. Let $\\{t_{0}+\varepsilon_{i}\\}$ be a sequence in
which $\varepsilon_{i}>0$ and $\varepsilon_{i}\to 0$. By the second hypothesis
and by taking large enough $i$, there exists a sequence of minimizing
geodesics $\sigma_{i}$ joining $[z]$ to
$\operatorname{exp}_{[z]}((t_{0}+\varepsilon_{i})v)$ and let
$\sigma_{i}^{\prime}(0)\in T_{[z]}M$ be the corresponding sequence of tangent
vectors at $[z]$. By the third hypothesis, $\sigma_{i}\rightarrow\gamma$ and
$\sigma_{i}^{\prime}(0)\rightarrow\gamma^{\prime}(0)$.

We show $d\operatorname{exp}_{[z]}$ is singular at $t_{0}\gamma^{\prime}(0)$.
Suppose otherwise for the sake of contradiction. Then, there exists a
neighborhood $W$ of $t_{0}\gamma^{\prime}(0)$ such that
$\operatorname{exp}_{[z]}\big{|}_{W}$ is a diffeomorphism. By definition of
$\sigma_{j}$,
$\gamma(t_{0}+\varepsilon_{j})=\sigma_{j}(t_{0}+\varepsilon_{j}^{\prime})$
where $\varepsilon_{j}^{\prime}\leq\varepsilon_{j}$, because $\sigma_{j}$ is
minimizing. Take $\varepsilon_{j}$ sufficiently small so that
$(t_{0}+\varepsilon_{j}^{\prime})\sigma_{j}^{\prime}(0)$ and
$(t_{0}+\varepsilon_{j})\gamma^{\prime}(0)$ belong to $W$. Then,

$\operatorname{exp}_{[z]}((t_{0}+\varepsilon_{j})\gamma^{\prime}(0))=\operatorname{exp}_{[z]}((t_{0}+\varepsilon_{j}^{\prime})\sigma_{j}^{\prime}(0)),$

hence
$(t_{0}+\varepsilon_{j})\gamma^{\prime}(0)=(t_{0}+\varepsilon_{j}^{\prime})\sigma_{j}^{\prime}(0)$,
that is $\gamma^{\prime}(0)=\sigma_{j}^{\prime}(0)$. This contradicts the
assumption that $\gamma$ is no longer minimizing beyond $t_{0}$. Thus, there
exists nonzero $u\in T_{[z]}M$ such that
$d_{t_{0}\gamma^{\prime}(0)}\operatorname{exp}_{[z]}u=0$. By Corollary 3.46 in
[16], there exists a Jacobi field $Y$ along $\gamma$ such that $Y(0)=0$,
$Y^{\prime}(0)=u$ and $Y(t_{0})=0$. In other words, $[z]$ and $[x]$ are
conjugate along $\gamma$. Now, let $\gamma^{r}\colon[0,d([x],[z])]\to M$ be
the unique minimal geodesic joining $[x]$ to $[z]$. Since $\gamma^{r}$ is the
reverse parametrization of $\gamma$, $[x]$ and $[z]$ are also conjugate along
$\gamma^{r}$. However, by the first hypothesis, $\gamma^{r}$ remains
minimizing shortly beyond $[z]$. This contradicts the fact that minimizing
geodesics lack conjugate points in their interior (Theorem 3.73 in [16]). ∎

###  A.2 Note on Stabilizers

The following proposition shows that stabilizers can only “shrink” locally.
The result is well-known (see for example Lemma 6.3 and Corollary 4.13 in
[27]) but we provide a proof.

######  Proposition 109.

Suppose $G\leq\operatorname{O}(d)$ is compact. Fix
$i\in\\{1,\dots,|\pi_{0}(G)|\\}$. Then, the following statements hold for
$x\in\mathbb{R}^{d}$:

  1. (a)

If $z\in U_{x}^{i}$, then $G_{z}\leq G_{x}$.

  2. (b)

If $z\in Q_{x}^{i}$, then $(G_{z})\leq(G_{x})$.

  3. (c)

If $x\in P(G)$ and $z\in N_{x}$, then $G_{x}\leq G_{z}$.

###### Proof.

For (a), if $z\in U_{x}^{i}$ and $g\in G_{z}$, then $z\in g^{-1}U_{x}^{i}\cap
U_{x}^{i}\neq\varnothing$ so by Lemma 43, $g^{-1}x=x$ and so $g\in G_{x}$ as
desired.

For (b) and by definition of $Q_{x}^{i}$, there exists $g$ such that $z\in
V_{gx}^{i}\subseteq U_{gx}^{i}$. Hence by (a), $G_{z}\leq G_{gx}=gG_{x}g^{-1}$
meaning $(G_{z})\leq(G_{x})$ as desired.

For (c) and by (a), we have that for $z\in U_{x}^{i}$, $G_{z}\leq G_{x}$. By
principality of $x$, we obtain $G_{z}=G_{x}$. It follows that $G_{x}$ fixes
$U_{x}^{i}$ and since $\operatorname{span}(U_{x}^{i})=N_{x}$, we obtain that
$G_{x}$ fixes $N_{x}$ as desired. ∎

###  A.3 Proof of Lemma 45

Before proceeding, we note that by Proposition 109, if $x\in P(G)$ and $z\in
V_{x}^{i}$, then $G_{x}=G_{z}$ and so $z\in P(G)$.

(a)$\Rightarrow$(b). From the above and since $P(G)\subseteq P(G_{0})$, it
holds that $x,z\in P(G_{0})$. Moreover, for any $t,y\in\mathbb{R}^{d}$ and by
Proposition 26, the minimal geodesics from $[t]_{0}$ to $[y]_{0}$ in
$\mathbb{R}^{d}/G_{0}$ are precisely the orbital projections of straightlines
from $t$ to $\arg\max_{p\in[y]_{0}}\langle p,t\rangle$.

Now, by Proposition 27, we have that $M_{0}:=P(G_{0})/G_{0}$ is a Riemannian
manifold and a geodesic space. Moreover, since $z\in V_{x}^{i}$ and
$V_{x}^{i}$ is open in $N_{x}$, there exists $\epsilon>0$ such that
$q:=z+(z-x)\varepsilon\in V_{x}^{i}$. By definition of $V_{x}^{i}$ and by
35(b) and for $t\in(q,z]$, we have that $[t,x]$ is the unique straighline
connecting $t$ to $\arg\max_{p\in[x]_{0}}\langle p,t\rangle=\\{x\\}$. It
follows that $M_{0}$ satisfies the first hypothesis of Lemma 108. The second
hypothesis follows from $M_{0}$ being a geodesic space.

For the third hypothesis, suppose $[x_{i}]_{0}\to[x]_{0}$ and $\sigma_{i}$ are
minimal geodesics joining $[z]_{0}\to[x_{i}]_{0}$. Then, $\sigma_{i}$ are
projections of straightlines connecting $z$ to elements of
$\arg\max_{p\in[x_{i}]_{0}}\langle p,z\rangle$. In the limit, every element of
$\arg\max_{p\in[x_{i}]_{0}}\langle p,z\rangle$ converges to $x$. This is
because for $p\neq x\in[x]_{0}$, we have $\langle z,p\rangle<\langle
z,x\rangle$ and so the inequality remains true in a neighborhood of $x$ and
$p$. As such, $\sigma_{i}$ converges to the projection of the unique
straightline from $z$ to $x$ and the third hypothesis holds.

It follows by Lemma 108 that the straigthline from $z$ to $x$ remains shortly
minimizing to $[z]_{0}$. The implication $x\in V_{z}^{i}$ now follows from
Lemma 39, 35(b) and 43(e).

(b)$\Rightarrow$(a). By Proposition 27, $P(G)$ is dense and by Lemma 44,
$Q_{x}^{i}$ is dense. As such, select $y\in P(G)\cap Q_{x}^{i}$ and after
translating assume $y\in P(G)\cap V_{x}^{i}$. By assumption, $x\in V_{y}^{i}$
and by the note in the beginning of the proof, we obtain that $x\in P(G)$ as
desired.

(b)$\Rightarrow$(c). This is immediate.

(c)$\Rightarrow$(a). As above, select $y\in P(G)\cap V_{x}^{i}$. By
assumption, $x\in Q_{y}^{i}$ so $x\in V_{gy}^{i}$ for some $g\in G$. Since
$gy\in P(G)$, we obtain $x\in P(G)$ as desired.

###  A.4 Proof of Lemma 95

(a)$\Rightarrow$(b). Suppose $x\in R(G)$. Since $G$ has trivial principal
isotropy group, we obtain that $G_{x}$ is finite. Denote $V_{x}:=V_{x}^{1}$.
Equip $G$ with a bi-invariant metric $\gamma_{G}$.

Claim 1: There exists an open neighborhood $U$ of the identity $e$ in $G$ such
that $U=U^{-1}$ and $\overline{U^{-1}U}\cap G_{x}=\\{e\\}$.

Proof: The claim is folklore but we provide a proof. Denote the multiplication
map by $m_{G}\colon G\times G\to G$. Take an open neighborhood $W$ of the
identity such that $\overline{W}\cap G_{x}=\\{e\\}$. Then, for a simple open
neighborhood $W_{1}\times W_{2}\subseteq m_{G}^{-1}(W)$, take $U=(W_{1}\cap
W_{2})\cap(W_{1}\cap W_{2})^{-1}$. $\blacksquare$

Claim 2: The multiplication map $F\colon U\times V_{x}\rightarrow U\cdot
V_{x}$ is a diffeomorphism.

Proof: Surjectivity and smoothness are immediate. For injectivity, take
$u_{j}\in U$ and $s_{j}\in V_{x}$ such that $u_{1}s_{1}=u_{2}s_{2}$. Then,
$s_{1}=u_{1}^{-1}u_{2}s_{2}$ implying $u_{1}^{-1}u_{2}V_{x}\cap
V_{x}\neq\varnothing$ and so by Lemma 43, $u_{1}^{-1}u_{2}\in G_{x}\cap
U^{-1}U=\\{e\\}$. Hence, $u_{1}=u_{2}$ and $s_{1}=s_{2}$. Lastly, it is a
submersion since at $(u,s)\in U\times V_{x}$, the image of the differential
$dF(u,s)$ is $T_{us}\oplus N_{us}$ so it has full rank. $\blacksquare$

Consider the smooth submersive projection $\pi_{2}\colon U\times V_{x}\to
V_{x}$. We obtain a smooth submersion $\pi:=\pi_{2}\circ F^{-1}\colon U\cdot
V_{x}\to V_{x}$ which restricts to the identity on $V_{x}$ and whose fibers
are given by $\pi^{-1}(q)=U\cdot q$ for $q\in V_{x}$.

Claim 3: Equip $V_{x}$ with the metric
$\gamma_{\pi(s)}(d\pi(X_{s}),d\pi(Y_{s})):=\langle X_{s},Y_{s}\rangle$, for
$X_{s},Y_{s}\in N_{s}$ and $s\in V_{x}$. Then, $\gamma$ is well-defined and
makes $\pi\colon(U\cdot V_{x},\langle{,}\rangle)\to(V_{x},\gamma)$ into a
Riemannian submersion.

Proof: First, we compute the differential of $\pi$. Let $\mathfrak{g}$ denote
the Lie algebra of $G$. Then, a tangent vector to $u\in U$ may be viewed as
$g_{p}\cdot u$ where $g_{p}\in\mathfrak{g}$. A local $\gamma_{G}$-geodesic at
$u$ in the direction of $g_{p}\cdot u$ has the form $\exp_{G}(tg_{p})\cdot u$.
By the product rule and with $s\in V_{x}$ and $s_{t}\in V_{x}=T_{s}V_{x}$, it
follows that

$dF(u,s)|_{(g_{p}u,s_{t})}=\frac{\partial}{\partial
t}\big{[}(\exp_{G}(tg_{p})\cdot
u)\cdot(s+ts_{t})\big{]}\Big{|}_{t=0}=g_{p}\cdot us+u\cdot s_{t}.$

Observe that $g_{p}\cdot us\in T_{us}$ and $u\cdot s_{t}\in N_{us}$. As such,
for $v\in\mathbb{R}^{d}$, we decompose $v=(us_{t})\oplus(g_{p}us)\in
N_{us}\oplus T_{us}$ where $s_{t}\in N_{s}$ and $g_{p}\in\mathfrak{g}$, and we
obtain $dF^{-1}(us)|_{v}=(g_{p}u,s_{t})$ and hence $d\pi(us)|_{v}=s_{t}$. It
follows that the kernel bundle, i.e. the vertical bundle, of $d\pi$ is the
orbit tangent bundle of $U\cdot V_{x}$ while the orthogonal complement of
that, i.e. the horizontal bundle, is the orbit normal bundle of $U\cdot
V_{x}$. Moreover, the following equivariance relationship holds

$d\pi(us)|_{us_{t}}=u\cdot d\pi(s)|_{s_{t}}$

Since $u$ is an isometry, we may equip $V_{x}$ with the well-defined metric
$\gamma_{\pi(s)}(d\pi(X_{s}),d\pi(Y_{s})):=\langle X_{s},Y_{s}\rangle$, for
$X_{s},Y_{s}\in N_{s}$ and $s\in V_{x}$. With this metric, $\pi\colon(U\cdot
V_{x},\langle{,}\rangle)\to(V_{x},\gamma)$ becomes a Riemannian submersion.
$\blacksquare$

Claim 4: For $p\in V_{x}$, there exists a neighborhood $U_{p}$ of $x$ in
$V_{x}$ such that the following holds for $q\in U_{p}$:

  * (i)

$G_{x}U_{p}\subseteq U_{p}$.

  * (ii)

Straightlines from $p$ to (defacto well-defined) distance minimizers in $Uq$
all lie in $U\cdot V_{x}$.

  * (iii)

If $g\in G$ is such that $\|p-gq\|\leq d(p,Uq)$, then $gq\in Uhq$ for some
$h\in G_{x}$ and it holds that $[p,gq]\subseteq U\cdot V_{x}$.

Proof: Take any sequences $c_{n}$ and $q_{n}$ such that $c_{n}$ are distance
minimizers from $p$ to $\overline{Uq_{n}}$ and $q_{n}\to x$ and $q_{n}\in
V_{x}$. Let $w_{n}\in\overline{U}$ be such that $c_{n}=w_{n}\cdot q_{n}$. We
may take a subsequence such that $w_{n}\to w\in\overline{U}$ and hence
$c_{n}\to c=w\cdot x\in\overline{Ux}$. Then, $c$ is a distance minimizer from
$p$ to $\overline{Ux}$ since for every $g\in\overline{U}$, we have
$d(p,gq_{n})\geq d(p,c_{n})$ and by taking the limit, we obtain $d(p,gx)\geq
d(p,c)$. Since $p\in V_{x}$, we have that $x$ is the unique distance minimizer
from $p$ to $Gx$. It follows that $c=x$ and $w\in\overline{U}\cap
G_{x}=\\{e\\}$. Next, by 35(c) and 43(e), $[p,x]\subseteq V_{x}$ and also
$[p,c_{n}]\to[p,x]$ uniformly. It follows that for large enough $n$,
$[p,c_{n}]$ lies in the open set $R\cdot V_{x}$ containing $[p,x]$. Then,
there exists an open neighborhood $U_{p}$ of $x$ in $V_{x}$ such that
assertion (ii) holds. We shrink $U_{p}$ so that $G_{x}U_{p}\subseteq U_{p}$.

Now, suppose there is a sequence $g_{n}\in G$ and $q_{n}\in U_{p}$ with
$q_{n}\to x$ such that $\|p-g_{n}q_{n}\|\leq d(p,Uq_{n})$. Then, after taking
subsequences, we may assume $g_{n}\to g\in G$ and by taking limits, we obtain
$\|p-gx\|\leq d(p,Ux)=\|p-x\|=d(p,Gx)$. By minimizer uniqueness, we get $gx=x$
and so $g\in G_{x}$. It follows that $g_{n}$ is eventually in $U\cdot G_{x}$.
Moreover, $[p,g_{n}q_{n}]$ is eventually in the open set $U\cdot V_{x}$
containing $[p,x]$. As such, we may shrink $U_{p}$ so that assertion (iii)
holds. Lastly, assertion (i) follows by shrinking $U_{p}$ again.
$\blacksquare$

Denote by $V_{x}^{\gamma}$ the space $V_{x}$ equipped with the metric
$\gamma$. For $q\in V_{x}$, we denote the point $\pi(q)$ in $V_{x}^{\gamma}$
by its fiber $Uq$.

Claim 5: Suppose $p\in V_{x}$ and $q\in U_{p}$. Then, minimal geodesics from
$Up$ to $Uq$ in $V_{x}^{\gamma}$ exist and are precisely the $\pi$-images of
straighline distance minimizers from $p$ to $Uq$ in $U\cdot V_{x}$.

Proof: Let $c_{\gamma}:[0,L(c)]\to V_{x}^{\gamma}$ be a unit-speed geodesic
joining $Up$ to $Uq$ where $L(c)$ denotes the length of $c$. Suppose that
$L(c)\leq d(p,Uq)$. By local lifting of geodesics (Proposition 2.109 in [16]),
there exists $0=t_{0}<t_{1}<\cdots<t_{n}=L(c)$ and for
$i\in\\{0,\dots,n-1\\}$, there exists $p_{i}\in\pi^{-1}(c_{\gamma}(t_{i}))\cap
V_{x}$ and $q_{i+1}\in\pi^{-1}(c_{\gamma}(t_{i+1}))$ such that
$[p_{i},q_{i+1}]$ are straighlines connecting $p_{i}$ to $Up_{i+1}$ entirely
lieing in $U\cdot V_{x}$ and horizontally lifting
$c_{\gamma}|_{[t_{i},t_{i+1}]}$ to $U\cdot V_{x}$ starting from $p_{i}$. We
take $p_{0}:=p$ and $p_{n}:=q$. In particular, observe that
$\sum_{i=0}^{n-1}\|q_{i+1}-p_{i}\|=L(c)$.

Take $u_{i}\in U$ such that $q_{i}=u_{i}p_{i}$. Then, by equivariance of
lifting, $q_{i+1}-p_{i}$ has the same direction as
$u_{i+1}(q_{i+2}-p_{i+1})=u_{i+1}q_{i+2}-q_{i+1}$ for $i\in\\{0,\dots,n-2\\}$.
It follows that there exists $\\{g_{i}\\}_{i=0}^{n}\in G^{n+1}$ with
$g_{0}=e$, such that the straighline segments $[g_{i}p_{i},g_{i}q_{i+1}]$ are
parallel with non-overlapping interiors and their union is $[p,g_{n}q]$ and
has the same length as $L(c)$. It follows that $L(c)=\|p-g_{n}q\|\leq d(p,Uq)$
and hence by the previous claim there exists $h\in G_{x}$ such that $g_{n}q\in
Uhq$ and also $[p,g_{n}q]\subseteq U\cdot V_{x}$. It follows that the very
initial lift of $c_{\gamma}$ starting from $p$ does not terminate and $Uhq=Uq$
i.e. $hq=q$ and $L(c)=d(p,Uq)$ and $c_{\gamma}$ has a straighline minimizing
lift as desired.

$\blacksquare$

Claim 6: The hypotheses of Lemma 108 hold for $V_{x}^{\gamma}$ with respect to
$Ux$ and $Uz$ for $z\in V_{x}$.

Proof: With $z\in V_{x}$, there exists $\varepsilon>0$ such that
$q=z+(z-x)\varepsilon\in V_{x}$. Then by 43(e), the straighline $[q,x]$
satisfies the first hypothesis. The second hypothesis follows by considering a
neighborhood $U_{z}$ as in the claim above. The third hypothesis follows by
taking limits of straight line minimizing lifts to the unique straightline
minimizing lift. $\blacksquare$

We now finish the proof. First, we note that for all $p_{1},p_{2}\in U$ and
since $U^{-1}=U$, we have

$d(p_{1},Up_{2})=d(U^{-1}p_{1},p_{2})=d(p_{2},Up_{1}).$ (16)

Let $z\in V_{x}$ so that $\|z-x\|=d(z,Ux)$. and let $U_{z}$ be as above. Then,
Lemma 108 entails that the length of the straightline $[z,x]$ remains equal to
$d(z,Up)$ for $p$ shortly beyond $x$. By definition of $U_{z}$, we have that
$[z,p]\subseteq U\cdot V_{x}$ and by (16), we obtain that
$\|p-z\|=d(z,Up)=d(p,Uz)$. Then, by 35(b), there exists a neighborhood $V$
around $x$ such that for $q\in V$, there is a unique straighline minimizer
from $q$ to $Uz$. It follows that $x\in V_{z}^{loc}$ as desired.

(b)$\Rightarrow$(a). Since $Q_{z}^{1}$ and $P(G)$ are $G$-invariant and dense,
pick $z\in V_{x}^{1}\cap P(G)$. Then, by Lemma 43, we get $Gz\cap
V_{x}^{1}=G_{x}z$. By assumption, we otain $x\in V_{hz}^{loc}$ for every $h\in
G_{x}$. By definition of $V_{hz}^{loc}$ and by Lemma 43, we obtain that the
set $B:=\arg\min_{q\in[z]}\|q-x\|\subseteq V_{x}^{1}$ is discrete and
$G_{x}$-transitive. By Proposition 109, we also have $G_{q}\leq G_{x}$ for
every $q\in B$. By transitivity and the orbit-stabilizer theorem, we deduce
that $G_{x}/G_{q}$ is discrete and $x\in R(G)$ as desired.

##  Appendix B Local Behavior at Regular Points

###  B.1 Technical Lemma

######  Definition 110.

Fix a compact group $G\leq\operatorname{O}(d)$. For a sort index
$i\in\\{1,\dots,|\pi_{0}(G)|\\}$ and $x,z\in\mathbb{R}^{d}$, we denote the set
of realizing orbit elements of $z$ with respect to $x$ by

$S^{i}(z,x):=\\{w\in[z]:\Psi_{i}([z],[x])=\langle w,x\rangle\\}.$

######  Remark 111.

It holds that $S^{i}(z,gx)=g\cdot S^{i}(z,x)$ and $S^{i}(z,x)=S^{i}(gz,x)$ for
all $x,z\in\mathbb{R}^{d}$ and $g\in G$. In particular, $S^{i}(z,x)$ is
$G_{x}$-invariant.

######  Lemma 112.

Suppose $G\leq\operatorname{O}(d)$ is compact. Suppose $x\in\mathbb{R}^{d}$
and $i\in\\{1,\dots,|\pi_{0}(G)|\\}$. Let $x_{n},y_{n}\to x$ with
$[x_{n}]\neq[y_{n}]$ and $x_{n},y_{n}\in N_{x}$. Suppose that the following
hypothesis holds:

  1. (H1)

For every $z\in V_{x}^{i}$, it holds that $x\in V_{z}^{loc}$.

  2. (H2)

There exists a set $S\subseteq V_{x}^{i}$ spanning $N_{x}$, such that for
every $q\in S$, the following limit exists

$L([q]):=\lim_{n\to\infty}S^{i}(q,x_{n})\cap\lim_{n\to\infty}S^{i}(q,y_{n})\neq\varnothing,$

and $q\in L([q])$.

Then, the following statements hold:

  1. (a)

After taking subsequences, the limits
$u_{0}:=\lim_{n\to\infty}\frac{x_{n}-y_{n}}{\|x_{n}-y_{n}\|}\in\mathbb{S}^{d-1}$
and
$c_{0}:=\liminf_{n\to\infty}\frac{\|x_{n}-y_{n}\|}{d([x_{n}],[y_{n}])}\in[1,\infty)$
exist.

  2. (b)

Let $u:=c_{0}u_{0}$. If $z\in Q_{x}^{i}$ and if after taking subsequences
$L([z])\neq\varnothing$ is nonempty, then the following convergence holds
after taking further subsequences

$w\in
L([z])\implies\frac{\Psi_{i}([x_{n}],[z])-\Psi_{i}([y_{n}],[z])}{d([x_{n}],[y_{n}])}\to\langle
w,u\rangle.$ (17)

######  Remark 113.

Note that $u$ only depends on $(x_{n})_{n\in\mathbb{N}}$ and
$(y_{n})_{n\in\mathbb{N}}$.

###### Proof.

Take arbitrary $i\in\\{1,\dots,|\pi_{0}(G)|\\}$ and $z\in Q_{x}^{i}$, and
assume that after taking subsequences $w\in L([z])\neq\varnothing$. By
$G$-invariance of (17) with respect to $z$, we may translate $z$ so that
$w=z\in L([z])$. Let $z^{x}_{n},z^{y}_{n}\to z$ where $z^{x}_{n}\in
S^{i}(z,x_{n})$ and $z^{y}_{n}\in S^{i}(z,y_{n})$.

By (H1) and since $z\in L([z])\subseteq S^{i}(z,x)\subseteq V_{x}^{i}$, we
have $x\in V_{z}^{loc}$. Then, by definition of $V_{z}^{loc}$, there exists
open neighborhoods $V$ of $x$ and $U$ of $z$ such that

$\forall q\in V,\ \ \Big{|}\arg\sup_{p\in[z]_{0}\cap U}\langle
p,q\rangle\Big{|}=1,$

so we may take subsequences so that $x_{n},y_{n}\in V$ and
$z^{x}_{n},z^{y}_{n}\in U$. Moreover, we define $v_{z}\colon V\to[z]$ in such
a way that $v_{z}(q)$ is the unique element in $\arg\sup_{p\in[z]_{0}\cap
U}\langle p,q\rangle$. In particular, since $z\in V_{x}^{i}$, we have
$v_{z}(x)=z$.

Note that $z^{x}_{n}=v_{z}(x_{n})$ and $z^{y}_{n}=v_{z}(y_{n})$ . Also,
$\|x_{n}-y_{n}\|\neq 0$ since $[x_{n}]\neq[y_{n}]$. We get that

$\frac{\Psi_{i}([x_{n}],[z])-\Psi_{i}([y_{n}],[z])}{d([x_{n}],[y_{n}])}=\frac{\|x_{n}-y_{n}\|}{d([x_{n}],[y_{n}])}\cdot\frac{\langle
x_{n},v_{z}(x_{n})\rangle-\langle
y_{n},v_{z}(y_{n})\rangle}{\|x_{n}-y_{n}\|}.$ (18)

Next, we have

$\frac{\langle x_{n},v_{z}(x_{n})\rangle-\langle
y_{n},v_{z}(y_{n})\rangle}{\|x_{n}-y_{n}\|}=\left\langle\frac{x_{n}-y_{n}}{\|x_{n}-y_{n}\|},v_{z}(y_{n})\right\rangle-\frac{\|v_{z}(x_{n})-v_{z}(y_{n})\|}{\|x_{n}-y_{n}\|}\left\langle
x_{n},\frac{v_{z}(x_{n})-v_{z}(y_{n})}{\|v_{z}(x_{n})-v_{z}(y_{n})\|}\right\rangle,$

where we take $\frac{0}{\|0\|}:=0$. By 35(b), the map $v_{z}(\cdot)$ is smooth
over $V$ and hence locally upper Lipschitz there. Then, since $x\in V$, there
exists $C>0$ such that after taking subsequences, we have

$\frac{\|v_{z}(x_{n})-v_{z}(y_{n})\|}{\|x_{n}-y_{n}\|}<C.$

Moreover, since $[z]$ is a smooth submanifold of $\mathbb{R}^{d}$ and
$v_{z}(x_{n}),v_{z}(y_{n})\to v_{z}(x)$, we get
$\frac{v_{z}(x_{n})-v_{z}(y_{n})}{\|v_{z}(x_{n})-v_{z}(y_{n})\|}\to t\in
T_{v_{z}(x)}$. Note further that $x\in N_{v_{z}(x)}$ since $v_{z}(x)$ is a
distance minimizer of $x$ to a neighborhood of $v_{z}(x)$ in $[z]_{0}$
(35(a)). It follows that

$\left|\frac{\|v_{z}(x_{n})-v_{z}(y_{n})\|}{\|x_{n}-y_{n}\|}\left\langle
x_{n},\frac{v_{z}(x_{n})-v_{z}(y_{n})}{\|v_{z}(x_{n})-v_{z}(y_{n})\|}\right\rangle\right|\leq
C\left|\left\langle
x_{n},\frac{v_{z}(x_{n})-v_{z}(y_{n})}{\|v_{z}(x_{n})-v_{z}(y_{n})\|}\right\rangle\right|\to
C\left|\langle x,t\rangle\right|=0.$

By taking subsequences, we have $\frac{x_{n}-y_{n}}{\|x_{n}-y_{n}\|}\to u_{0}$
for some $u_{0}\in\mathbb{R}^{d}$ with $\|u_{0}\|=1$. It follows that

$\frac{\langle x_{n},v_{z}(x_{n})\rangle-\langle
y_{n},v_{z}(y_{n})\rangle}{\|x_{n}-y_{n}\|}\to\langle
v_{z}(x),u_{0}\rangle=\langle z,u_{0}\rangle.$ (19)

Next, we claim that
$1\leq\liminf_{n\to\infty}\frac{\|x_{n}-y_{n}\|}{d([x_{n}],[y_{n}])}$ is upper
bounded. To this end, observe that so far, $z\in Q_{x}^{i}$ with
$L([z])\neq\varnothing$ has been arbitrary and $u_{0}\in N_{x}$ does not
depend on $z$. In particular, $z$ may as well have been chosen to be in the
set $S$ given by (H2). For $z^{\prime}\in S$, we have
$v_{z^{\prime}}(x)=z^{\prime}$ so that $\langle
v_{z^{\prime}}(x),u_{0}\rangle=\langle z^{\prime},u_{0}\rangle$. Since $S$ is
spanning, we can pick $z^{\prime}$ so that $\langle
v_{z^{\prime}}(x),u_{0}\rangle\neq 0$. Now, by Theorem 31, we have

$\left|\frac{\Psi_{i}([x_{n}],[z^{\prime}])-\Psi_{i}([y_{n}],[z^{\prime}])}{d([x_{n}],[y_{n}])}\right|\leq\|z^{\prime}\|<\infty.$

On the other hand, by (19) and since $\langle
v_{z^{\prime}}(x),u_{0}\rangle\neq 0$ and $z^{\prime}\in L([z^{\prime}])$,
there exists $d>0$ such that after taking a subsequence,

$\left|\frac{\langle x_{n},v_{z^{\prime}}(x_{n})\rangle-\langle
y_{n},v_{z^{\prime}}(y_{n})\rangle}{\|x_{n}-y_{n}\|}\right|>d.$

Hence, by (18), we get
$1\leq\frac{\|x_{n}-y_{n}\|}{d([x_{n}],[y_{n}])}\leq\frac{D}{d}<\infty$ for
all $n$ which proves the claim. We take a further subsequence so that
$\frac{\|x_{n}-y_{n}\|}{d([x_{n}],[y_{n}])}\to
c_{0}:=\liminf_{n\to\infty}\frac{\|x_{n}-y_{n}\|}{d([x_{n}],[y_{n}])}\in[1,\infty)$.
The result now follows by taking $u=c_{0}\cdot u_{0}$ and combining equations
(18) and (19).

∎

###  B.2 Proof of Lemma 61

Fix arbitrary $x\in P(G)$, $i\in\\{1,\dots,|\pi_{0}(G)|\\}$ and
$x_{n},y_{n}\to x$ with $[x_{n}]\neq[y_{n}]$ and $x_{n},y_{n}\in N_{x}$. By
Lemma 112, we only need to show that $v_{z}^{i}(x)\in L([z])$ for all $z\in
Q_{x}^{i}$ and to verify (H1) and (H2).

With $z\in V_{x}^{i}$, then by Lemma 45, we have $x\in V_{z}^{i}\subseteq
Q_{z}^{i}$. Then, (H1) follows from the characterization 44(c); note that in
the right-hand side of the characterization, $[q]_{0}=[z]_{0}$ by Lemma 38 and
$x\in V_{[z]_{0}}^{i}$. Next, $v_{z}^{i}(x)=z$ and $v_{z}^{i}$ is smooth over
$Q_{z}^{i}$. Since $x\in Q_{z}^{i}$, we obtain that
$S^{i}(z,x_{n})=\\{v_{z}^{i}(x_{n})\\}$ and similarly for $y_{n}$, and by
taking limits in $n$, we get $L([z])=\\{v_{z}^{i}(x)\\}$. Since $z\in
V_{x}^{i}$ was arbitrary and $V_{x}^{i}$ is open in $N_{x}$ (Lemma 43), (H2)
follows by taking any spanning subset of $V_{x}^{i}$ and noting that
$L([z])=\\{z\\}$ for $z\in V_{x}^{i}$.

##  Appendix C Finite-Index Stabilizer Groups - Proof of Theorem 71

We will need two propositions before proceeding with the proof. The following
proposition follows from Proposition 2.29 and the discussion preceeding
Example 2C.4 in [21]:

######  Proposition 114.

$\mathbb{Z}/2\mathbb{Z}$ is the only nontrivial group that can act freely on
classical projective spaces $\mathbb{C}P^{l-1}$ and $\mathbb{H}P^{l-1}$. In
the complex projective case, it is necessary that $l-1$ is odd and in the
quaternionic projective case, it is necessary that $l=2$.

The following proposition follows from Lemma 5 in [33] and Lemma 2.1 and the
discussion following Remark 3.1 in [18]:

######  Proposition 115.

Suppose $G\leq\operatorname{O}(d)$ is compact. Let $\mathbb{S}$ denote the
sphere in $\mathbb{R}^{d}$. If $G$ is not transitive over $\mathbb{S}$ and if
$\mathbb{S}/G$ is a manifold, then the following hold:

  1. (a)

$G$ has trivial principal isotropy group.

  2. (b)

The identity component $G_{0}$ of $G$ satisfies $G_{0}\cong
S^{1}\subseteq\mathbb{C}$ or $G_{0}\cong S^{3}\subseteq\mathbb{H}$ with
actions given by left scalar multiplication over
$\mathbb{R}^{d}\cong_{\mathbb{R}}\mathbb{C}^{n}$ or
$\mathbb{R}^{d}\cong_{\mathbb{R}}\mathbb{H}^{n}$ respectively.

###### Proof of Theorem 71.

Step 1. Invariant Subspace of Finite Orbits. Since $G$ has finitely many
connected components, there are finitely many finite-index stabilizers. Let
$W$ be the union of their $1$-eigenspaces. Then, $x\in W$ if and only if $x$
has finite orbit. Observe that $W$ is a linear space: if $x,y\in W$ and
$a,b\in\mathbb{R}$, then $[ax+by]\subseteq a[x]+b[y]$ has finite size meaning
$ax+by\in W$. Moreover, $W$ is $G$-invariant: $x$ has finite orbit implies
$[x]\subseteq W$. If $W=\mathbb{R}^{d}$, then $G$ is a finite group:
otherwise, principal stabilizers have finite index and so $G_{0}$ belongs to
every principal stabilizer hence every stabilizer by normality, meaning
$G_{0}=\\{\operatorname{Id}_{\mathbb{R}^{d}}\\}$. As such, case (a) follows
when $W=\mathbb{R}^{d}$.

Step 2. Principal Invariant Subspace. To this end, assume
$W\neq\mathbb{R}^{d}$. It follows that $\mathbb{R}^{d}$ has an orthogonal
$G$-invariant decomposition $W\oplus_{G}V$ where every nonzero $y\in V$ is
principal.

Let $G_{V}\leq\operatorname{O}(V)$ be the restriction of $G$ to $V$. Denote
the restriction by $\rho\colon G\to G_{V}$. The kernel of $\rho$ is trivial:
if $g\in G$ fixes $V$, then $g$ belongs to every prinicpal stabilizer and
hence $g$ fixes $\mathbb{R}^{d}$ i.e. $g=\operatorname{Id}_{\mathbb{R}^{d}}$.
As such, $\rho$ is a group isomorphism.

Morover, all stabilizers in $G_{V}$ of nonzero elements in $V$ form the
collection $\\{\rho(G_{p})\\}_{G_{p}\in(G_{P})}$ and this forms a single
conjugacy class in $G_{V}$. It follows that $G_{V}$ is principal over $V$ and
hence $\mathbb{S}_{V}/G_{V}$ is a manifold, where $\mathbb{S}_{V}$ is the
sphere in $V$. Let $G_{V}^{0}$ denote the identity component of $G_{V}$.

Step 3. Transitive Case. If the action of $G_{V}$ is transitive over
$\mathbb{S}_{V}$, then so is the action of $G_{V}^{0}$ and so every principal
isotropy group of $G_{V}$ (hence $G$) intersects every connected component of
$G_{V}$ (hence $G$). It follows that every orbit of $G$ has exactly one
connected component. Then, by orbit finiteness, $G$ acts trivially on $W$ and
case (b) follows.

Step 4. Non-Transitive Case.  If the action of $G_{V}$ is not transitive, then
since $\mathbb{S}_{V}/G_{V}$ is a manifold and by Proposition 115, it follows
that $G_{V}$ has trivial principal isotropy group and the connected component
$G_{V}^{0}$ of $G_{V}$ satisfies $G_{V}^{0}\cong S^{1}\subseteq\mathbb{C}$ or
$G_{V}^{0}\cong S^{3}\subseteq\mathbb{H}$ with actions given by left scalar
multiplication over $\mathbb{R}^{d}\cong_{\mathbb{R}}\mathbb{C}^{n}$ or
$\mathbb{R}^{d}\cong_{\mathbb{R}}\mathbb{H}^{n}$ respectively. The quotients
are classical projective spaces denoted by $\mathbb{C}P^{n-1}$ and
$\mathbb{H}P^{n-1}$ respectively.

When $G_{V}$ (hence $G$) is connected, cases (c) and (d) follow since in these
cases, the only finite-index stabilizer is the whole group $G$.

Now, suppose $g\in G_{V}$ is such that $g\notin G_{V}^{0}$. Then, by
principality, $\overline{f}:=g|_{\mathbb{S}_{V}}$ is free. By normality of
$G_{V}^{0}$ and for every $g_{0}\in G_{V}^{0}$, there exists
$g^{\prime}_{0}\in G_{V}^{0}$ such that
$\overline{f}g_{0}=g_{0}^{\prime}\overline{f}$. Since $\overline{f}\notin
G_{V}^{0}$, we obtain that $\overline{f}$ maps a $G_{V}^{0}$ orbit to a
distinct $G_{V}^{0}$ orbit. As such, $\overline{f}$ induces a free map $f$ on
$\mathbb{S}_{V}/G_{V}^{0}$. Let $p\geq 2$ be the minimal integer such that
$\overline{f}^{p}\in G_{V}^{0}$. Then,
$\mathbb{Z}/\mathbb{Z}p=\\{f^{i}\\}_{i=1}^{p}$ acts freely on
$\mathbb{S}_{V}/G_{V}^{0}$. By Proposition 114, it necessarily holds that
$p=2$, that is, $f$ is a free involution.

Step 5. Complex Case. First, we consider the case $G_{V}^{0}\cong S^{1}$.
Proposition 114 entails that $V\cong_{\mathbb{R}}\mathbb{C}^{2n}$. Since $f$
is a free involution, $V\cong_{\mathbb{R}}(\mathbb{C}\oplus\mathbb{C})^{n}$
where over each copy $\overline{f}(\mathbb{C}\oplus 0)=0\oplus\mathbb{C}$ and
vice versa.

To this end, we analyze the case $n=1$. There exists $w_{1},w_{2}\in S^{1}$
and $\varphi_{1},\varphi_{2}$ where $\varphi_{j}$ is either identity or
conjugation such that
$\overline{f}(z_{1},z_{2})=(w_{2}\varphi_{2}z_{2},w_{1}\varphi_{1}z_{1})$.

Also, if $\overline{f}^{2}=\operatorname{Id}_{\mathbb{C}^{2}}$, then
$\overline{f}$ is orthogonal and symmetric. By freeness of $\overline{f}$, the
spectrum is given by $\\{-1,-1\\}$. As such, we have
$\overline{f}=-\operatorname{Id}_{\mathbb{C}^{2}}$ so $f$ fixes
$\mathbb{C}P^{1}$, which contradicts the freeness of $f$. As such,
$\overline{f}^{2}$ is free.

Now, since $\varphi_{j}$ is a multiplicative homomorphism, we have

$\overline{f}^{2}(z_{1},z_{2})=(w_{2}\varphi_{2}w_{1}\varphi_{2}\varphi_{1}z_{1},w_{1}\varphi_{1}w_{2}\varphi_{1}\varphi_{2}z_{2}).$

We need $\varphi_{2}\varphi_{1}=1$ since otherwise, we solve
$z_{1}=w_{2}\varphi_{2}w_{1}\overline{z_{1}}$ for $z_{1}\in S^{1}$ to get a
fixed point of $\overline{f}^{2}$.

If $\varphi_{1}=\varphi_{2}=1$, then
$\overline{f}(z_{1},z_{2})=(w_{2}z_{2},w_{1}z_{1})$ induces $f$ with a fixed
orbit: pick $g\in S^{1}$ such that $g=w_{2}\overline{g}w_{1}$ and pick
$z_{1},z_{2}\in S^{1}$ in such a way that $z_{2}=\overline{g}w_{1}z_{1}$;
then, $gz_{1}=w_{2}\overline{g}w_{1}z_{1}=w_{2}z_{2}$ so
$\overline{f}(z_{1},z_{2})=g\cdot(z_{1},z_{2})$. This contradicts the freeness
of $f$.

Hence,
$\overline{f}(z_{1},z_{2})=(w_{2}\overline{z}_{2},w_{1}\overline{z}_{1})$ so
that
$\overline{f}^{2}(z_{1},z_{2})=(w_{2}\overline{w}_{1}z_{1},w_{1}\overline{w}_{2}z_{2})$.
Now, $f^{2}$ fixes orbits so $\overline{f}^{2}$ belongs to $G_{V}^{0}$ by
freeness and hence $w_{2}\overline{w}_{1}=w_{1}\overline{w}_{2}$ is real but
not equal to $1$ since
$\overline{f}^{2}\neq\operatorname{Id}_{\mathbb{C}^{2}}$. Hence,
$w_{2}\overline{w}_{1}=-1$ and $w_{2}=-w_{1}$ so by a translation
$(z_{1},z_{2})\mapsto w_{2}\cdot(z_{1},z_{2})$, we obtain
$\overline{f}(z_{1},z_{2})=(-\overline{z}_{2},\overline{z}_{1})$ as desired.

Step 6. Quaternionic Case. Next, we tackle the case $\mathbb{H}P^{1}$. Again
by similar arguments as before, we have $\overline{f}(\mathbb{H}\oplus
0)=0\oplus\mathbb{H}$ and vice versa, and we have that $\overline{f}^{2}$ is
free. Moreover, $\overline{f}$ maps every nonzero $S^{3}$-orbit to a distinct
$S^{3}$-orbit while $\overline{f}^{2}$ maps every $S^{3}$-orbit into itself.

By Van Elfrinkhof’s formula [32], there exists unit quaternions
$w_{1},w_{2},p_{1},p_{2}\in S^{3}$ and $\varphi_{1},\varphi_{2}$ where
$\varphi_{j}$ is either identity or conjugation such that

$\overline{f}(q_{1},q_{2})=(p_{1}\varphi_{1}q_{2}p_{2},w_{1}\varphi_{2}q_{1}w_{2}).$

If both $\varphi_{1}$ and $\varphi_{2}$ are identity, then $f$ has a fixed
orbit: pick any $q_{1}\in S^{3}$ and pick $g\in S^{3}$ such that
$gq_{1}=p_{1}\overline{g}w_{1}q_{1}w_{2}p_{2}$ (this is equivalent to
$(\overline{p}_{1}g)^{2}=\overline{p}_{1}w_{1}q_{1}w_{2}p_{2}\overline{q}_{1}$)
and take $q_{2}=\overline{g}w_{1}q_{1}w_{2}$ so that $gq_{2}=w_{1}q_{1}w_{2}$.
Then, $gq_{1}=p_{1}\overline{g}w_{1}q_{1}w_{2}p_{2}=p_{1}q_{2}p_{2}$ so
$\overline{f}(q_{1},q_{2})=g\cdot(q_{1},q_{2})$. This contradicts the freeness
of $f$.

If say $\varphi_{2}=1$ while $\varphi_{1}$ is conugation, then
$\overline{f}(q_{1},q_{2})=(p_{1}\overline{q}_{2}p_{2},w_{1}q_{1}w_{2})$ and
so
$\overline{f}^{2}(q_{1},q_{2})=(p_{1}\overline{w}_{2}\overline{q}_{1}\overline{w}_{1}p_{2},w_{1}p_{1}\overline{q}_{2}p_{2}w_{2})$.
Then, we may find $q_{1},q_{2}\in S^{3}$ such that
$q_{1}=p_{1}\overline{w}_{2}\overline{q}_{1}\overline{w}_{1}p_{2}$
(equivalently
$(w_{2}\overline{p}_{1}q_{1})^{2}=w_{2}\overline{p}_{1}\overline{w}_{1}p_{2}$)
and $q_{2}=w_{1}p_{1}\overline{q}_{2}p_{2}w_{2}$. This contradicts the
freeness of $\overline{f}^{2}$.

As such,

$\overline{f}(q_{1},q_{2})=(p_{1}\overline{q}_{2}p_{2},w_{1}\overline{q}_{1}w_{2}),$

and

$\overline{f}^{2}(q_{1},q_{2})=(p_{1}\overline{w}_{2}q_{1}\overline{w}_{1}p_{2},w_{1}\overline{p}_{2}q_{2}\overline{p}_{1}w_{2}).$

Since $f$ is an involution of $\mathbb{H}P^{1}$, then for every
$q_{1},q_{2}\in S^{3}$, there exists $g\in S^{3}$, such that
$\overline{f}^{2}(q_{1},q_{2})=g\cdot(q_{1},q_{2})$. It follows that

$p_{1}\overline{w}_{2}q_{1}\overline{w}_{1}p_{2}\overline{q}_{1}=w_{1}\overline{p}_{2}q_{2}\overline{p}_{1}w_{2}\overline{q}_{2}$
(20)

for all $q_{1},q_{2}\in S^{3}$. In particular,
$p_{1}\overline{w}_{2}q_{1}\overline{w}_{1}p_{2}\overline{q}_{1}$ is constant
in $q_{1}\in S^{3}$. By setting $q_{1}\in\\{1,w_{1}\\}$, we obtain
$p_{1}\overline{w}_{2}\overline{w}_{1}p_{2}=p_{1}\overline{w}_{2}p_{2}\overline{w}_{1}$
meaning $\overline{w}_{1}p_{2}=p_{2}\overline{w}_{1}$. In commutator terms,
$[\overline{w}_{1},p_{2}]=0$. By conjugation and symmetry, we have

$[\overline{w}_{1},p_{2}]=[\overline{w}_{2},p_{1}]=[w_{1},\overline{p}_{2}]=[w_{2},\overline{p}_{1}]=0.$
(21)

By setting $q_{1}\in\\{w_{2},w_{2}\overline{p}_{1}\\}$, we obtain
$p_{1}\overline{w}_{1}p_{2}\overline{w}_{2}=\overline{w}_{1}p_{2}p_{1}\overline{w}_{2}$
meaning
$p_{1}\overline{w}_{1}p_{2}=\overline{w}_{1}p_{2}p_{1}=p_{2}\overline{w}_{1}p_{1}$.
Similarly, by setting $q_{2}\in\\{w_{2},p_{2}\\}$, we obtain
$w_{1}\overline{p}_{2}w_{2}\overline{p}_{1}=w_{1}\overline{p}_{1}w_{2}\overline{p}_{2}$
meaning $p_{1}\overline{w}_{2}p_{2}=p_{2}\overline{w}_{2}p_{1}$. By symmetry,
we obtain

$\begin{cases}p_{1}\overline{w}_{2}p_{2}=p_{2}\overline{w}_{2}p_{1}\\\
p_{1}\overline{w}_{1}p_{2}=p_{2}\overline{w}_{1}p_{1}\\\
w_{1}\overline{p}_{2}w_{2}=w_{2}\overline{p}_{2}w_{1}\\\
w_{1}\overline{p}_{1}w_{2}=w_{2}\overline{p}_{1}w_{1}\end{cases}$ (22)

By setting $q_{1}=1$ and $q_{2}=p_{2}\overline{w}_{1}$, we also get

$p_{1}\overline{w}_{2}\overline{w}_{1}p_{2}=\overline{p}_{1}w_{2}w_{1}\overline{p}_{2}.$
(23)

We use equations (21), (22) and (23) to get

$\displaystyle\overline{p}_{1}w_{2}w_{1}\overline{p}_{2}$
$\displaystyle=p_{1}\overline{w}_{2}\overline{w}_{1}p_{2}$
$\displaystyle=\overline{w}_{2}p_{1}\overline{w}_{1}p_{2}$
$\displaystyle=\overline{w}_{2}p_{2}\overline{w}_{1}p_{1}$
$\displaystyle=\overline{w}_{1}p_{2}\overline{w}_{2}p_{1}$
$\displaystyle=p_{2}\overline{w}_{1}\overline{w}_{2}p_{1}.$

As such, $p_{2}\overline{w}_{1}\overline{w}_{2}p_{1}\in\mathbb{R}$ so
$p_{2}\overline{w}_{1}\overline{w}_{2}p_{1}=\pm 1$. Suppose
$p_{2}\overline{w}_{1}\overline{w}_{2}p_{1}=1$. Then, $w_{2}w_{1}=p_{1}p_{2}$
and so

$\displaystyle\overline{f}^{2}(0,1)$
$\displaystyle=(0,w_{1}\overline{p}_{2}\overline{p}_{1}w_{2})=(0,w_{1}\overline{w_{2}w_{1}}w_{2})=(0,1).$

By freeness of $\overline{f}^{2}$ and by symmetry, we obtain
$p_{2}\overline{w}_{1}\overline{w}_{2}p_{1}=-1$ and
$w_{2}\overline{p}_{1}\overline{p}_{2}w_{1}=-1$. Recall by (20) that
$g(q_{2}):=w_{1}\overline{p}_{2}q_{2}\overline{p}_{1}w_{2}\overline{q}_{2}$ is
constant for all $q_{2}\in S^{3}$. Since
$g(p_{2}\overline{w}_{1})=\overline{p}_{1}w_{2}w_{1}\overline{p}_{2}=\overline{-1}=-1$.
It follows that $-w_{2}\overline{q}_{2}w_{1}=p_{1}\overline{q}_{2}p_{2}$ for
all $q_{2}\in S^{3}$.

As such,

$\overline{f}(q_{1},q_{2})=(-w_{2}\overline{q}_{2}w_{1},w_{1}\overline{q}_{1}w_{2}).$

Now, take $q_{1}=jw_{1}$, $g=i$ and
$q_{2}=\overline{g}w_{1}\overline{q}_{1}w_{2}$. Then,
$gq_{2}=w_{1}\overline{q}_{1}w_{2}$. Hence, we have the following chain of
equivalences:

$\displaystyle\overline{f}(q_{1},q_{2})=g\cdot(q_{1},q_{2})$
$\displaystyle\iff gq_{1}=-w_{2}\overline{q}_{2}w_{1}$ $\displaystyle\iff
gq_{1}=-w_{2}\overline{w}_{2}q_{1}\overline{w}_{1}gw_{1}$ $\displaystyle\iff
g(q_{1}\overline{w}_{1})=-(q_{1}\overline{w}_{1})g$ $\displaystyle\iff
ij=-ji.$

The last statement holds true and this contradicts the freeness of $f$. The
result follows.

∎

##  Appendix D Non-Differentiability of Bilipschitz Invariants

In this section, we extend Theorem 21 in [9] to the case of compact group
acting on finite-dimensional real Hilbert spaces. We show that for compact
groups, diffentiability of an invariant map at a nonprincipal point forces any
local lower Lipschitz bound there to be zero. In what follows, suppose that
$G\leq\operatorname{O}(d)$ is compact.

######  Proposition 116 (Lemma 20 in [9]).

Suppose $f\colon\mathbb{R}^{d}\to\mathbb{R}^{n}$ is $G$-invariant and
differentiable at $x\in\mathbb{R}^{d}$. Then, for each $g\in G$, it holds that
$f$ is differentiable at $gx$ with $Df(gx)=Df(x)\circ g^{-1}$.

######  Theorem 117.

Suppose $x\in P(G)^{c}$, and consider any $G$-invariant map
$f\colon\mathbb{R}^{d}\to\mathbb{R}^{n}$ that is differentiable at $x$. Then,
the following hold

  * (a)

There exists a unit vector $v\in N_{x}\cap x^{\perp}$ such that $Df(x)v=0$.

  * (b)

The induced map $f^{\downarrow}\colon\mathbb{R}^{d}/G\to\mathbb{R}^{d}$ is not
lower Lipschitz.

  * (c)

If $x\in\mathbb{S}^{d-1}$, the restriction of the induced map
$f^{\downarrow}|_{\mathbb{S}^{d-1}}$ is not lower Lipschitz.

###### Proof.

For (a), we begin by taking $G_{p}\leq G_{x}$ (Proposition 109) to be a
principal stabilizer of some $p\in P(G)\cap V_{x}^{1}$. Pick $g\in
G_{x}\setminus G_{p}$. By the same argument as in Theorem 21(a) in [9], it
suffices to show $\ker(g-\operatorname{id})^{\perp}\cap N_{x}$ is nonzero. By
Proposition 21, we have that $T_{x}$ and $N_{x}$ are invariant under the
action of $g$, so $g$ splits as a block matrix and its $1$-eigenspace
$E:=\ker(g-\operatorname{id})=(E\cap T_{x})\oplus(E\cap N_{x})$ is an
orthogonal direct sum. It follows that $E^{\perp}\cap N_{x}=\\{0\\}$ if and
only if $N_{x}\subseteq E$ which is false since $gp\neq p$.

For (b) and (c), the same arguments as in the proof of Theorem 21 in [9] hold.
The only modification is that due to 35(b) applied to $z=x$, we have $x\in
V_{x}^{1}$, and due to (a), it remains true that for small enough
$t\in\mathbb{R}$, we have $x+tv\in V_{x}^{1}$ so that $d([x+tv],[x])=|t|$. ∎