# Almost equivariant maps for td-groups

Bartels, A  WWU Münster
Mathematisches Institut
Einsteinstr. 62, D-48149 Münster, Germany<EMAIL_ADDRESS>http://www.math.uni-muenster.de/u/bartelsa and  Lück, W  Mathematisches
Institut der Universität Bonn
Endenicher Allee 60
53115 Bonn, Germany<EMAIL_ADDRESS>http://www.him.uni-
bonn.de/lueck

(Date: April 2024)

###### Abstract.

We construct certain maps from buildings associated to td-groups to a space
closely related to the classifying numerable $G$-space for the family
${\mathcal{C}\mathrm{vcy}}$ of covirtually cyclic subgroups. These maps are
used elsewhere to study the K-theory of Hecke algebras in the spirit of the
Farrell–Jones conjecture.

###### Key words and phrases:

CAT(0)-geometry, flow spaces

###### 1991 Mathematics Subject Classification:

20F67, 51F99, 37D99

##  1\. Introduction

The Farrell–Jones conjecture [13] originated in the surgery theory and has
applications to the classification of manifolds, notably it implies (in
dimension $\geq 5$) Borel’s conjecture on the topological rigidity of
aspherical manifolds. The conjecture concerns the K- and L-groups of group
rings and expresses these in terms of an equivariant homology theory. It can
be viewed as reducing computations to the case of group rings for virtually
cyclic groups. Further information on the conjecture can be found for instance
in [19].

The main result from this paper are used in [5] to obtain computations for the
K-theory of Hecke algebras that are in spirit similar to the Farrell–Jones
conjecture. It can viewed as extending results from [3] which are a central
ingredient to the proof of the Farrell–Jones conjecture for
$\operatorname{CAT}(0)$-groups [4] from the setting of discrete groups to td-
groups.

###  1.a. Discrete case

Let $\Gamma$ be a discrete groups. Typically a $\Gamma$-space cannot be both
compact and $\Gamma$-CW-complex with small isotropy groups. Compromises
between these two properties are central to axiomatic results for the
Farrell–Jones conjecture, see for example [2, Sec. 2]. For
$\operatorname{CAT}(0)$-groups such a compromise was established in [3, Main
Thm] and [23, Thm 3.4].

For a collection $\mathcal{F}$ of subgroups and $N\in\mathbb{N}$ we consider
the $n$+$1$-fold join
$E^{N}_{\mathcal{F}}(\Gamma):=\ast_{i=0}^{N}(\coprod_{V\in\mathcal{F}}\Gamma/V)$111The
infinite join $\ast_{i=0}^{\infty}(\coprod_{V\in\mathcal{F}}\Gamma/V)$ is a
model for the classifying $\Gamma$-CW-complex for the family consisting of all
subgroups of $\Gamma$ that are subconjugated to one of the $V\in\mathcal{F}$,
compare [8, App. A1].. As $E^{N}_{\mathcal{F}}(\Gamma)$ is a simplicial
complex we can equip it with the $\ell^{\infty}$-metric $d_{E}$. We note that
this metric is $\Gamma$-invariant. Let $X$ be a $\operatorname{CAT}(0)$-space
of finite covering dimension with a cocompact proper isometric
$\Gamma$-action. We fix a basepoint in $X$ and write $B_{R}$ for the closed
ball of radius $R$ around the basepoint. Let $\pi_{R}\colon X\to B_{R}$ be the
radial projection. Let ${\mathcal{C}\mathrm{vcy}}$ be the family of subgroups
of $\Gamma$ that admit a map to a cyclic group with finite kernel.

######  Theorem 1.1.

There is $N\in\mathbb{N}$ such that for all finite $M\subseteq\Gamma$ and
$\epsilon>0$ there is $\mathcal{V}\subseteq{\mathcal{C}\mathrm{vcy}}$ finite
such that for all $L>0$ we find $R>0$ and a (continuous) map $f\colon X\to
E^{N}_{\mathcal{V}}(G)$ satisfying:

  1. (i)

for $x\in B_{R+L},g\in M$ we have $d_{E}(f(gx),gf(x))<\epsilon$;

  2. (ii)

for $x\in B_{R+L}$, $R^{\prime}\geq R$ we have
$d_{E}(f(x),f(\pi_{R^{\prime}}(x)))<\epsilon$.

Theorem 1.1 is the discrete precursor to our main result in the totally
disconnected case, see Theorem 1.2. We discuss in Remark 1.7 how it is implied
by the main result. Theorem 1.1 has not been stated before, but it can be
viewed as a reformulation of the results from [3, 23] cited above.

There is a homotopy $\Gamma$-action on $B_{R}$ where $g\in\Gamma$ acts as
$x\mapsto\pi_{R}(gx)$. Roughly speaking (i) says that $f$ is almost
$G$-equivariant and (ii) ensures that the tracks of the homotopies of the
homotopy action on $B_{R}$ have small images in $E^{N}_{\mathcal{V}}(G)$.

###  1.b. The setup

Throughout this paper we fix a td-group $G$, i.e., a locally compact second
countable topological Hausdorff group. We also fix an action of $G$ on a
locally compact $\operatorname{CAT}(0)$-space $X$ of finite covering
dimension. We assume that the action is by isometries, continuous, cocompact,
smooth and proper. In particular, all isotropy groups $G_{x}$ for the action
are compact open in $G$. The main result will also depend on a further more
technical assumption. Informally, the assumption is that isotropy groups of
geodesics in the space of geodesics with bounded period for the $G$-action get
only smaller in small neighborhoods, at least on a suitable fundamental
domain. Technically this is formulated as Assumption 2.7 using the flow space
for $X$. The main example where Assumption 2.7 is satisfied is the action of a
reductive $p$-adic group on its associated extended Bruhat-Tits building, see
Appendix A. This is the main example we are interested in.

###  1.c. The space $J_{\mathcal{F}}^{N}(G)$

Let $\mathcal{F}$ be a collection of closed subgroups of $G$. As in the
discrete case we can consider for $N\in\mathbb{N}$ the $N$+$1$-fold join

$J_{\mathcal{F}}^{N}(G):={\ast}_{i=0}^{N}\Big{(}\coprod_{V\in\mathcal{F}}G/V\Big{)}.$

For closed subgroups $V,V^{\prime}$ of $G$ the product $G/V\times
G/V^{\prime}$ can as a $G$-space not necessarily be written as a coproduct of
orbits. For this reason $J_{\mathcal{F}}^{N}(G)$ is not $G$-CW-complex. But it
is still a numerable $G$-space in the sense of [18, Def. 2.1]222We note that
$\operatorname{colim}_{N\to\infty}J_{\mathcal{F}}^{N}(G)$ is a model for the
classifying numerable $G$-space for the family $\mathcal{F}$, see [18, Def.
2.3] and [8, App. A1]. We will not need this fact, but it motivates the
definition. . In contrast to $E_{\mathcal{F}}^{N}(\Gamma)$, there is in
general no $G$-invariant metric on $J_{\mathcal{F}}^{N}(G)$. In fact, for a
closed (but neither open nor compact) subgroup $V$ of $G$ there may be no
$G$-invariant metric on the orbit $G/V$ that generates the topology. This is a
more substantial difficulty to formulating Theorem 1.1 for td-groups. We will
explain our solution to this difficulty next.

###  1.d. $V$-foliated distance in $G$

We can equip $G$ with a left invariant proper metric $d_{G}$ that generates
the topology of $G$, see [14, Thm. 4.5] or [1, Thm. 1.1]. Let $V$ be a closed
subgroup of $G$. As a replacement for the in general not existing
$G$-invariant metric on $G/V$ we will use the following $V$-foliated distance
in $G$. For $g,g^{\prime}\in G$, $\beta,\eta>0$ we write

$d_{V\text{-}\mathrm{fol}}(g,g^{\prime})<(\beta,\eta)$

if there is $v\in V$ with $d_{G}(e,v)=d_{G}(g,gv)<\beta$ and
$d_{G}(gv,g^{\prime})<\eta$. Note that $d_{V\text{-}\mathrm{fol}}$ is left
$G$-invariant in the sense that

$d_{V\text{-}\mathrm{fol}}(g,g^{\prime})<(\beta,\eta)\Longleftrightarrow
d_{V\text{-}\mathrm{fol}}(g^{\prime\prime}g,g^{\prime\prime}g^{\prime})<(\beta,\eta)$

holds for all $g,g^{\prime},g^{\prime\prime}\in G$.

Two elements $g,g^{\prime}$ in $G$ satisfy $gV=g^{\prime}V$ if and only if
there exists $\beta>0$ such that for every $\eta>0$ we have
$d_{V\text{-}\mathrm{fol}}(g,g^{\prime})<(\beta,\eta)$. One might be tempted
to consider

$d_{G/V}(gV,g^{\prime}V):=\inf\\{\eta\mid
d_{V\text{-}\mathrm{fol}}(g,g^{\prime})<(\beta,\eta)\;\text{for some
$\beta>0$}\\},$

but this infimum can be $0$ for $gV\neq g^{\prime}V$. This happens for example
for $V$ the subgroup of $\operatorname{SL}_{2}(\mathbb{Q}_{p})$ consisting of
diagonal matrices $g=e$ and $g^{\prime}$ unipotent.

###  1.e. The space $J_{\mathcal{F}}^{N}(G)^{\wedge}$

Let $\mathcal{F}$ be a collection of closed subgroups of $G$. For
$N\in\mathbb{N}$ let

$J_{\mathcal{F}}^{N}(G)^{\wedge}:={\ast}_{i=0}^{N}\Big{(}\coprod_{V\in\mathcal{F}}G\Big{)}.$

The projections $G\to G/V$ induce a $G$-equivariant map
$J_{\mathcal{F}}^{N}(G)^{\wedge}\to J_{\mathcal{F}}^{N}(G)$.

As $\coprod_{V\in\mathcal{F}}G=G\times\mathcal{F}$ we can write elements in
$J_{\mathcal{F}}^{N}(G)^{\wedge}$ as
$[t_{0}(g_{0},V_{0}),\dots,t_{n}(g_{n},V_{n})]$ with $t_{i}\in[0,1]$,
$g_{i}\in G$, $V_{i}\in\mathcal{F}$ such that $\sum t_{i}=1$. In this notation
we have
$[t_{0}(g_{0},V_{0}),\dots,t_{n}(g_{n},V_{n})]=[t^{\prime}_{0}(g^{\prime}_{0},V^{\prime}_{0}),\dots,t^{\prime}_{n}(g^{\prime}_{n},V^{\prime}_{n})]$
if and only if $t_{i}=t^{\prime}_{i}$ for all $i$ and
$(g_{i},V_{i})=(g^{\prime}_{i},V^{\prime}_{i})$ for all $i$ with $t_{i}\neq
0\neq t^{\prime}_{i}$.

###  1.f. $J$-foliated distance

As discussed in Subsection 1.c there is in general no $G$-invariant metric on
$J_{\mathcal{F}}^{N}(G)$. As a replacement we work with the following notion
of foliated distance on $J_{\mathcal{F}}^{N}(G)^{\wedge}$. For

$y=[t_{0}(g_{0},V_{0}),\ldots,t_{N}(g_{N},V_{N})],\;y^{\prime}=[t^{\prime}_{0}(g_{0}^{\prime},V^{\prime}_{0}),\ldots,t^{\prime}_{N}(g^{\prime}_{N},V^{\prime}_{N})]\in
J^{N}_{\mathcal{F}}(G)^{\wedge}$

and $\beta,\eta,\epsilon>0$ we write

$d_{J\text{-}\mathrm{fol}}(y,y^{\prime})<(\beta,\eta,\epsilon)$

if $|t_{i}-t^{\prime}_{i}|<\epsilon$ for all $i$ and, in addition, for all $i$
with $\max\\{t_{i},t^{\prime}_{i}\\}\geq\epsilon$ we have

$V_{i}=V^{\prime}_{i},\quad\text{and}\quad
d_{V_{i}\text{-}\mathrm{fol}}(g_{i},g^{\prime}_{i})<(\beta,\eta).$

There is a map $J^{N}_{\mathcal{F}}(G)^{\wedge}\to\ast_{i=0}^{N}\mathcal{V}$,
$[t_{0}(g_{0},V_{0}),\ldots,t_{N}(g_{N},V_{N})]\mapsto[t_{0}V_{0},\ldots,t_{N}V_{N}]$.
The first requirement implies that the images of $y$ and $y^{\prime}$ in the
join $\ast_{i=0}^{N}\mathcal{V}$ are of distance $<\epsilon$ with respect to
the $\ell^{\infty}$-metric. Two points $y$ and $y^{\prime}$ in
$J_{\mathcal{F}}^{N}(G)^{\wedge}$ have the same image under the projection
$J_{\mathcal{F}}^{N}(G)^{\wedge}\to J_{\mathcal{F}}^{N}(G)$ if and only if
there exists $\beta>0$ such that for any $\epsilon>0$ and any $\eta>0$ we have
$d_{J\text{-}\mathrm{fol}}(z,z^{\prime})<(\beta,\eta,\epsilon)$.

###  1.g. Statement of main result

We write now ${\mathcal{C}\mathrm{vcy}}$ for the family of subgroups $V$ of
$G$ which are _covirtually cyclic_ , i.e., $V$ is compact or there exists an
exact sequence of topological groups $1\to
K\xrightarrow{i}V\to\mathbb{Z}\xrightarrow{p}1$, where $i$ is the inclusion of
a compact open subgroup $K$ of $V$ and $\mathbb{Z}$ is equipped with the
discrete topology.

As before we fix a base point $b\in X$ and write $B_{R}$ for the closed ball
of radius $R$ around $b$ in $X$. We write $\pi_{R}\colon X\to B_{R}$ for the
radial projection.

######  Theorem 1.2 (Main Theorem).

Suppose that Assumption 2.7 holds.

There is $N\in\mathbb{N}$ such that for all $M\subseteq G$ compact and
$\epsilon>0$ there are $\beta>0$ and
$\mathcal{V}\subseteq{\mathcal{C}\mathrm{vcy}}$ finite with the following
property: For all $\eta>0$ and all $L>0$ we find $R>0$ and a (not necessarily
continuous) map $f\colon X\to J_{\mathcal{V}}^{N}(G)^{\wedge}$ satisfying:

  1. (i)

for $x\in B_{R+L}$, $g\in M$ we have
$d_{J\text{-}\mathrm{fol}}(f(gx),gf(x))<(\beta,\eta,\epsilon)$;

  2. (ii)

for $x\in B_{R+L}$, $R^{\prime}\geq R$ we have
$d_{J\text{-}\mathrm{fol}}(f(x),f(\pi_{R^{\prime}}(x)))<(\beta,\eta,\epsilon)$;

  3. (iii)

there is $\rho>0$ such that for all $x,x^{\prime}\in X$ with
$d_{X}(x,x^{\prime})<\rho$ we have
$d_{J\text{-}\mathrm{fol}}(f(x),f(x^{\prime}))<(\beta,\eta,\epsilon)$.

######  Remark 1.3 (Quantifiers).

Using quantifiers the beginning of Theorem 1.2 reads as

$\exists N\;\forall
M,\epsilon\;\exists\beta,\mathcal{V}\;\forall\eta,L\;\exists R,f\;\text{such
that}\ldots$

######  Remark 1.4 (Failure of continuity).

The map $f$ appearing in Theorem 1.2 is not necessarily continuous, but this
should not be viewed as a serious problem; (iii) is a sufficient replacement
for continuity.

This issue arise in Proposition 6.8. We discuss in Remark 6.10 how it might be
circumvented with more careful bookkeeping.

######  Remark 1.5 (Strategy of the proof of Theorem 1.2).

The proof of Theorem 1.2 will use the flow space $\mathit{FS}$ from [3] that
mimics the geodesic flow on non-positively curved manifolds. More precisely
the map $f$ from Theorem 1.2 will be constructed as a composition

$X\xrightarrow{f_{0}}\mathit{FS}\xrightarrow{f_{1}}J^{N}_{\mathcal{V}}(G).$

The map $f_{0}$ is constructed in Theorem 4.1. This uses the dynamic
properties of the flow on $\mathit{FS}$ coming from the
$\operatorname{CAT}(0)$-geometry of $X$. The map $f_{1}$ is constructed in
Theorem 4.3 and uses an adaptation of the long and thin covers for flow spaces
from [6, 15] to the case of td-groups.

######  Remark 1.6 (About Assumption 2.7).

We do not know whether our main theorem fails in the absence of Assumption
2.7. We do not know whether or whether not Assumption 2.7 is always satisfied.
It is not difficult to check that Assumption 2.7 holds automatically if $G$ is
discrete. It is not difficult to check that Assumption 2.7 implies that for
$\ell>0$ the collection of all $V_{c}$ with $\tau_{c}\leq\ell$ contains up to
conjugation only finitely many subgroups. We do not know whether or not the
converse holds.

######  Remark 1.7 (Back to the discrete case).

As any discrete group $\Gamma$ is also a td-group we can apply Theorem 1.2 in
the situation of Subsection 1.a. Write $p\colon
J_{\mathcal{V}}^{N}(\Gamma)^{\wedge}\to
J_{\mathcal{V}}^{N}(\Gamma)=E_{\mathcal{V}}^{N}(\Gamma)$ for the canonical
projection. As $\Gamma$ is discrete there is $\eta>0$ such that
$d_{\Gamma}(g,g^{\prime})<\eta$ implies $g=g^{\prime}$. For such $\eta$ and
all $\beta,\epsilon>0$, $y,y^{\prime}\in J_{\mathcal{V}}^{N}(\Gamma)^{\wedge}$
we then have

$d_{J\text{-}\mathrm{fol}}(y,y^{\prime})<(\beta,\eta,\epsilon)\implies
d_{E}(p(y),p(y^{\prime}))<\epsilon.$

Therefore we can simply compose $f$ from Theorem 1.2 with $p$ to obtain
Theorem 1.1. Note that (iii) from Theorem 1.2 implies that $p\circ f$ is
continuous (in fact uniformly).

###  1.h. Acknowledgments

We thank Linus Kramer and Stefan Witzel for helpful comments and discussions.
We are grateful to the referee for further helpful comments and a list of
careful corrections.

This work has been supported by the ERC Advanced Grant “KL2MG-interactions”
(no. 662400) of the second author granted by the European Research Council, by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –
Project-ID 427320536 - SFB 1442, as well as under Germany’s Excellence
Strategy - GZ 2047/1, Projekt-ID 390685813, Hausdorff Center for Mathematics
at Bonn, and EXC 2044 - 390685587, Mathematics Münster: Dynamics - Geometry -
Structure.

The paper is organized as follows:

###### Contents

  1. 1 Introduction
    1. 1.a Discrete case
    2. 1.b The setup
    3. 1.c The space $J_{\mathcal{F}}^{N}(G)$
    4. 1.d $V$-foliated distance in $G$
    5. 1.e The space $J_{\mathcal{F}}^{N}(G)^{\wedge}$
    6. 1.f $J$-foliated distance
    7. 1.g Statement of main result
    8. 1.h Acknowledgments
  2. 2 The flow space $\mathit{FS}$
    1. 2.a Construction of the flow space
    2. 2.b Basic facts about the flow space
    3. 2.c Foliated distance in $\mathit{FS}$
    4. 2.d The groups $K_{c}$ and $V_{c}$
  3. 3 Triangle inequalities for $d_{V\text{-}\mathrm{fol}}$ and $d_{J\text{-}\mathrm{fol}}$
  4. 4 Factorization over the flow space
    1. 4.a From $X$ to $\mathit{FS}$
    2. 4.b From $\mathit{FS}$ to $|J^{N}_{\mathcal{V}}(G)|^{\wedge}$
    3. 4.c Proof of Main Theorem using Theorems 4.1 and 4.3
  5. 5 The map to the flow space
  6. 6 Three properties of the flow space
    1. 6.a Long thin covers
    2. 6.b Partitions of unity for long thin covers
    3. 6.c Dimension of long thin covers
    4. 6.d The local structure of $\mathit{FS}$
    5. 6.e Proof of Theorem 4.3 using Propositions 6.4, 6.6 and 6.8
  7. 7 Partition of unity
  8. 8 Dimension of long thin covers
  9. 9 Local structure
    1. 9.a Neighborhoods in $\mathit{FS}$ mapping to $G$
    2. 9.b Proof of Proposition 6.8
  10. A The Bruhat-tits building for reductive $p$-adic groups
  11. B Basics about group actions

##  2\. The flow space $\mathit{FS}$

###  2.a. Construction of the flow space

Given a metric space $Z$, denote by $\mathit{FS}=\mathit{FS}(Z)$ the
associated flow space defined in [3, Section 1]. It consists of all
generalized geodesics. A _generalized geodesic_ is a continuous map
$c\colon\mathbb{R}\to Z$ whose restriction to some interval333By an interval
we mean a set of the form $[a,b]$, $[a,+\infty)$, $(-\infty,b]$ or
$(-\infty,+\infty)$ is an isometric embedding and is locally constant on the
complement of this interval. We do allow that $c$ is constant. The metric on
$\mathit{FS}$ is given by

$d_{\mathit{FS}}(c,c^{\prime}):=\int_{\mathbb{R}}\frac{d_{Z}(c(t),c^{\prime}(t))}{2e^{|t|}}\,dt.$

We recall from [3, Prop. 1.7] that this metric generates the topology of
uniform convergence on compact subsets. The flow $\Phi$ on $\mathit{FS}$ is
defined by

$(\Phi_{\tau}c)(t):=c(t+\tau).$

We also write $\mathit{FS}_{\infty}=\mathit{FS}_{\infty}(Z)$ for the subspace
of $\mathit{FS}$ consisting of all generalized geodesics that are bi-infinite
geodesics, i.e., are nowhere locally constant.

###  2.b. Basic facts about the flow space

For later reference we recall some facts about $\mathit{FS}$ from [3].

######  Lemma 2.1.

Let $(Z,d_{Z})$ be a metric space.

  1. (i)

The map $\Phi$ is a continuous flow and we have for $c,d\in\mathit{FS}(Z)$ and
$\tau,\sigma\in\mathbb{R}$

$d_{\mathit{FS}}\bigl{(}\Phi_{\tau}(c),\Phi_{\sigma}(d)\bigr{)}\leq
e^{|\tau|}\cdot d_{\mathit{FS}}(c,d)+|\sigma-\tau|;$

  2. (ii)

For fixed $\alpha$ the map
$\mathit{FS}\times[-\alpha,\alpha]\to\mathit{FS},\;(c,t)\mapsto\Phi_{t}(c)$ is
uniformly continuous.

###### Proof.

Assertion (i) is proved in [3, Lemma 1.3] and implies assertion (ii). ∎

######  Lemma 2.2.

Suppose that $Z$ is a proper metric space. Then $\mathit{FS}(Z)$ is a proper
metric space, and for any $t\in\mathbb{R}$ the evaluation map $\mathit{FS}\to
Z$, $c\mapsto c(t)$ is proper and uniformly continuous.

###### Proof.

See [3, Prop. 1.9 and Lem. 1.10]. ∎

######  Lemma 2.3.

If $G$ acts cocompactly, isometrically, or properly respectively on the proper
metric space $Z$, then the $G$-action on $\mathit{FS}$ is cocompact, isometric
or proper respectively.

###### Proof.

Obviously the $G$-actions on $\mathit{FS}$ is isometric if $G$ acts
isometrically on $Z$. We conclude from Lemma B.1 (vii) and (xi) and Lemma 2.2
that the $G$-action on $\mathit{FS}$ is proper or cocompact respectively if
$G$ acts proper or cocompact respectively on $Z$. ∎

###  2.c. Foliated distance in $\mathit{FS}$

For $c,c^{\prime}\in\mathit{FS}$, $\alpha,\delta>0$ we write

$d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime})<(\alpha,\delta)$

to mean that there is $t\in[-\alpha,\alpha]$ with
$d_{\mathit{FS}}(\Phi_{t}(c),c^{\prime})<\delta$. We set

$U^{\mathrm{fol}}_{\alpha,\delta}(c):=\\{c^{\prime}\in\mathit{FS}\mid
d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime})<(\alpha,\delta)\\}.$

######  Lemma 2.4 (Basics about foliated distance).

  1. (i)

For $c,c^{\prime}\in\mathit{FS}$, we get

$d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime})<(\alpha,\delta)\;\implies\;d_{\mathit{FS}}(c^{\prime},c)<\alpha+\delta;$

  2. (ii)

For $c,c^{\prime}\in\mathit{FS}$ and $g\in G$, we have

$d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime})<(\alpha,\delta)\Longleftrightarrow
d_{\mathit{FS}\text{-}\mathrm{fol}}(gc,gc^{\prime})<(\alpha,\delta).$

###### Proof.

(i) This follows from the triangle inequality, since $\Phi$ has at most unit
speed, i.e., $d_{\mathit{FS}}(\Phi_{t}(c),c)\leq|t|$ for all $c\in\mathit{FS}$
and $t\in\mathbb{R}$, by Lemma 2.1 (i).
(ii) Recall that $d_{\mathit{FS}}$ left $G$-invariant and $\Phi$ is compatible
with the $G$-action and hence
$d_{\mathit{FS}}(\Phi_{t}(gc),gc^{\prime})=d_{\mathit{FS}}(\Phi_{t}(c),c^{\prime})$
for all $c,c^{\prime}\in\mathit{FS}$, $g\in G$ and $t\in\mathbb{R}$. ∎

######  Lemma 2.5 (Symmetry and triangle inequality for the foliated
distance).

  1. (i)

For $\alpha>0$, $\delta>0$ there is $\epsilon>0$ such that for all
$c,c^{\prime}\in\mathit{FS}$

$d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime})<(\alpha,\epsilon)\;\implies\;d_{\mathit{FS}\text{-}\mathrm{fol}}(c^{\prime},c)<(\alpha,\delta);$

  2. (ii)

For $\alpha>0$, $\delta>0$ there is $\epsilon>0$ such that for all
$c,c^{\prime},c^{\prime\prime}\in\mathit{FS}$

$d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime}),d_{\mathit{FS}\text{-}\mathrm{fol}}(c^{\prime},c^{\prime\prime})<(\alpha,\epsilon)\;\implies\;d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime\prime})<(2\alpha,\delta).$

###### Proof.

(i) Given $\alpha>0$, $\delta>0$, we conclude from Lemma 2.1 (i), that there
is $\epsilon>0$ such that
$d_{\mathit{FS}}(\Phi_{t}(c),\Phi_{t}(c^{\prime}))<\delta$, whenever
$t\in[-\alpha,\alpha]$ and $d_{\mathit{FS}}(c,c^{\prime})<\epsilon$. For (i),
suppose now
$d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime})<(\alpha,\epsilon)$. Then
there is $t\in[-\alpha,\alpha]$ with
$d_{\mathit{FS}}(\Phi_{t}(c),c^{\prime})<\epsilon$. This implies
$d_{\mathit{FS}}(c,\Phi_{-t}(c^{\prime}))=d_{\mathit{FS}}(\Phi_{-t}(\Phi_{t}(c)),\Phi_{-t}(c^{\prime}))<\delta$
and therefore
$d_{\mathit{FS}\text{-}\mathrm{fol}}(c^{\prime},c)<(\alpha,\delta)$.
(ii) Given $\alpha>0$, $\delta>0$, we find again $\epsilon>0$ as in (i). If
$d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime}),d_{\mathit{FS}\text{-}\mathrm{fol}}(c^{\prime},c^{\prime\prime})<(\alpha,\epsilon)$,
then there exists $t,t^{\prime}\in[-\alpha,\alpha]$ satisfying
$d_{\mathit{FS}}(\Phi_{t}(c),c^{\prime}),d_{\mathit{FS}}(\Phi_{t^{\prime}}(c),c^{\prime\prime})<\epsilon$.
This yields $d_{\mathit{FS}}(\Phi_{t+t^{\prime}}(c),c^{\prime\prime})\leq
d_{\mathit{FS}}(\Phi_{t^{\prime}}(\Phi_{t}(c)),\Phi_{t^{\prime}}(c^{\prime}))+d_{\mathit{FS}}(\Phi_{t^{\prime}}(c^{\prime}),c^{\prime\prime})<\delta+\epsilon$.
Hence
$d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime\prime})<(2\alpha,\delta+\epsilon)$.
Using $\delta^{\prime}:=\delta/2$ in place of $\delta$ and asking in addition
for $\epsilon<\delta/2$ gives (ii). ∎

###  2.d. The groups $K_{c}$ and $V_{c}$

For $c\in\mathit{FS}(X)$ we set

  1. (2.6a)

$K_{c}:=\;G_{c}=\;\\{g\in G\mid gc=c\\}=\;\\{g\in G\mid gc(t)=c(t)\;\text{for
all}\;t\in\mathbb{R}\\}$;

  2. (2.6b)

$V_{c}\;:=\;\\{g\in G\mid\exists t\in\mathbb{R}\,:\,gc=\Phi_{t}(c)\\}$;

  3. (2.6c)

$\tau_{c}\;\;:=\;\inf\\{t>0\mid\exists v\in V_{c}\setminus
K_{c},\text{with}\;\Phi_{t}(c)=vc\\}$.

We use $\inf\emptyset=\infty$. If $\tau_{c}<\infty$ then we say that $c$ is
_periodic_. We have $K_{c}\subseteq V_{c}$ as the flow is $G$-equivariant.

######  Assumption 2.7.

There exists $\mathit{FS}_{0}\subseteq\mathit{FS}$ compact such that

  1. (2.7a)

$G\cdot\mathit{FS}_{0}=\mathit{FS}$;

  2. (2.7b)

for $\ell>0$ and $c_{0}\in\mathit{FS}_{0}$ there exists an open neighborhood
$U$ of $c_{0}$ in $\mathit{FS}_{0}$ such that for all $c\in U$ with
$\tau_{c}\leq\ell$ we have $V_{c}\subseteq V_{c_{0}}$.

######  Lemma 2.8.

Let $c\in\mathit{FS}$ be periodic. Then $c\in\mathit{FS}_{\infty}$. Moreover,
there is $v\in V_{c}$ with $vc=\Phi_{\tau_{c}}(c)$ and any such $v$ together
with $K_{c}$ generates $V_{c}$.

###### Proof.

Consider $c\not\in\mathit{FS}_{\infty}$. Then $c$ is constant on some interval
$(-\infty,a_{-})$ or $(a_{+},\infty)$. This implies that for all $v\in V_{c}$
there is $t$ with $vc(t)=c(t)$. In particular, $v$ fixes an endpoint of the
image of $c$ which is a finite geodesic or a geodesic ray. Since $v$ also
fixes the image of $c$ as a set, it must in fact fix $c$. Hence $K_{c}=V_{c}$.
If $c$ is periodic we have $K_{c}\subsetneq V_{c}$ and so we must have
$c\in\mathit{FS}_{\infty}$.

In particular, $t\mapsto\Phi_{t}(c)$ is injective. It follows that for $v\in
V_{c}$ there is a unique $t_{v}\in\mathbb{R}$ with $vc=\Phi_{t_{v}}(c)$. We
obtain a group homomorphism $V_{c}\to\mathbb{R}$, $v\mapsto t_{v}$ whose
kernel is $K_{c}$. Denote the image of this homomorphism by $\Gamma_{c}$. We
claim that $\Gamma_{c}$ is discrete.

To prove this claim, suppose that there are $(v_{n})_{n\in\mathbb{N}}$ with
$t_{v_{n}}\to 0$ as $n\to\infty$. As $c\in\mathit{FS}_{\infty}$,
$c\colon\mathbb{R}\to X$ is injective. Now for any $s\in\mathbb{R}$ we have
$v_{n}(c(s))=(v_{n}c)(s)=(\Phi_{t_{n}}(c))=c(s+t_{n})\to c(s)$, so $c(s)$ is
an accumulation point of its $G$-orbit. But $G\curvearrowright X$ is smooth
and proper, so orbits are discrete. Thus $\Gamma_{c}$ is discrete.

Thus $\tau_{c}=\min\Gamma_{c}\cap\mathbb{R}_{>0}$ and there is $v\in V_{c}$
with $t_{v}=\tau_{c}$. Moreover, $\Gamma_{c}$ is infinite cyclic and generated
by $\tau_{c}$. Thus any $v\in V_{c}$ with $t_{v}=\tau_{c}$ will together with
$K_{c}$ generate $V_{c}$. ∎

##  3\. Triangle inequalities for $d_{V\text{-}\mathrm{fol}}$ and
$d_{J\text{-}\mathrm{fol}}$

We have already proven a version of the triangle inequality for
$d_{\mathit{FS}\text{-}\mathrm{fol}}$ on the flow space $\mathit{FS}$ in Lemma
2.5. We will need versions of the triangle inequality for
$d_{V\text{-}\mathrm{fol}}$ and $d_{J\text{-}\mathrm{fol}}$ as well.

######  Lemma 3.1.

  1. (3.1a)

Let $M\subseteq G$ be compact. For any $\epsilon>0$ there is $\delta>0$ such
that for all $g,g^{\prime}\in G$, $v\in M$ we have

$d_{G}(g,g^{\prime})<\delta\implies d_{G}(gv,g^{\prime}v)<\epsilon;$

  2. (3.1b)

Let $\alpha\geq 0$. Then for any $\epsilon>0$ there is $\delta>0$ such that
for any closed subgroup $V$ of $G$ and $g,g^{\prime},g^{\prime\prime}\in G$ we
have

$d_{V\text{-}\mathrm{fol}}(g,g^{\prime}),d_{V\text{-}\mathrm{fol}}(g^{\prime},g^{\prime\prime})\leq(\alpha,\delta)\quad\implies\quad
d_{V\text{-}\mathrm{fol}}(g,g^{\prime\prime})\leq(2\alpha,\epsilon).$

###### Proof.

Assume the first statement fails for given $M\subseteq G$ compact and
$\epsilon>0$. Then there are sequences $g_{n},g^{\prime}_{n}\in G$, $v_{n}\in
M$ with $d_{G}(g_{n},g^{\prime}_{n})<1/n$ and
$d_{G}(g_{n}v_{n},g^{\prime}_{n}v_{n})\geq\epsilon$. Let
$x_{n}:=g_{n}^{-1}g^{\prime}_{n}$. Then
$d_{G}(e,x_{n})=d_{G}(g_{n},g^{\prime}_{n})<1/n$ and thus
$\lim_{n\to\infty}x_{n}=e$. By passing to a subsequence, we can arrange
$\lim_{n\to\infty}v_{n}=v$ for some $v\in M$. This implies
$\lim_{n\to\infty}v_{n}^{-1}x_{n}v_{n}=v^{-1}ev=e$. Hence
$\lim_{n\to\infty}d_{G}(g_{n}v_{n},g^{\prime}_{n}v_{n})=\lim_{n\to\infty}d_{G}(e,v_{n}^{-1}x_{n}v_{n})=0$,
a contradiction.

For the second statement, let $\alpha>0$ and $\epsilon>0$ be given. Using the
compactness of the closed $\alpha$-ball in $G$ we find, using (3.1a),
$\delta>0$ such that

$d_{G}(g,g^{\prime})<\delta,d_{G}(v,e)<\alpha\implies
d_{G}(gv,g^{\prime}v)<\epsilon/2.$

After decreasing $\delta$ we may assume $\delta<\epsilon/2$. Now if
$d_{V\text{-}\mathrm{fol}}(g,g^{\prime}),d_{V\text{-}\mathrm{fol}}(g^{\prime},g^{\prime\prime})\leq(\alpha,\delta)$
then there are $v,v^{\prime}\in G$ with
$d_{G}(gv,g^{\prime}),d_{G}(g^{\prime}v^{\prime},g^{\prime\prime})<\delta$,
$d_{G}(v,e),d_{G}(v^{\prime},e)<\alpha$. Our choice of $\delta$ implies
$d_{G}(gvv^{\prime},g^{\prime}v^{\prime})<\epsilon/2$. Now
$d_{G}(vv^{\prime},e)\leq d_{G}(vv^{\prime},v)+d_{G}(v,e)\leq 2\alpha$ and
hence $d_{G}(gvv^{\prime},g^{\prime\prime})\leq
d_{G}(gvv^{\prime},g^{\prime}v^{\prime})+d_{G}(g^{\prime}v^{\prime},g^{\prime\prime})<\epsilon/2+\delta<\epsilon$.
Thus $d_{V\text{-}\mathrm{fol}}(g,g^{\prime\prime})<(2\alpha,\epsilon)$. ∎

######  Lemma 3.2.

Let $\mathcal{V}\subseteq{\mathcal{C}\mathrm{vcy}}$ be finite. Fix $\beta>0$.
Then for all $\eta^{\prime}>0$ there is $\eta>0$ with the property that for
every $\epsilon>0$ and every $y,y^{\prime},y^{\prime\prime}$ in
$J_{\mathcal{V}}^{N}(G)^{\wedge}$ we have

$d_{J\text{-}\mathrm{fol}}(y,y^{\prime}),d_{J\text{-}\mathrm{fol}}(y^{\prime},y^{\prime\prime})<(\beta,\eta,\epsilon)\quad\implies\quad
d_{J\text{-}\mathrm{fol}}(y,y^{\prime\prime})<(2\beta,\eta^{\prime},2\epsilon).$

###### Proof.

We use $\eta^{\prime}:=\delta^{\prime}$ from (3.1b) with $\delta:=\eta$. Write

$\displaystyle y$ $\displaystyle=$
$\displaystyle[t_{0}(g_{0},V_{0}),\ldots,t_{N}(g_{N},V_{N})];$ $\displaystyle
y^{\prime}$ $\displaystyle=$
$\displaystyle[t^{\prime}_{0}(g^{\prime}_{0},V^{\prime}_{0}),\ldots,t^{\prime}_{N}(g^{\prime}_{N},V^{\prime}_{N})];$
$\displaystyle y^{\prime\prime}$ $\displaystyle=$
$\displaystyle[t^{\prime\prime}_{0}(g^{\prime\prime}_{0},V^{\prime\prime}_{0}),\ldots,t^{\prime\prime}_{N}(g^{\prime\prime}_{N},V^{\prime\prime}_{N})].$

As $|t_{i}-t^{\prime}_{i}|,|t^{\prime}_{i}-t^{\prime\prime}_{i}|<\epsilon$ for
all $i$, we have $|t_{i}-t^{\prime\prime}_{i}|<2\epsilon$ for all $i$.

Suppose that $\max\\{t_{i},t^{\prime\prime}_{i}\\}\geq 2\epsilon$. As
$|t_{i}-t^{\prime}_{i}|,|t^{\prime}_{i}-t^{\prime\prime}_{i}|<\epsilon$, this
implies both $\max\\{t_{i},t^{\prime}_{i}\\}\geq\epsilon$ and
$\max\\{t^{\prime}_{i},t^{\prime\prime}_{i}\\}\geq\epsilon$. Thus
$V_{i}=V^{\prime}_{i}=V^{\prime\prime}_{i}$ and
$d_{V_{i}\text{-}\mathrm{fol}}(g_{i},g^{\prime}_{i})<(\beta,\eta)$ and
$d_{V_{i}\text{-}\mathrm{fol}}(g^{\prime}_{i},g^{\prime\prime}_{i})<(\beta,\eta)$.
Now (3.1b) gives
$d_{V_{i}\text{-}\mathrm{fol}}(g_{i},g^{\prime\prime}_{i})<(2\beta,\eta^{\prime})$.
∎

##  4\. Factorization over the flow space

In this section we prove our main Theorem 1.2 modulo two results about the
flow space $\mathit{FS}$.

###  4.a. From $X$ to $\mathit{FS}$

######  Theorem 4.1.

For all $M\subseteq G$ compact there is $\alpha>0$ with the following
property. For all $\delta>0$, $L>0$ there exists $R>0$ and a uniformly
continuous map $f_{0}\colon X\to\mathit{FS}$ such that

  1. (i)

for $x\in B_{R+L}$, $g\in M$ we have
$d_{\mathit{FS}\text{-}\mathrm{fol}}(f_{0}(gx),gf_{0}(x))<(\alpha,\delta)$;

  2. (ii)

for $x\in B_{R+L}$, $R^{\prime}\geq R$ we have
$d_{\mathit{FS}\text{-}\mathrm{fol}}(f_{0}(x),f_{0}(\pi_{R^{\prime}}(x)))<(\alpha,\delta)$,
where $\pi_{R^{\prime}}$ denotes the radial projection onto $B_{R+L}$;

######  Remark 4.2.

Using quantifiers the beginning of Theorem 4.1 reads as

$\forall M\;\exists\alpha\;\forall\delta,L\;\exists R,f_{0}\;\text{such
that}\ldots$

The proof of Theorem 4.1 is given at the end of Section 5.

###  4.b. From $\mathit{FS}$ to $|J^{N}_{\mathcal{V}}(G)|^{\wedge}$

######  Theorem 4.3.

Suppose that Assumption 2.7 holds.

There is $N\in\mathbb{N}$ such that for any $\alpha>0$ and any $\epsilon>0$
there are $\beta>0$ and $\mathcal{V}\subseteq{\mathcal{C}\mathrm{vcy}}$ finite
such that for any $\eta>0$ there are $\delta>0$, $f_{1}\colon\mathit{FS}\to
J_{\mathcal{V}}^{N}(G)^{\wedge}$, satisfying the following properties.

  1. (i)

For $c,c^{\prime}\in\mathit{FS}$ with
$d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime})<(\alpha,\delta)$ we have
$d_{J\text{-}\mathrm{fol}}(f_{1}(c),f_{1}(c^{\prime}))<(\beta,\eta,\epsilon)$;

  2. (ii)

For $c\in\mathit{FS}$, $g\in G$ we have
$d_{J\text{-}\mathrm{fol}}(f_{1}(gc),gf_{1}(c))<(\beta,\eta,\epsilon)$.

######  Remark 4.4.

Using quantifiers the beginning of Theorem 4.3 reads as

$\exists
N\;\forall\alpha,\epsilon\;\exists\beta,\mathcal{V}\;\forall\eta\;\exists\delta,f_{1}\;\text{such
that}\ldots$

The proof of Theorem 4.3 (modulo three results proven later) is given at the
end of Section 6.

###  4.c. Proof of Main Theorem using Theorems 4.1 and 4.3

We restate our Main Theorem 1.2 from the introduction.

######  Theorem (Main Theorem).

Suppose that Assumption 2.7 holds.

There is $N\in\mathbb{N}$ such that for all $M\subseteq G$ compact and
$\epsilon>0$ there are $\beta>0$ and
$\mathcal{V}\subseteq{\mathcal{C}\mathrm{vcy}}$ finite with the following
property: For all $\eta>0$ and all $L>0$ we find $R>0$ and a (not necessarily
continuous) map $f\colon X\to J_{\mathcal{V}}^{N}(G)^{\wedge}$ satisfying:

  1. (i)

for $x\in B_{R+L}$, $g\in M$ we have
$d_{J\text{-}\mathrm{fol}}(f(gx),gf(x))<(\beta,\eta,\epsilon)$;

  2. (ii)

for $x\in B_{R+L}$, $R^{\prime}\geq R$ we have
$d_{J\text{-}\mathrm{fol}}(f(x),f(\pi_{R^{\prime}}(x)))<(\beta,\eta,\epsilon)$;

  3. (iii)

there is $\rho>0$ such that for all $x,x^{\prime}\in X$ with
$d_{X}(x,x^{\prime})<\rho$ we have
$d_{J\text{-}\mathrm{fol}}(f(x),f(x^{\prime}))<(\beta,\eta,\epsilon)$.

###### Proof of Main Theorem.

Let $N$ be the number from Theorem 4.3. Let $M\subseteq G$ be compact and
$\epsilon>0$. Theorem 4.1 gives us a number $\alpha>0$. Theorem 4.3 gives us
$\beta/2>0$ and $\mathcal{V}\subseteq{\mathcal{C}\mathrm{vcy}}$ finite. Let
$\eta>0$ be given. Because of Lemma 3.2 we can find $0<\eta_{0}\leq\eta$ with
the property that every $y,y^{\prime},y^{\prime\prime}$ in
$J_{\mathcal{F}}^{N}(G)^{\wedge}$ we have

  1. (4.5a)

$d_{J\text{-}\mathrm{fol}}(y,y^{\prime}),d_{J\text{-}\mathrm{fol}}(y^{\prime},y^{\prime\prime})<(\beta/2,\eta_{0},\epsilon/2)\implies
d_{J\text{-}\mathrm{fol}}(y,y^{\prime\prime})<(\beta,\eta,\epsilon)$.

Theorem 4.3 gives us $\delta>0$ and $f_{1}\colon\mathit{FS}\to
J^{N}_{\mathcal{V}}(G)^{\wedge}$ satisfying

  1. (4.5b)

if $c,c^{\prime}\in\mathit{FS}$ fullfils
$d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime})<(\alpha,\delta)$, then we
have
$d_{J\text{-}\mathrm{fol}}(f_{1}(c),f_{1}(c^{\prime}))<(\beta/2,\eta_{0},\epsilon/2)$;

  2. (4.5c)

for $c\in\mathit{FS}$, $g\in G$ we have
$d_{J\text{-}\mathrm{fol}}(f_{1}(gc),gf_{1}(c))<(\beta/2,\eta_{0},\epsilon/2)$.

Let $L>0$ be given. Theorem 4.1 gives us $R>0$ and $f_{0}\colon
X\to\mathit{FS}$ uniformly continuous satisfying

  1. (4.5d)

for $x\in B_{R+L}$, $g\in M$ we have
$d_{\mathit{FS}\text{-}\mathrm{fol}}(f_{0}(gx),gf_{0}(x))<(\alpha,\delta)$;

  2. (4.5e)

for $x\in B_{R+L}$, $R\leq t$ we have
$d_{\mathit{FS}\text{-}\mathrm{fol}}(f_{0}(x),f_{0}(\pi_{t}(x)))<(\alpha,\delta)$.

We set now $f:=f_{1}\circ f_{0}$. It remains to verify the three assertions
from Theorem 1.2.
(i) Let $x\in B_{R+L}$, $g\in M$. Then
$d_{\mathit{FS}\text{-}\mathrm{fol}}(f_{0}(gx),gf_{0}(x))<(\alpha,\delta)$ by
(4.5d). Thus (4.5b) implies
$d_{J\text{-}\mathrm{fol}}(f_{1}(f_{0}(gx)),f_{1}(gf_{0}(x)))<(\beta/2,\eta_{0},\epsilon/2)$.
On the other hand (4.5c) implies
$d_{J\text{-}\mathrm{fol}}\bigl{(}f_{1}(gf_{0}(x)),gf_{1}(f_{0}(x))\bigr{)}<(\beta/2,\eta_{0},\epsilon/2)$.
Now we conclude

$d_{J\text{-}\mathrm{fol}}(f(gx),gf(x))=d_{J\text{-}\mathrm{fol}}\bigl{(}f_{1}(f_{0}(gx)),gf_{1}(f_{0}(x))\bigr{)}<(\beta,\eta,\epsilon)$

from (4.5a).
(ii) Let $x\in B_{R+L}$, $R^{\prime}\geq R$. Then
$d_{\mathit{FS}\text{-}\mathrm{fol}}(f_{0}(x),f_{0}(\pi_{R^{\prime}}(x)))<(\alpha,\delta)$
by (4.5e). We conclude

$d_{J\text{-}\mathrm{fol}}(f(x),f(\pi_{R^{\prime}}(x)))=d_{J\text{-}\mathrm{fol}}(f_{1}(f_{0}(x)),f_{1}(f_{0}(\pi_{R^{\prime}}(x))))\\\
<(\beta/2,\eta_{0},\epsilon/2)\leq(\beta,\eta,\epsilon)$

from (4.5b).
(iii) Since $f_{0}$ is uniformly continuous there is $\rho>0$ such that
$d_{\mathit{FS}}(f_{0}(x),f_{0}(x^{\prime}))<\delta$ (and in particular
$d_{\mathit{FS}\text{-}\mathrm{fol}}(f_{0}(x),f_{0}(x^{\prime}))<(\alpha,\delta)$)
for all $x,x^{\prime}\in X$ with $d_{X}(x,x^{\prime})<\rho$. Using (4.5b) we
obtain

$d_{J\text{-}\mathrm{fol}}(f(x),f^{\prime}(x))<(\beta/2,\eta_{0},\epsilon/2)\leq(\beta,\eta,\epsilon)$

for all $x,x^{\prime}\in X$ with $d_{X}(x,x^{\prime})<\rho$. ∎

##  5\. The map to the flow space

The proof of Theorem 4.1, given in this section, will follow closely arguments
from similar results for actions of discrete groups in [3].

For $x,x^{\prime}\in X$ we write $c_{x,x^{\prime}}\in\mathit{FS}$ for the
generalized geodesic from $x$ to $x^{\prime}$, i.e., for the generalized
geodesic characterized by

$\displaystyle c_{x,x^{\prime}}(t)$ $\displaystyle=$ $\displaystyle x\quad
t\in(-\infty,0],$ $\displaystyle c_{x,x^{\prime}}(t)$ $\displaystyle=$
$\displaystyle x\quad t\in[d(x,x^{\prime}),+\infty).$

Recall that we fixed a base point $b$ and write $B_{R}$ for the closed ball of
radius $R$ around $b$ in $X$. Recall also that $\pi_{R}\colon X\to B_{R}$
denotes the radial projection.

######  Lemma 5.1.

The map $X\to\mathit{FS}$, $x\mapsto c_{b,x}$ is uniformly continuous.

###### Proof.

The map is continuous by [9, II.1.4 (1) on page 160] and [3, Proposition 1.7].
As it is equivariant for the cocompact actions of $G$, it is also uniformly
continuous. (Alternatively, uniform continuity can also be checked directly
without using the $G$-action.) ∎

######  Lemma 5.2.

For all $\delta>0$ there is $\Delta>0$ such that for all $R^{\prime},T$ with
$R^{\prime}\geq T+\Delta$, $x\in X$ we have

$d_{\mathit{FS}}(\Phi_{T}(c_{b,x}),\Phi_{T}(c_{b,\pi_{R^{\prime}}(x)}))<\delta.$

###### Proof.

Choose $\Delta>0$ such that
$\int_{\Delta}^{\infty}\frac{s}{2e^{|s|}}ds<\delta$ holds. Let $x\in X$. For
$s+T\leq R^{\prime}$ we have

$\big{(}\Phi_{T}(c_{b,x})\big{)}(s)=c_{b,x}(s+T)=c_{b,\pi_{R^{\prime}}(x)}(s+T)=\big{(}\Phi_{T}(c_{b,\pi_{R^{\prime}}(x)})\big{)}(s),$

while for $s+T\geq R^{\prime}$ we have

$\displaystyle d_{X}\big{(}\big{(}\Phi_{T}(c_{b,x})$
$\displaystyle\big{)}(s),\big{(}\Phi_{T}(c_{b,\pi_{R^{\prime}}(x)})\big{)}(s)\big{)}$
$\displaystyle=$
$\displaystyle\;d_{X}\big{(}c_{b,x}(s+T),c_{b,\pi_{R^{\prime}}(x)}(s+T)\big{)}=s+T-R^{\prime}=s-(R^{\prime}-T).$

Thus, for $R^{\prime}\geq T+\Delta$,

$\displaystyle d_{\mathit{FS}}(\Phi_{T}(c_{b,x}),$
$\displaystyle\Phi_{T}(c_{b,\pi_{R^{\prime}}(x)}))$
$\displaystyle=\int_{-\infty}^{\infty}\frac{d_{X}\big{(}\big{(}\Phi_{T}(c_{b,x})\big{)}(s),\big{(}\Phi_{T}(c_{b,\pi_{R^{\prime}}(x)})\big{)}(s)\big{)}}{2e^{|s|}}ds$
$\displaystyle=\int_{R^{\prime}-T}^{\infty}\frac{s-(R^{\prime}-T)}{2e^{|s|}}ds$
$\displaystyle\leq\int_{\Delta}^{\infty}\frac{s}{2e^{|s|}}ds$
$\displaystyle\leq\delta.\qed$

######  Lemma 5.3.

Let $r^{\prime}$, $L$, $\alpha>0$, $r^{\prime\prime}>\alpha$. Let
$T:=r^{\prime}+r^{\prime\prime}$, $R:=r^{\prime\prime}+2r^{\prime}+\alpha$.
Let $x,x_{1},x_{2}\in X$ with $d_{X}(x_{1},x_{2})\leq\alpha$,
$d_{X}(x_{1},x)\leq R+L$. Set $\tau:=d_{X}(x,x_{1})-d_{X}(x,x_{2})$. Then for
all $t\in[-r^{\prime},r^{\prime}]$

$d_{X}(c_{x_{1},x}(T+\tau+t),c_{x_{2},x}(T+t))\leq\frac{2\alpha(L+2r^{\prime}+2\alpha)}{r^{\prime\prime}}.$

###### Proof.

This is an application of the $\operatorname{CAT}(0)$-condition, see [3, Lemma
3.6(i)]444Strictly speaking, this reference gives
$d_{X}(c_{x_{1},x}(T+t),c_{x_{2},x}(T+t+\tau))\leq\frac{2\alpha(L+2r^{\prime}+\alpha)}{r^{\prime\prime}}$.
The statement is not quite symmetric in $x_{1}$ and $x_{2}$ as we only know
$d_{X}(x,x_{2})\leq d_{X}(x,x_{1})+d_{X}(x_{1},x_{2})\leq R+L+\alpha$. Thus in
applying [3, Lemma 3.6(i)] we need to use $L^{\prime}:=L+\alpha$. In any
event, the exact estimate is not important.. ∎

######  Lemma 5.4.

For all $\alpha>0$, $\Delta>0$, $L>0$ and $\delta>0$ there are $R>0$, $0\leq
T\leq R-\Delta$ such that for all $x,x_{1},x_{2}\in X$ with
$d_{X}(x_{1},x_{2})\leq\alpha$, $d_{X}(x_{1},x)\leq R+L$ we have

$d_{\mathit{FS}\text{-}\mathrm{fol}}(\Phi_{T}(c_{x_{1},x}),\Phi_{T}(c_{x_{2},x}))<(\alpha,\delta).$

###### Proof.

Pick $1>\delta^{\prime}>0,r^{\prime}>\Delta,r^{\prime\prime}>\alpha$ such that

$\int_{-\infty}^{-r^{\prime}}\frac{2|t|+1}{2e^{|t|}}dt<\frac{\delta}{3},\quad\int_{-r^{\prime}}^{r^{\prime}}\frac{\delta^{\prime}}{2e^{|t|}}dt<\frac{\delta}{3}\quad\text{and}\quad\frac{2\alpha(L+2r^{\prime}+2\alpha)}{r^{\prime\prime}}<\delta^{\prime}.$

Set $R:=2r^{\prime}+r^{\prime\prime}+\alpha$ and
$T:=r^{\prime}+r^{\prime\prime}$. Then $0\leq T=R-r^{\prime}-\alpha\leq
R-\Delta$. Set $\tau:=d_{X}(x,x_{1})-d_{X}(x,x_{2})$. Then
$\tau\in[-\alpha,\alpha]$ as $d_{X}(x_{1},x_{2})\leq\alpha$. By Lemma 5.3 we
have for all $t\in[-r^{\prime},r^{\prime}]$

$d_{X}(c_{x_{1},x}(T+\tau+t),c_{x_{2},x}(T+t))<\delta^{\prime}.$

For $t\in(-\infty,-r^{\prime}]$ we have

$\displaystyle d_{X}(c_{x_{1},x}(T+\tau+t),c_{x_{2},x}$ $\displaystyle(T+t))$
$\displaystyle\leq$ $\displaystyle\quad
d_{X}(c_{x_{1},x}(T+\tau+t),c_{x_{1},x}(T+\tau-r^{\prime}))$
$\displaystyle+\quad
d_{X}(c_{x_{1},x}(T+\tau-r^{\prime}),c_{x_{2},x}(T-r^{\prime}))$
$\displaystyle+\quad d_{X}(c_{x_{2},x}(T-r^{\prime}),c_{x_{2},x}(T+t))$
$\displaystyle<$
$\displaystyle\quad|t+r^{\prime}|+\delta^{\prime}+|t+r^{\prime}|$
$\displaystyle=$ $\displaystyle\quad 2|t+r^{\prime}|+\delta^{\prime}\leq
2|t|+\delta^{\prime}<2|t|+1.$

Similarly, for $t\in[r^{\prime},\infty)$ we have

$d_{X}(c_{x_{1},x}(T+t),c_{x_{2},x}(T+\tau+t))<2|t|+1.$

We can now estimate

$\displaystyle d_{\mathit{FS}}$
$\displaystyle\big{(}\Phi_{T+\tau}(c_{x_{1},x}),\Phi_{T}(c_{x_{2},x})\big{)}$
$\displaystyle=\int_{-\infty}^{\infty}\frac{d_{X}(c_{x_{1},x}(T+\tau+t),c_{x_{2},x}(T+t))}{2e^{|t|}}dt$
$\displaystyle<\int_{-\infty}^{-r^{\prime}}\frac{2|t|+1}{2e^{|t|}}dt+\int_{-r^{\prime}}^{r^{\prime}}\frac{\delta^{\prime}}{2e^{|t|}}dt+\int_{r^{\prime}}^{+\infty}\frac{2|t|+1}{2e^{|t|}}dt$
$\displaystyle<\frac{\delta}{3}+\frac{\delta}{3}+\frac{\delta}{3}\quad=\quad\delta.$

Therefore
$d_{\mathit{FS}\text{-}\mathrm{fol}}(\Phi_{T}(c_{x_{1},x}),\Phi_{T}(c_{x_{2},x}))<(\alpha,\delta)$.
∎

###### Proof of Theorem 4.1.

Let $M\subseteq G$ be compact. By compactness of $M\cdot b$ there is
$\alpha>0$ such that $d_{X}(b,gb)\leq\alpha$ for all $g\in M$. Let $\delta>0$,
$L>0$ be given. Let $\Delta$ be the number from Lemma 5.2. Let $R>0$ and
$0\leq T\leq R-\Delta$ be the numbers from Lemma 5.4. Let $f_{0}\colon
X\to\mathit{FS}$ be the map $x\mapsto\Phi_{T}(c_{b,x})$. It is uniformly
continuous by Lemma 2.1 and 5.1. We can now verify the two assertions from
Theorem 4.1.
(i) Let $g\in M$ and $x\in B_{R+L}$. Then $f_{0}(gx)=\Phi_{T}(c_{gb,gx})$ and
$gf_{0}(x)=\Phi_{T}(c_{b,gx})$. As $g\in M$, we have $d_{X}(b,gb)<\alpha$.
Also $d_{X}(gb,gx)=d_{X}(b,x)\leq R+L$. We can therefore apply Lemma 5.4 and
conclude

$d_{\mathit{FS}\text{-}\mathrm{fol}}(f_{0}(gx),gf_{0}(x))=d_{\mathit{FS}\text{-}\mathrm{fol}}(\Phi_{T}(c_{gb,gx}),\Phi_{T}(c_{b,gx}))<(\alpha,\delta).$

(ii) Let $x\in B_{R+L}$ and $R^{\prime}\geq R$. Then
$f_{0}(x)=\Phi_{T}(c_{b,x})$ and
$f_{0}(\pi_{R^{\prime}}(x))=\Phi_{T}(c_{b,\pi_{R^{\prime}}(x)})$. As $T\leq
R-\Delta$ we have $R^{\prime}\geq R\geq T+\Delta$. We can therefore apply
Lemma 5.2 and conclude
$d_{\mathit{FS}}\bigl{(}f_{0}(x),f_{0}(\pi_{R^{\prime}}(x))\bigr{)}=d_{\mathit{FS}}(\Phi_{T}(c_{b,x}),\Phi_{T}(c_{b,\pi_{R^{\prime}}(x)}))<\delta$.
In particular we get

$d_{\mathit{FS}\text{-}\mathrm{fol}}\bigl{(}f_{0}(x),f_{0}(\pi_{R^{\prime}}(x))\bigr{)}<(\alpha,\delta).$

∎

##  6\. Three properties of the flow space

Recall from Subsection 4.c and Section 5 that to prove our main Theorem 1.2 it
remains to prove Theorem 4.3 about maps $\mathit{FS}\to
J_{\mathcal{V}}^{N}(G)$. In this section we formulate three propositions about
the flow space and use them to prove Theorem 4.3. These three propositions
will be proved in the forthcoming sections.

###  6.a. Long thin covers

######  Definition 6.1 ($\alpha$-long cover).

A cover $\mathcal{U}$ of the flow space $\mathit{FS}$ is said to be _$\alpha$
-long_ if for any $c_{0}\in\mathit{FS}$ there exists $U\in\mathcal{U}$ such
that $\Phi_{[-\alpha,\alpha]}(c_{0})\subseteq U$.

Typically such covers are thin in directions transversal to the flow and are
often referred to as long thin covers.

######  Remark 6.2 (Discrete versus totally disconnected).

For a discrete group $\Gamma$ it is possible to construct $G$-equivariant maps
from the flow space $\mathit{FS}$ to $G$-simplicial complexes (such as
$E_{\mathcal{V}}^{N}(\Gamma)$) via the $\alpha$-long
${\mathcal{C}\mathrm{vcy}}$-covers of $\mathit{FS}$ from [6, Sec. 5] or [15].
Here a ${\mathcal{C}\mathrm{vcy}}$-cover consists of open subsets $U$ of
$\mathit{FS}$ for which there is $V\in{\mathcal{C}\mathrm{vcy}}$ such that:
$gU\cap U\neq\emptyset$ iff $g\in V$ and then $gU=U$. For a td-group $G$ such
$U$ will not exist; elements of $g\in G\setminus V$ that are close to $V$ will
typically produce $gU\cap U\neq\emptyset$. The way out is to consider $G\cdot
U$. In the discrete case there is a map $G\cdot U\to G/V$ and this generalizes
to the totally disconnected case. In fact we will even construct $U\to G$, but
this will typically not be equivariant and create some $(\beta,\eta)$-errors.
Some equalities from the case of discrete groups, we will here be replaced
with foliated bounds for the distance. This applies for instance to the map
$\mathit{FS}\to J_{\mathcal{V}}^{N}(G)^{\wedge}$.

Since $G$-action commutes with the flow, we get also a flow on
$G\backslash\mathit{FS}$.

######  Lemma 6.3.

  1. (i)

Let $\mathcal{U}$ be a $\alpha$-long cover of the flow space $\mathit{FS}$
such that each element $U\in\mathcal{U}$ is $G$-invariant. Then
$\mathcal{U}^{\prime}:=\\{p(U)\mid U\in\mathcal{U}\\}$ is a $\alpha$-long
covering of $G\backslash\mathit{FS}$;

  2. (ii)

Let $\mathcal{V}$ be a $\alpha$-long covering of $G\backslash\mathit{FS}$.
Then $\mathcal{V}^{\prime}=\\{p^{-1}(V)\mid V\in\mathcal{V}$ is a
$\alpha$-long cover of the flow space $\mathit{FS}$ such that each element
$V\in\mathcal{U}$ is $G$-invariant.

###### Proof.

This is obvious from the definitions. ∎

###  6.b. Partitions of unity for long thin covers

######  Proposition 6.4 (Partition of unity).

For all $\alpha>0$, ${\varepsilon}>0$, $N\in\mathbb{N}$ there is
$\alpha^{\prime}>0$ such that the following holds. Let $\mathcal{U}$ be a
$\alpha^{\prime}$-long cover of dimension $\leq N$ by $G$-invariant open
subsets of $\mathit{FS}$. Then there exists a partition of unity
$\\{t_{U}\colon\mathit{FS}\to[0,1]\mid U\in\mathcal{U}\\}$ subordinate to
$\mathcal{U}$ and $\delta>0$ such that

  1. (i)

for $U\in\mathcal{U}$, $c,c^{\prime}\in\mathit{FS}$ with
$d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime})<(\alpha,\delta)$ we have

$|t_{U}(c)-t_{U}(c^{\prime})|<{\varepsilon};$

  2. (ii)

the $t_{U}$ are $G$-invariant.

###### Proof.

This follows from Lemma 6.3 and from Proposition 7.1 applied to
$G\backslash\mathit{FS}$. ∎

######  Remark 6.5.

Using quantifiers the beginning of Proposition reads as

$\forall\alpha,\epsilon,N\;\exists\alpha^{\prime}\;\forall\,\mathcal{U}\;\exists\\{t_{U}\\},\delta\;\text{such
that}\ldots$

###  6.c. Dimension of long thin covers

######  Proposition 6.6 (Dimension of long thin covers).

There is $N\in\mathbb{N}$ such that for any $\alpha^{\prime}>0$ there is
$\alpha^{\prime\prime}$ such that the following is true. Let $\mathcal{W}$ be
an $\alpha^{\prime\prime}$-long cover of $\mathit{FS}$ by $G$-invariant open
subsets. Then there exist collections $\mathcal{U}_{0},\ldots,\mathcal{U}_{N}$
of open $G$-invariant subsets of $\mathit{FS}$ such that

  1. (i)

$\mathcal{U}:=\mathcal{U}_{0}\sqcup\ldots\sqcup\mathcal{U}_{N}$ is an
$\alpha^{\prime}$-long cover of $\mathit{FS}$, in particular
$\mathcal{U}_{i}\cap\mathcal{U}_{j}=\emptyset$ for $i\not=j$;

  2. (ii)

for each $i$ the open sets in $\mathcal{U}_{i}$ are pairwise disjoint;

  3. (iii)

for each $U\in\mathcal{U}=\mathcal{U}_{0}\sqcup\ldots\sqcup\mathcal{U}_{N}$
there is $W\in\mathcal{W}$ with $U\subseteq W$.

###### Proof.

This follows from Lemma 6.3 and Proposition 8.1 applied to
$G\backslash\mathit{FS}$. ∎

######  Remark 6.7.

Using quantifiers the beginning of Proposition 6.6 reads as

$\exists
N\;\forall\alpha^{\prime}\;\exists\alpha^{\prime\prime}\;\forall\mathcal{W}\;\exists\mathcal{U}_{0},\ldots,\mathcal{U}_{N}\;\text{such
that}\ldots$

###  6.d. The local structure of $\mathit{FS}$

######  Proposition 6.8 (Local structure).

Suppose that Assumption 2.7 holds.

For all $\alpha^{\prime\prime}>0$ there are $\beta>0$ and
$\mathcal{V}\subseteq{\mathcal{C}\mathrm{vcy}}$ finite with the following
property. For all $\eta>0$ and all $c_{0}\in\mathit{FS}$ there exist
$U\subseteq\mathit{FS}$ open, $h\colon U\to G$, $V\in\mathcal{V}$ and
$\delta^{\prime\prime}>0$ such that

  1. (i)

for some neighborhood $U_{0}$ of the orbit $Gc_{0}$ we have
$\Phi_{[-\alpha^{\prime\prime},\alpha^{\prime\prime}]}(U_{0})\subseteq U$;

  2. (ii)

$U$ is $G$-invariant;

  3. (iii)

for $c,c^{\prime}\in U$ we have

$d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime})<(\alpha^{\prime\prime},\delta^{\prime\prime})\implies
d_{V\text{-}\mathrm{fol}}(h(c),h(c^{\prime}))<(\beta,\eta);$

  4. (iv)

for $c\in U$, $g\in G$ we have;

$d_{V\text{-}\mathrm{fol}}(h(gc),gh(c))<(\beta,\eta)$

Proposition 6.8 is proven in Section 9.

######  Remark 6.9.

Using quantifiers the beginning of Proposition 6.8 reads as

$\forall\alpha^{\prime\prime}\;\exists\beta,\mathcal{V}\;\forall\eta,c_{0}\;\exists
U,h,\delta^{\prime\prime}\;\text{such that}\ldots$

######  Remark 6.10 (Failure of continuity).

The failure of continuity in Theorem 1.2 comes from the failure of continuity
of $h$ in Proposition 6.8. The action of $G$ on the flow space $\mathit{FS}$
is not free and so maps $h\colon U\to G$ defined on open subsets of the flow
space are necessarily a bit pathological. The action of $G$ on the flow space
is proper and it seems possible to construct maps $U\to G/K(V)$ that are
continuous where $K(V)$ is the maximal compact subgroup of $V$. It should then
be possible to obtain continuous maps in Theorem 1.2 after one replaces
$J_{\mathcal{V}}^{N}(G)^{\wedge}$ with
$\ast_{i=0}^{n}\big{(}\coprod_{V\in\mathcal{V}}G/K(V)\big{)}$. We will not
work this out in detail here.

###  6.e. Proof of Theorem 4.3 using Propositions 6.4, 6.6 and 6.8

###### Proof of Theorem 4.3.

We will use for $N$ the number appearing in Proposition 6.6. Given $\alpha>0$
and $\epsilon>0$ Proposition 6.4 gives us a number $\alpha^{\prime}$ and
Proposition 6.6 gives us then a number $\alpha^{\prime\prime}>0$. We can
assume $\alpha^{\prime\prime}\geq\alpha$. Next Proposition 6.8 gives us a
number $\beta>0$ and $\mathcal{V}\subseteq{\mathcal{C}\mathrm{vcy}}$ finite.
Let now $\eta>0$ be given. From Proposition 6.8 we get for every
$c_{0}\in\mathit{FS}$ an open subset $U(c_{0})\subseteq\mathit{FS}$, an open
neighborhood $U_{0}(c_{0})$ of the orbit $Gc_{0}$, a map $h(c_{0})\colon
U(c_{0})\to G$, an element $V(c_{0})\in\mathcal{V}$ and
$\delta^{\prime\prime}(c_{0})>0$ such that the assertions (i), (ii), (iii) and
(iv) hold. Since $G$ acts cocompactly on $\mathit{FS}$, we can find a finite
subset $I\subseteq\mathit{FS}$ such that $\mathit{FS}=\bigcup_{c_{0}\in
I}U_{0}(c_{0})$ holds. Define
$\delta^{\prime\prime}:=\min\\{\delta^{\prime\prime}(c_{0})\mid c_{0}\in I\\}$
and $\mathcal{W}=\\{U(c_{0})\mid c_{0}\in I\\}$. Then $\mathcal{W}$ an
$\alpha^{\prime\prime}$-long cover of $\mathit{FS}$ by $G$-invariant open
subsets and comes for every $W\in\mathcal{W}$ with maps $h_{W}\colon W\to G$
and an element $V_{W}\in\mathcal{V}$ satisfying

  1. (6.10a)

for $W\in\mathcal{W}$ and $c,c^{\prime}\in W$ we have

$d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime})<(\alpha^{\prime\prime},\delta^{\prime\prime})\implies
d_{V_{W}-{\mathrm{fol}}}(h_{W}(c),h_{W}(c^{\prime}))<(\beta,\eta);$

  2. (6.10b)

for $W\in\mathcal{W}$, $c\in W$ and $g\in G$ we have

$d_{V_{W}-{\mathrm{fol}}}(h_{W}(gc),gh_{W}(c))<(\beta,\eta).$

We apply Proposition 6.6 to the cover $\mathcal{W}$ and obtain collections
$\mathcal{U}_{0},\ldots,\mathcal{U}_{N}$ of open $G$-invariant subsets of
$\mathit{FS}$ such that

  1. (6.10c)

$\mathcal{U}:=\mathcal{U}_{0}\sqcup\ldots\sqcup\mathcal{U}_{N}$ is an
$\alpha^{\prime}$-long cover of $\mathit{FS}$;

  2. (6.10d)

for each $i$ the open sets in $\mathcal{U}_{i}$ are pairwise disjoint;

  3. (6.10e)

for each $U\in\mathcal{U}=\mathcal{U}_{0}\sqcup\ldots\sqcup\mathcal{U}_{N}$,
there is $W_{U}\in\mathcal{W}$ with $U\subseteq W_{U}$; we pick such a $W_{U}$
for each $U$ and set $V_{U}:=V_{W_{U}}$, $h_{U}:=h_{W_{U}}$.

Proposition 6.4 yields a $G$-invariant partition of unity
$\\{t_{U}\colon\mathit{FS}\to[0,1]\mid U\in\mathcal{U}\\}$ subordinate to
$\mathcal{U}$ and $\delta>0$, such that

  1. (6.10f)

for $U\in\mathcal{U}$, $c,c^{\prime}\in\mathit{FS}$ with
$d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime})<(\alpha,\delta)$ we have

$|t_{U}(c)-t_{U}(c^{\prime})|<{\epsilon}.$

We can assume $\delta\leq\delta^{\prime\prime}$. We now define
$f_{1}\colon\mathit{FS}\to J^{N}_{\mathcal{V}}(G)^{\wedge}$ by

$c\mapsto[t_{U_{0}}(c)(h_{{U_{0}}}(c),V_{{U_{0}}}),\ldots,t_{U_{N}}(c)(h_{U_{N}}(c),V_{{U_{N}}})],$

where for $i=0,\ldots,N$, $U_{i}\in\mathcal{U}_{i}$ is determined by $c\in
U_{i}$. There is at most one such $U_{i}$ by (6.10d). If there is no
$U_{i}\in\mathcal{U}_{i}$ containing $c$, then $t_{U}(c)=0$ for all
$U\in\mathcal{U}_{i}$ and the choice of $U_{i}$ is without effect. It remains
to verify the two assertions of Theorem 4.3.
(i) Let $c,c^{\prime}\in\mathit{FS}$ with
$d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime})<(\alpha,\delta)$. We write

$\displaystyle f_{1}(c)$ $\displaystyle=$
$\displaystyle[t_{U_{0}}(c)(h_{{U_{0}}}(c),V_{{U_{0}}}),\ldots,t_{U_{N}}(c)(h_{U_{N}}(c^{\prime}),V_{{U_{N}}})];$
$\displaystyle f_{1}(c^{\prime})$ $\displaystyle=$
$\displaystyle[t_{U^{\prime}_{0}}(c^{\prime})(h_{{U^{\prime}_{0}}}(c^{\prime}),V_{{U^{\prime}_{0}}}),\ldots,t_{U^{\prime}_{N}}(c^{\prime})(h_{U^{\prime}_{N}}(c^{\prime}),V_{{U^{\prime}_{N}}})].$

If $\max\\{t_{U_{i}}(c),t_{U^{\prime}_{i}}(c^{\prime})\\}\geq\epsilon$ then,
by (6.10f), we necessarily have $U_{i}=U^{\prime}_{i}$ and
$|t_{U_{i}}(c)-t_{U^{\prime}_{i}}(c^{\prime})|<\epsilon$. Moreover, by (6.10a)
$d_{V_{U_{i}}-{\mathrm{fol}}}(h_{U_{i}}(c),h_{U^{\prime}_{i}}(c^{\prime}))<(\beta,\eta)$.
If $\max\\{t_{U_{i}}(c),t_{U^{\prime}_{i}}(c^{\prime})\\}<\epsilon$, then
$|t_{U_{i}}(c)-t_{U^{\prime}_{i}}(c^{\prime})|<\epsilon$, as
$t_{U_{i}}(c),t_{U^{\prime}_{i}}(c^{\prime})\in[0,1]$. Thus
$d_{J\text{-}\mathrm{fol}}(f_{1}(c),f_{1}(c^{\prime}))<(\beta,\eta,\epsilon)$.
(ii) Let $c\in\mathit{FS}$ and $g\in G$. We write

$\displaystyle f_{1}(c)$ $\displaystyle=$
$\displaystyle[t_{U_{0}}(c)(h_{{U_{0}}}(c),V_{{U_{0}}}),\ldots,t_{U_{N}}(c)(h_{U_{N}}(c^{\prime}),V_{{U_{N}}})];$
$\displaystyle f_{1}(gc)$ $\displaystyle=$
$\displaystyle[t_{U^{\prime}_{0}}(gc)(h_{{U^{\prime}_{0}}},(gc)V_{{U^{\prime}_{0}}}),\ldots,t_{U^{\prime}_{N}}(gc)(h_{U^{\prime}_{N}}(gc),V_{{U^{\prime}_{N}}})].$

Then

$\displaystyle gf_{1}(c)$ $\displaystyle=$
$\displaystyle[t_{U_{0}}(c)(gh_{{U_{0}}}(c),V_{{U_{0}}}),\ldots,t_{U_{N}}(c)(gh_{U_{N}}(c),V_{{U^{\prime}_{N}}})].$

As the $U_{i}$ and the $t_{U_{i}}$ are all $G$-invariant we have
$U_{i}=U^{\prime}_{i}$, $t_{U_{i}}(c)=t_{U^{\prime}_{i}}(gc)$ for all $i$.
Moreover, by (6.10b),
$d_{V_{U_{i}}\text{-}\mathrm{fol}}(h_{U_{i}}(gc),gh_{U_{i}}(c))<(\beta,\eta)$.
In particular,
$d_{J\text{-}\mathrm{fol}}(f_{1}(gc),gf_{1}(c))<(\beta,\eta,\epsilon)$. ∎

In order to prove our main Theorem 1.2, it suffices to prove Propositions 6.4,
6.6 and 6.8. This we will do in the forthcoming sections.

##  7\. Partition of unity

In this section we finish the proof Proposition 6.4 by proving Proposition 7.1
below.

######  Proposition 7.1 (Partition of unity).

Let $Z$ be a compact metric space with a flow $\Phi$. For all $\alpha>0$,
$\epsilon>0$, $N\in\mathbb{N}$ there is $\widehat{\alpha}>0$ such that the
following holds. Let $\mathcal{U}$ be a $\widehat{\alpha}$-long cover of $Z$
of dimension $\leq N$ by open subsets. Then there exists a partition of unity
$\\{t_{U}\colon Z\to[0,1]\mid U\in\mathcal{U}\\}$ subordinate to $\mathcal{U}$
and $\delta>0$ such that for $U\in\mathcal{U}$, $z,z^{\prime}\in Z$ with
$d_{\mathit{FS}\text{-}\mathrm{fol}}(z,z^{\prime})<(\alpha,\delta)$ we have

$|t_{U}(z)-t_{U}(z^{\prime})|<{\varepsilon}.$

######  Remark 7.2.

Using quantifiers the beginning of Proposition 7.1 reads as

$\forall\alpha,\epsilon,N\;\exists\widehat{\alpha}\;\forall\,\mathcal{U}\;\exists\\{t_{U}\\},\delta>0\;\forall\,U,z,z^{\prime}\;\text{we
have}\ldots$

######  Lemma 7.3.

Let $\widehat{\alpha}>0$. Let $K\subseteq Z$ be compact and $U$ be an open
neighborhood of $K$. Then there exists a (continuous) map $f\colon Z\to[0,1]$
satisfying

  1. (i)

$f|_{K}\equiv 1$;

  2. (ii)

$f|_{Z\setminus\Phi_{[-\widehat{\alpha},\widehat{\alpha}]}U}\equiv 0$;

  3. (iii)

for $s\in\mathbb{R}$, $z\in Z$ we have
$|f(\Phi_{s}(z))-f(z)|\leq\frac{|s|}{\widehat{\alpha}}$.

###### Proof.

As $Z$ is a metric space, there is $\varphi\colon Z\to[0,1]$ with
$\varphi|_{K}\equiv 1$ and $\varphi|_{Z\setminus U}\equiv 0$. Define $F\colon
Z\times\mathbb{R}\to[0,1]$ by

$F(z,t):=\begin{cases}(1-\frac{|t|}{\widehat{\alpha}})\varphi(\Phi_{t}(z))&t\in[-\widehat{\alpha},\widehat{\alpha}];\\\
0&\text{else}.\end{cases}$

Put

$f(z):=\sup\\{F(z,t)\mid t\in\mathbb{R}\\}=\max\\{F(z,t)\mid
t\in[-\widehat{\alpha},\widehat{\alpha}]\\}.$

We verify that $f$ is continuous. Assume it is not. Then there exist
$\epsilon>0$, $z\in Z$, and a sequence $(z_{n})_{n\geq 0}$ in $Z$ such that
$z_{n}\to z$ in $Z$ and $|f(z_{n})-f(z)|>\epsilon$ holds for all $n\geq 0$.
Choose $\tau_{n}\in[-\widehat{\alpha},\widehat{\alpha}]$ with
$f(z_{n})=F(z_{n},\tau_{n})$. We can arrange after passing to subsequences
that there exists $\tau\in[-\widehat{\alpha},\widehat{\alpha}]$ satisfying
$\lim_{n\to\infty}\tau_{n}=\tau$. Then $f(z_{n})=F(z_{n},\tau_{n})\to
F(z,\tau)\leq f(z)$. Hence there is a natural number $N$ such that
$f(z_{n})<f(z)+\epsilon$ holds for $n\geq N$. On the other hand, choose
$\tau^{\prime}\in[-\widehat{\alpha},\widehat{\alpha}]$ with
$f(z)=F(z,\tau^{\prime})$. Then $f(z_{n})\geq F(z_{n},\tau^{\prime})\to f(z)$
and hence there is a natural number $N^{\prime}$ such that
$f(z_{n})>f(z)-\epsilon$ for $n\geq N^{\prime}$. This is a contradiction.

It remains to check (i), (ii) and (iii).
(i) Let $z\in K$. Then $F(z,0)=\varphi(z)=1$. Thus $f(z)=1$.
(ii) Let $z\in Z\setminus\Phi_{[-\widehat{\alpha},\widehat{\alpha}]}(U)$. Then
$\Phi_{t}(z)\not\in U$ for all $t\in[-\widehat{\alpha},\widehat{\alpha}]$.
Thus $F(z,t)=0$ for all $t\in[-\widehat{\alpha},\widehat{\alpha}]$. Therefore
$f(z)=0$.
(iii) Next we show for all $s,t\in\mathbb{R}$ and $z\in Z$.

(7.4) $\Big{|}F(\Phi_{s}(z),t)-F(z,t)\big{|}\leq\frac{|s|}{\widehat{\alpha}}.$

If $s$ and $s+t$ belong to $[-\widehat{\alpha},\widehat{\alpha}]$, this
follows from

$\displaystyle\bigl{|}F(\Phi_{s}(z),t)-F(z,t+s)\bigr{|}$ $\displaystyle=$
$\displaystyle\left|(1-\frac{|t|}{\widehat{\alpha}})\varphi(\Phi_{t}(\Phi_{s}(z)))-(1-\frac{|t+s|}{\widehat{\alpha}})\varphi(\Phi_{t+s}(z))\right|$
$\displaystyle=$
$\displaystyle\left|(1-\frac{|t|}{\widehat{\alpha}})\varphi(\Phi_{t+s}(z))-(1-\frac{|t+s|}{\widehat{\alpha}})\varphi(\Phi_{t+s}(z))\right|$
$\displaystyle=$
$\displaystyle\left|\frac{|t|-|t+s|}{\widehat{\alpha}}\varphi(\Phi_{t+s}(z))\right|$
$\displaystyle=$
$\displaystyle\frac{\bigl{|}|t|-|t+s|\bigr{|}}{\widehat{\alpha}}\varphi(\Phi_{t+s}(z))$
$\displaystyle\leq$
$\displaystyle\frac{\bigl{|}|t|-|t+s|\bigr{|}}{\widehat{\alpha}}$
$\displaystyle\leq$ $\displaystyle\frac{|s|}{\widehat{\alpha}}.$

Suppose that $s\in[-\widehat{\alpha},\widehat{\alpha}]$ and
$(s+t)\notin[-\widehat{\alpha},\widehat{\alpha}]$. Then $F(z,t+s)=0$ and
$\widehat{\alpha}\leq|t+s|\leq|s|+[t]$. Now(7.4) follows from

$\bigl{|}F(\Phi_{s}(z),t)|=(1-\frac{|t|}{\widehat{\alpha}})\varphi(\Phi_{t}(z))\leq\frac{\widehat{\alpha}-|t|}{\widehat{\alpha}}\leq\frac{|s|}{\widehat{\alpha}}.$

Suppose that $(s+t)\in[-\widehat{\alpha},\widehat{\alpha}]$ and
$t\notin[-\widehat{\alpha},\widehat{\alpha}]$. Then $F(\Phi_{s}(z),t)=0$ and
$\widehat{\alpha}\leq|t|=|(s+t)-s|\leq|s+t|+|s|$. Now (7.4) follows from

$\bigl{|}F((z),s+t)|=(1-\frac{|s+t|}{\widehat{\alpha}})\varphi(z)\leq\frac{\widehat{\alpha}-|s+t|}{\widehat{\alpha}}\leq\frac{|s|}{\widehat{\alpha}}.$

If $(s+t)\notin[-\widehat{\alpha},\widehat{\alpha}]$ and
$t\notin[-\widehat{\alpha},\widehat{\alpha}]$, then
$F(\Phi_{s}(z),t)=F(z,t+s)=0$ and hence (7.4) is true. This finishes the proof
of (7.4).

From the definitions we conclude that there exists $t_{0}$ and
$t_{1}\in\mathbb{R}$ such that for all $t\in\mathbb{R}$ we have

(7.5) $\displaystyle f(\Phi_{s}(z))$ $\displaystyle=$ $\displaystyle
F(\Phi_{s}(z),t_{0});$ (7.6) $\displaystyle F(\Phi_{s}(z),t)$
$\displaystyle\leq$ $\displaystyle F(\Phi_{s}(z),t_{0});$ (7.7) $\displaystyle
f(z)$ $\displaystyle=$ $\displaystyle F(z,t_{1});$ (7.8) $\displaystyle
F(z,t)$ $\displaystyle\leq$ $\displaystyle F(z,t_{1}).$

We estimate

$f(z)\stackrel{{\scriptstyle\eqref{lem:slow-bump-
fct:auxiliary_4}}}{{=}}F(z,t_{1})\stackrel{{\scriptstyle\eqref{lem:slow-bump-
fct:auxiliary_1}}}{{\leq}}F(\Phi_{s}(z),t_{1}-s)+\frac{|s|}{\widehat{\alpha}}\stackrel{{\scriptstyle\eqref{lem:slow-
bump-
fct:auxiliary_3}}}{{\leq}}F(\Phi_{s}(z),t_{0})+\frac{|s|}{\widehat{\alpha}}\stackrel{{\scriptstyle\eqref{lem:slow-
bump-fct:auxiliary_2}}}{{=}}f(\Phi_{s}(z))+\frac{|s|}{\widehat{\alpha}},$

and

$f(\Phi_{s}(z))\stackrel{{\scriptstyle\eqref{lem:slow-bump-
fct:auxiliary_2}}}{{=}}F(\Phi_{s}(z),t_{0})\stackrel{{\scriptstyle\eqref{lem:slow-
bump-
fct:auxiliary_1}}}{{\leq}}F(z,t_{0}+s)+\frac{|s|}{\widehat{\alpha}}\stackrel{{\scriptstyle\eqref{lem:slow-
bump-
fct:auxiliary_5}}}{{\leq}}F(z,t_{1})+\frac{|s|}{\widehat{\alpha}}\stackrel{{\scriptstyle\eqref{lem:slow-
bump-fct:auxiliary_4}}}{{\leq}}f(z)+\frac{|s|}{\widehat{\alpha}}.$

This finishes the proof of Lemma 7.3. ∎

###### Proof of Proposition 7.1.

Let $\alpha,\epsilon$ and $N$ be given. Pick $\widehat{\alpha}>0$ such that

$\frac{(2N+3)\alpha}{\widehat{\alpha}}<\epsilon/2.$

For $U\in\mathcal{U}$ let $U^{-\widehat{\alpha}}:=\\{z\in
Z\mid\Phi_{[-\widehat{\alpha},\widehat{\alpha}]}(z)\subseteq U\\}$. As
$\mathcal{U}$ is $\widehat{\alpha}$-long, we can find for every $z\in Z$ an
open neighborhood $U_{0}(z)$ and an element $U(z)\in\mathcal{U}$ such that
$U_{0}(z)\subseteq U^{-\widehat{\alpha}}$. Since $Z$ is compact, we can find a
finite subset $I\subseteq Z$ such that $Z=\bigcup_{z\in Z}U_{0}(z)$. By
replacing $\mathcal{U}$ with $\\{U(z)\mid z\in I\\}$, we can arrange that both
$\mathcal{U}$ and
$\mathcal{U}^{-\widehat{\alpha}}:=\\{U^{-\widehat{\alpha}}\mid
U\in\mathcal{U}\\}$ are $N$-dimensional finite coverings of $Z$. As $Z$ is
compact we can find a Lebesgue number $\ell>0$ for $\mathcal{U}$. Define a
compact subset $K_{U}\subseteq U$ by $K_{U}=\\{z\in Z\mid d_{Z}(z,Z\setminus
U)\geq\ell\\}$. Then $\\{K_{U}\mid U\in\mathcal{U}\\}$ covers $Z$. For each
$U\in\mathcal{U}$ we now choose $f_{U}$ as in Lemma 7.3, i.e., such that

  1. (7.9a)

$f_{U}|_{K_{U}}\equiv 1$;

  2. (7.9b)

$f_{U}|_{Z\setminus U}\equiv 0$;

  3. (7.9c)

for $\tau\in\mathbb{R}$, $z\in Z$ we have
$|f_{U}(z)-f_{U}(\Phi_{\tau}(z))|\leq\frac{|\tau|}{\widehat{\alpha}}$.

As $\mathcal{U}$ is finite, we can normalize the $f_{U}$ to obtain
$t_{U}\colon Z\to[0,1]$ with

$t_{U}(z):=\frac{f_{U}(z)}{\sum_{U^{\prime}\in\mathcal{U}}f_{U^{\prime}}(z)}.$

Then $\\{t_{U}\mid U\in\mathcal{U}\\}$ is a partition of unity subordinate to
$\mathcal{U}$. For $\tau\in[-\alpha,\alpha]$ and $z\in Z$, we next want to
estimate $t_{U}(z)-t_{U}(\Phi_{\tau}(z))$. We abbreviate $x_{V}:=f_{V}(z)$,
$x^{\prime}_{V}:=f_{V}(\Phi_{\tau}(u))$ for $V\in\mathcal{U}$. By (7.9a)
$x_{V}=1$, $x^{\prime}_{V^{\prime}}=1$ for at least one $V,V^{\prime}$. In
particular $\sum_{V}x_{V},\sum_{V}x^{\prime}_{V}\geq 1$. By (7.9b) and since
the dimension of $\mathcal{U}$ is at most $N$, we have $x_{V}\neq 0$ for at
most $N+1$ different $V\in\mathcal{V}$, and similarly for $x^{\prime}_{V}$. By
(7.9c) $|x_{V}-x^{\prime}_{V}|\leq\frac{\tau}{\widehat{\alpha}}$. Using all
this we compute for $\tau\in[-\alpha,\alpha]$ and $z\in Z$, where $V$ and
$V^{\prime}$ run through $\mathcal{U}$:

$\displaystyle|t_{U}(z)-t_{U}(\Phi_{\tau}(z))|$ $\displaystyle=$
$\displaystyle\Bigg{|}\frac{x_{U}}{\sum_{V}x_{V}}-\frac{x^{\prime}_{U}}{\sum_{V^{\prime}}x^{\prime}_{V^{\prime}}}\Bigg{|}$
$\displaystyle=$
$\displaystyle\Bigg{|}\frac{\sum_{V^{\prime}}x_{U}x^{\prime}_{V^{\prime}}}{\sum_{V,V^{\prime}}x_{V}x^{\prime}_{V^{\prime}}}-\frac{\sum_{V}x^{\prime}_{U}x_{V}}{\sum_{V,V^{\prime}}x_{V}x^{\prime}_{V^{\prime}}}\Bigg{|}$
$\displaystyle\leq$
$\displaystyle\frac{\sum_{V}\big{|}x_{U}x^{\prime}_{V}-x_{U}x_{V}+x_{U}x_{V}-x^{\prime}_{U}x_{V}\big{|}}{\sum_{V,V^{\prime}}x_{V}x^{\prime}_{V^{\prime}}}$
$\displaystyle\leq$
$\displaystyle\frac{\sum_{V}x_{U}\big{|}x^{\prime}_{V}-x_{V}\big{|}+\big{|}x_{U}-x^{\prime}_{U}\big{|}x_{V}}{\sum_{V,V^{\prime}}x_{V}x^{\prime}_{V^{\prime}}}$
$\displaystyle\leq$
$\displaystyle\frac{\sum_{V}\big{|}x_{V}-x^{\prime}_{V}\big{|}}{\sum_{V,V^{\prime}}x_{V}x^{\prime}_{V^{\prime}}}+\frac{\big{|}x_{U}-x^{\prime}_{U}\big{|}}{\sum_{V^{\prime}}x^{\prime}_{V^{\prime}}}$
$\displaystyle\leq$
$\displaystyle{\sum_{V}\big{|}x_{V}-x^{\prime}_{V}\big{|}}+{\big{|}x_{U}-x^{\prime}_{U}\big{|}}$
$\displaystyle\leq$ $\displaystyle\bigl{(}\bigl{|}\\{V\in\mathcal{U}\mid
x_{V}\not=0\\}|+\bigl{|}\\{V\in\mathcal{U}\mid
x^{\prime}_{V}\not=0\\}|+1\bigr{)}\cdot\max\\{\big{|}x_{V}-x^{\prime}_{V}\big{|}\mid
V,V^{\prime}\in\mathcal{U}\\}$ $\displaystyle\leq$
$\displaystyle(2(N+1)+1)\frac{\tau}{\widehat{\alpha}}<\frac{\epsilon}{2}.$

As $Z$ is compact the $t_{U}$ are uniformly continuous. Since $\mathcal{U}$ is
finite, there is $\delta>0$ such that for $U\in\mathcal{U}$, $z,z^{\prime}\in
Z$, we have

$d_{Z}(z,z^{\prime})<\delta\;\implies\;|t_{U}(z)-t_{U}(z^{\prime})|<\frac{\epsilon}{2}.$

Thus

$d_{\mathit{FS}\text{-}\mathrm{fol}}(z,z^{\prime})<(\alpha,\delta)\;\implies\;|t_{U}(z)-t_{U}(z^{\prime})|<\epsilon.\qed$

##  8\. Dimension of long thin covers

In this section we finish the proof of Proposition 6.6 by proving Proposition
8.1 below.

######  Proposition 8.1.

There is $N$ such that for any $\alpha>0$ there is $\widehat{\alpha}>0$ such
that the following is true. Let $\mathcal{W}$ be an $\widehat{\alpha}$-long
cover of $G\backslash\mathit{FS}$ by open subsets. Then there exists
collections $\mathcal{U}_{0},\ldots,\mathcal{U}_{N}$ of open subsets of
$G\backslash\mathit{FS}$ such that

  1. (i)

$\mathcal{U}:=\mathcal{U}_{0}\sqcup\ldots\sqcup\mathcal{U}_{N}$ is an
$\alpha$-long cover of $G\backslash\mathit{FS}$;

  2. (ii)

for each $i$ the open sets in $\mathcal{U}_{i}$ are pairwise disjoint;

  3. (iii)

for each $U\in\mathcal{U}=\mathcal{U}_{0}\sqcup\ldots\sqcup\mathcal{U}_{N}$
there is $W\in\mathcal{W}$ with $U\subseteq W$.

######  Remark 8.2.

Using quantifiers the beginning of Proposition 8.1 reads as

$\exists
N\;\forall\alpha\;\exists\widehat{\alpha}\;\forall\mathcal{W}\;\exists\mathcal{U}_{0},\ldots,\mathcal{U}_{N}\;\text{such
that}\ldots$

###### Proof of Proposition 8.1.

This follows by combining Proposition 8.4 and Lemma 8.5 below. ∎

######  Lemma 8.3.

$G\backslash\mathit{FS}$ is of finite dimension, compact and metrizable.

###### Proof.

Recall that $G$ acts on $\mathit{FS}$ cocompactly, isometrically, and properly
and that $\mathit{FS}$ is a proper metric space, see Lemmas 2.2, and Lemma
2.3. Hence $G\backslash\mathit{FS}$ is compact and metrizable. A formula for a
metric is

$d_{G\backslash\mathit{FS}}([c],[c^{\prime}])=\min\\{d_{\mathit{FS}}(gc,c^{\prime})\mid
g\in G\\}.$

By [3, Prop. 2.9] the dimension of
$\mathit{FS}\setminus\mathit{FS}^{\mathbb{R}}$ is finite. As
$\mathit{FS}^{\mathbb{R}}\cong X$ is also finite dimensional, the sum theorem
from dimension theory [12, Cor. 1.5.5] now implies that $\mathit{FS}$ is
finite dimensional. Another result from dimension theory [12, Thm. 1.12.7]
asserts that for open maps $A\to B$ with discrete fibers the dimension of $A$
and $B$ agree555Strictly speaking the two results cited from [12] are about
inductive dimension, not covering dimension. However, it is not difficult to
check that $\mathit{FS}$ is separable, so there is no difference between
covering dimension and inductive dimension [12, Thm. 1.7.7].. The quotient map
$\mathit{FS}\to G\backslash\mathit{FS}$ is open, but as the action of $G$ on
$\mathit{FS}$ is not smooth, the fibers of the quotient map are not discrete.
For $R>0$ let $\mathit{FS}_{R}$ be the subspace of $\mathit{FS}$ consisting of
all generalized geodesics $c\colon\mathbb{R}\to X$ that are locally constant
on the complement of $[-R,R]$. As $\mathit{FS}_{R}$ is a closed subspace of
$\mathit{FS}$ we have $\dim\mathit{FS}_{R}\leq\dim\mathit{FS}$. The action of
$G$ on $\mathit{FS}_{R}$ is smooth, so $\mathit{FS}_{R}\to
G\backslash\mathit{FS}_{R}$ has discrete fibers and $\dim
G\backslash\mathit{FS}_{R}=\dim\mathit{FS}_{R}\leq\dim\mathit{FS}$ is finite.
There is a canonical retract $p_{R}\colon\mathit{FS}\to\mathit{FS}_{R}$ that
sends $c$ to the restriction of $c$ to $[-R,R]$, more precisely to the
generalized geodesic that agrees with $c$ on $[-R,R]$ and is locally constant
on the complement. It is not difficult to check that the fibers of $p_{R}$ are
of uniformly bounded diameter $\epsilon_{R}$ with $\epsilon_{R}\to 0$ as
$R\to\infty$. The fibers of the induced map $\overline{p}_{R}\colon
G\backslash\mathit{FS}\to G\backslash\mathit{FS}_{R}$ have the same property.
Write $\mathcal{W}_{R,\delta}$ for the open cover of
$G\backslash\mathit{FS}_{R}$ by all balls of radius $\delta$. We can refine
$\mathcal{W}_{R,\delta}$ to an open cover $\mathcal{W}^{\prime}_{R,\delta}$ of
dimension $\leq\dim\mathit{FS}$, i.e., every point of
$G\backslash\mathit{FS}_{R}$ is contained in at most $\dim\mathit{FS}+1$ sets
in $\mathcal{W}^{\prime}_{R,\delta}$. Now let $\mathcal{U}$ an open cover of
$G\backslash\mathit{FS}$. By compactness $\mathcal{U}$ has a positive Lebesgue
number. We then find $\delta>0$ and $R>0$ such that the pull-back
$\overline{p}_{R}^{*}(\mathcal{W}_{R,\delta})$ refines $\mathcal{U}$. It
follows that $\overline{p}_{R}^{*}(\mathcal{W}^{\prime}_{R,\delta})$ is a
refinement of $\mathcal{U}$ of dimension $\leq\dim\mathit{FS}$. Thus $\dim
G\backslash\mathit{FS}\leq\dim\mathit{FS}<\infty$. ∎

######  Proposition 8.4.

There is $N$ such that for any $\beta>0$ there is $\widehat{\alpha}>0$ such
that the following is true. Let $\mathcal{W}$ be an $\widehat{\alpha}$-long
cover of $G\backslash\mathit{FS}$ by open subsets. Then there exists an open
cover $\mathcal{V}$ of $G\backslash\mathit{FS}$ such that

  1. (i)

$\mathcal{V}$ is an $\beta$-long;

  2. (ii)

$\dim\mathcal{V}\leq N$;

  3. (iii)

for each $U\in\mathcal{V}$ there is $W\in\mathcal{W}$ with $U\subseteq W$.

###### Proof.

The main result from [16] almost gives this. More precisely Lemma 8.3 allows
us to apply [16, Thm. 1.1] to $G\backslash\mathit{FS}$. Thus there exists $N$
only depending on the dimension of $G\backslash\mathit{FS}$ such that for
given $\beta>0$ there exists a cover $\mathcal{V}$ of $G\backslash\mathit{FS}$
satisfying (i) and (ii).

We will argue below that the construction from [16] in fact also gives (iii).

We point out that we apply [16] to the quotient $G\backslash\mathit{FS}$ which
no longer carries a group action. In [16] an equivariant situation is
considered, but we use the special case of [16] where the group acting on the
flow space is trivial.

Given $\beta>0$, let $\hat{\alpha}:=20\beta$. Suppose that $\mathcal{W}$ is an
$\hat{\alpha}$-long cover. Then for any $c\in G\backslash\mathit{FS}$ there is
$W\in\mathcal{W}$ with $\Phi_{[-\hat{\alpha},\hat{\alpha}]}(c)\subseteq W$. As
$W$ is open there is $\delta>0$ such that the $\delta$-neighborhood of
$\Phi_{[-\hat{\alpha},\hat{\alpha}]}(c)$ is still contained in $W$. As
$G\backslash\mathit{FS}$ is compact we can choose $\delta>0$ uniformly, that
is for any $c\in G\backslash\mathit{FS}$ there is $W\in\mathcal{W}$ containing
the $\delta$-neighborhood of $\Phi_{[-\hat{\alpha},\hat{\alpha}]}(c)$.

The period of $c\in G\backslash\mathit{FS}$ is
$\tau(c):=\inf\\{t>0\mid\Phi_{t}(c)=c\\}$. If $\Phi_{t}(c)\neq c$ for all
$t>0$, then $\tau(c)=\infty$. Let

$\big{(}G\backslash\mathit{FS}\big{)}_{>\hat{\alpha}}:=\\{c\in
G\backslash\mathit{FS}\mid\tau(c)\in(\hat{\alpha},\infty)\\}.$

Let $\delta>0$ be given. By [16, Thm. 5.3] there exists an $\beta$-long cover
$\mathcal{V}_{1}$ of $\big{(}G\backslash\mathit{FS}\big{)}_{>\hat{\alpha}}$ of
dimension $\leq N_{1}$, where $N_{1}$ depends only on the dimension of
$G\backslash\mathit{FS}$. Moreover, as explained in the last line of the proof
of [16, Thm. 5.3], every $V\in\mathcal{V}_{1}$ is contained in the
$\delta$-neighborhood of $\Phi_{[\hat{\alpha},\hat{\alpha}]}(c)$ for some $c$,
and therefore in some $W\in\mathcal{W}$.

Next consider

$\big{(}G\backslash\mathit{FS}\big{)}_{\leq\hat{\alpha}}:=\\{c\in
G\backslash\mathit{FS}\mid\tau(c)\in[0,\hat{\alpha}]\\}.$

By [16, Lem. 7.6] there exists an open cover $\mathcal{V}_{2}$ of
$\big{(}G\backslash\mathit{FS}\big{)}_{\leq\hat{\alpha}}$ of dimension $\leq
N_{2}$, where $N_{2}$ again depends only on the dimension of of
$G\backslash\mathit{FS}$. Moreover, for each $c$ of period $\leq\hat{\alpha}$
there is $V\in\mathcal{U}_{2}$ with $\Phi_{\mathbb{R}}(c)\subseteq V$, and
each $V\in\mathcal{V}_{2}$ is contained in the $\delta$-neighborhood of
$\Phi_{\mathbb{R}}(c)$. In fact, by construction, see the last line of the
proof of [16, Lem. 7.6], $c$ can here be chosen to be of period
$\leq\hat{\alpha}$. In particular, $V$ is contained in the
$\delta$-neighborhood of $\Phi_{[0,\hat{\alpha}]}(c)=\Phi_{\mathbb{R}}(c)$,
and therefore in some $W\in\mathcal{W}$.

We have
$G\backslash\mathit{FS}=\big{(}G\backslash\mathit{FS}\big{)}_{\leq\hat{\alpha}}\sqcup\big{(}G\backslash\mathit{FS}\big{)}_{>\hat{\alpha}}$,
where the first set if closed by [16, Lem. 7.1] and the second consequently
open. In particular, the $V\in\mathcal{V}_{1}$ are open in
$G\backslash\mathit{FS}$. We can use [16, Lem. 2.7] to extend the
$V\in\mathcal{V}_{2}$ to open subsets of $G\backslash\mathit{FS}$ while
preserving the properties of $\mathcal{V}_{2}$. The union of the two covers is
now the needed cover $\mathcal{V}$. ∎

######  Lemma 8.5.

Fix a number $N$. For any $\alpha>0$ there is $\beta>0$ with following
property. Let $\mathcal{V}$ be an $\beta$-long cover of
$G\backslash\mathit{FS}$. Assume that $\dim\mathcal{V}\leq N$. Then there
exists collections $\mathcal{U}_{0},\ldots,\mathcal{U}_{N}$ of open subsets of
$G\backslash\mathit{FS}$ such that

  1. (i)

$\mathcal{U}:=\mathcal{U}_{0}\sqcup\ldots\sqcup\mathcal{U}_{N}$ is an
$\alpha$-long cover of $G\backslash\mathit{FS}$;

  2. (ii)

for each $i$ the open sets in $\mathcal{U}_{i}$ are pairwise disjoint;

  3. (iii)

for each $U\in\mathcal{U}=\mathcal{U}_{0}\sqcup\ldots\sqcup\mathcal{U}_{N}$
there is $V\in\mathcal{V}$ with $U\subseteq V$.

The proof is not difficult. We translate between open covers and maps to
simplicial complexes and for the latter we use barycentric subdivision.

###### Proof of Lemma 8.5.

The metric on $\mathit{FS}$ has the property that
$d_{\mathit{FS}}(\Phi_{t}(c),c)\leq|t|$ for all $c\in\mathit{FS}$ and
$t\in\mathbb{R}$. It is not difficult to check that there is metric
$d_{G\backslash\mathit{FS}}$ with the same property. For $\lambda>0$ we define
a metric $d_{\lambda}$ on $G\backslash\mathit{FS}$ as follows. For
$c,c^{\prime}\in G\backslash\mathit{FS}$ set

$d_{\lambda}(c,c^{\prime}):=\inf\sum_{i=0}^{n}|t_{i}|+\lambda
d_{G\backslash\mathit{FS}}(\Phi_{t_{i}}(c_{i}),c_{i+1})$

where the infimum is taken over all finite sequences
$c=c_{0},\dots,c_{n+1}=c^{\prime}$, $t_{0},\dots,t_{n}\in\mathbb{R}$.
Compactness of $G\backslash\mathit{FS}$ can be used to check that for an
$\beta$-long cover $\mathcal{V}$ there is $\lambda>0$ such that the Lebesgue
number of $\mathcal{V}$ with respect to $d_{\lambda}$ is $\geq\beta$. Let now
$\Lambda$ be the nerve of $\mathcal{V}$, i.e., the simplicial complex that has
a vertex $v_{V}$ for each $V\in\mathcal{V}$ and where
$v_{V_{0}},\dots,v_{V_{n}}$ span a simplex iff
$V_{0}\cap\dots\cap{V_{n}}\neq\emptyset$. The dimension of $\mathcal{V}$ is
exactly the dimension of $\Lambda$. We equip $|\Lambda|$ with the
$l^{1}$-metric $d^{1}$. There is now a map $f\colon
G\backslash\mathit{FS}\to|\Lambda|$ satisfying

(8.6) $d_{\lambda}(c,c^{\prime})\leq\frac{\beta}{4N}\implies
d^{1}(f(c),f(c^{\prime}))\leq\frac{16N^{2}}{\beta}d_{\lambda}(c,c^{\prime}),$

see [7, Prop. 5.3]. By its construction the map $f$ has the following
property: the preimage of the open star of $v_{U}$ is exactly $U$. Let now
$\Lambda^{\prime}$ be the barycentric subdivision of $\Lambda$. The vertices
of $\Lambda^{\prime}$ correspond to the simplices of $\Lambda$. For
$j=0,\dots,N$ let $I_{j}$ be the set of simplices of $\Lambda^{\prime}$ to
$j$-simplices of $\Lambda$ and $\tilde{\mathcal{U}}_{j}$ be the collection of
open stars around simplices in $I_{j}$. Then
$\tilde{\mathcal{U}}:=\tilde{\mathcal{U}}_{0}\sqcup\dots\sqcup\tilde{\mathcal{U}}_{N}$
is an open cover $|\Lambda|=|\Lambda^{\prime}|$ of positive Lebesgue number
$L$, where $L$ depends only on the dimension of $\Lambda$. Moreover, for each
$j$ the open sets in $\tilde{\mathcal{U}}_{j}$ are pairwise disjoint. We now
set $\mathcal{U}_{j}:=f^{*}\tilde{\mathcal{U}}_{j}$ and
$\mathcal{U}:=f^{*}\mathcal{U}$. Using estimate 8.6 we see that the Lebesgue
number of $\mathcal{U}$ with respect to $d_{\Lambda}$ is at least
$\min\\{\frac{\beta}{4N},\frac{L\beta}{16N^{2}}\\}$. In particular, if we
choose $\beta\geq\max\\{4N\alpha,\frac{16N^{2}\alpha}{L}\\}$, then the
Lebesgue number of $\mathcal{U}$ with respect to $d_{\lambda}$ is at least
$\alpha$. Thus $\mathcal{U}$ is $\alpha$-long. Finally, each open star for
$\Lambda^{\prime}$ is contained in an open star for $\Lambda$. Thus each $U$
from $\mathcal{U}$ is contained in some $V$ from $\mathcal{V}$. ∎

######  Remark 8.7.

A more careful analysis of the arguments from [16] reveals that the
constructions there also lead to coloured covers. This leads to a more direct
proof of Proposition 8.1 and renders Lemma 8.5 superfluous.

##  9\. Local structure

In section we will prove Proposition 6.8.

###  9.a. Neighborhoods in $\mathit{FS}$ mapping to $G$

######  Lemma 9.1.

Let $\mathit{FS}_{0}\subseteq\mathit{FS}$ be compact. For all $\alpha>0$ there
is $\beta>0$ such that for $g\in G$, $c\in\mathit{FS}_{0}$ we have

$d_{\mathit{FS}}(gc,c)<\alpha\quad\implies\quad d_{G}(g,e)<\beta.$

###### Proof.

Assume this fails for a given $\alpha>0$. Then there are sequences
$(c_{n})_{n\geq 0}$ in $\mathit{FS}_{0}$, and $(g_{n})_{n\geq 0}$ in $G$ with
$d_{\mathit{FS}}(g_{n}c_{n},c_{n})<\alpha$ but

$\lim_{n\to\infty}d_{G}(g_{n},e)=\infty.$

After passing to a subsequence, we can assume $c_{0}=\lim_{n\to\infty}c_{n}$
for some $c_{0}\in\mathit{FS}_{0}$. We can choose a constant $C>0$ such that
$d_{\mathit{FS}}(c_{n},c_{0})=d_{\mathit{FS}}(g_{n}c_{n},g_{n}c_{0})\leq C$
for $n\geq 0$. Then the $g_{n}c_{0}$ are elements of the closed ball $K$ of
radius $\alpha+2C$ around $c_{0}$. Since $\mathit{FS}$ is a proper metric
space by Lemma 2.2, $K$ is compact. The set $\\{g\in G\mid g\cdot K\cap
K\not=\emptyset\\}$ is a compact subset of $G$ by Lemma B.1 (iii) and contains
the sequence $(g_{n})_{n\geq 0}$. After passing to subsequences again, we can
assume $\lim_{n\to\infty}g_{n}=g$ for some $g\in G$. Hence
$\lim_{n\to\infty}d_{G}(g_{n},e)=d_{G}(g,e)$. This contradicts
$\lim_{n\to\infty}d_{G}(g_{n},e)=\infty$. ∎

We have defined $U^{{\mathrm{fol}}}_{\alpha,\delta}(c_{0})$ in Subsection 2.c
and $V_{c_{0}}$ in (2.6b).

######  Proposition 9.2.

Let $\mathit{FS}_{0}\subseteq\mathit{FS}$ be compact. For all $\alpha>0$ there
is $\beta>0$ such that the following is true: For all $\eta>0$,
$c_{0}\in\mathit{FS}_{0}$, there are $\delta>0$ and a (not necessarily
continuous) map $h\colon G\cdot U^{{\mathrm{fol}}}_{\alpha,\delta}(c_{0})\to
G$ satisfying

  1. (i)

for $c,c^{\prime}\in G\cdot U^{{\mathrm{fol}}}_{\alpha,\delta}(c_{0})$ we have

$d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime})<(\alpha,\delta)\quad\implies\quad
d_{V_{c_{0}}-{\mathrm{fol}}}(h(c),h(c^{\prime}))<(\beta,\eta);$

  2. (ii)

for $g\in G$, $c\in G\cdot U^{{\mathrm{fol}}}_{\alpha,\delta}(c_{0})$ we have

$d_{V_{c_{0}}-{\mathrm{fol}}}(h(gc),gh(c))<(\beta,\eta).$

###### Proof.

Let $\alpha>0$ be given. By Lemma 9.1 there is $\beta>0$ such that for $g\in
G$, $c\in\mathit{FS}_{0}$ we have

(9.3) $d_{\mathit{FS}}(gc,c)<3\alpha\quad\implies\quad d_{G}(g,e)<\beta.$

Next let $\eta>0$ and $c_{0}\in\mathit{FS}_{0}$ be given. For $n\in\mathbb{N}$
choose $h_{n}\colon G\cdot U_{\alpha,1/n}^{{\mathrm{fol}}}(c_{0})\to G$ such
that $c\in h_{n}(c)\cdot U_{\alpha,1/n}^{\mathrm{fol}}(c_{0})$ for all $c\in
G\cdot U_{\alpha,\delta_{n}}^{{\mathrm{fol}}}(c_{0})$. We will show that for
all sufficiently large $n$ the map $h_{n}$ satisfies (i) and (ii).
(i) Assume there are infinitely many $n$ such that (i) fails. Then there is
$I\subseteq\mathbb{N}$ infinite and for $n\in I$ there are
$c_{n},c^{\prime}_{n}\in\mathit{FS}$ with $c_{n}\in h_{n}(c_{n})\cdot
U_{\alpha,1/n}^{\mathrm{fol}}(c_{0})$, $c^{\prime}_{n}\in
h_{n}(c^{\prime}_{n})\cdot U_{\alpha,1/n}^{\mathrm{fol}}(c_{0})$,
$d_{\mathit{FS}\text{-}\mathrm{fol}}(c_{n},c_{n}^{\prime})<(\alpha,1/n)$ such
that

$d_{V_{c_{0}}-{\mathrm{fol}}}(h_{n}(c_{n}),h_{n}(c^{\prime}_{n}))<(\beta,\eta)$

fails. Lemma 2.5 implies that we can arrange by possibly replacing $I$ by a
smaller infinite subset that we have

$d_{\mathit{FS}\text{-}\mathrm{fol}}(h_{n}(c_{n})c_{0},h_{n}(c^{\prime}_{n})c_{0})<(3\alpha,1/n).$

Then, with $a_{n}:=h_{n}(c^{\prime}_{n})^{-1}h_{n}(c_{n})$,

$d_{\mathit{FS}\text{-}\mathrm{fol}}(a_{n}c_{0},c_{0})<(3\alpha,1/n).$

We conclude from Lemma 2.4 (i) that $a_{n}c_{0}$ stays in some closed ball $K$
around $c_{0}$ with respect to $d_{G}$. Since $\mathit{FS}$ is a proper metric
space, $K$ is compact. Since $\mathit{FS}$ is a proper $G$-space by Lemma 2.3,
we conclude from Lemma B.1 (iii) that $\\{g\in G\mid g\cdot K\cap
K\not=\emptyset\\}$ is a compact subset of $G$ and contains the sequence
$(a_{n})_{n\geq 0}$. Hence we can arrange by passing to subsequences that
$\lim_{n\to\infty}a_{n}=a$ holds in $G$ for some $a\in G$.

We can choose $\tau_{n}\in[-3\alpha,3\alpha]$ for $n\geq 0$ satisfying
$d_{\mathit{FS}}(a_{n}\Phi_{\tau_{n}}(c_{0}),c_{0})<1/n$. This implies
$\lim_{n\to\infty}a_{n}\Phi_{\tau_{n}}(c_{0})=c_{0}$. By passing to
subsequences again, we can arrange $\lim_{n\to\infty}\tau_{n}=\tau$ for some
$\tau\in[-3\alpha,3\alpha]$. We conclude
$\lim_{n\to\infty}a_{n}\Phi_{\tau_{n}}(c_{0})=a\Phi_{\tau}(c_{0})$ from Lemma
2.1 (ii). This shows $a\Phi_{\tau}(c_{0})=c_{0}$ and hence $a\in V_{c_{0}}$.
As the flow is of at most unit speed, see Lemma 2.1 (i), and $d_{\mathit{FS}}$
is left $G$-invariant, we get

$d_{\mathit{FS}}(ac_{0},c_{0})=d_{\mathit{FS}}(ac_{0},a\Phi_{\tau}(c_{0}))=d_{\mathit{FS}}(c_{0},\Phi_{\tau}(c_{0}))\leq|\tau|\leq
3\alpha.$

Hence $d_{G}(a,e)\leq\beta$ by 9.3. Since $\lim_{n\to\infty}a_{n}=a$ and $a\in
V_{c_{0}}$ hold, there exists a natural number $N$ such that
$d_{V_{c_{0}}-{\mathrm{fol}}}(a_{n},e)<(\beta,\eta)$ holds for $n\in I$ with
$n\geq N$. Since $D_{G}$ and hence $d_{V_{c_{0}}-{\mathrm{fol}}}$ are left
$G$-invariant, we get
$d_{V_{c_{0}}-{\mathrm{fol}}}(h_{n}(c_{n}),h_{n}(c^{\prime}_{n}))<(\beta,\eta)$
for $n\in I$ with $n\geq N$, a contradiction.
(ii) Assume there are infinitely many $n$ such that (ii) fails. Then there is
$I\subseteq\mathbb{N}$ infinite and for $n\in I$ there are
$c_{n}\in\mathit{FS}$, $g_{n}\in G$ with $c_{n}\in h_{n}(c_{n})\cdot
U_{\alpha,1/n}^{\mathrm{fol}}(c_{0})$, $g_{n}c_{n}\in h_{n}(g_{n}c_{n})\cdot
U_{\alpha,1/n}^{\mathrm{fol}}(c_{0})$, such that

$d_{V_{c_{0}}-{\mathrm{fol}}}(h_{n}(g_{n}c_{n}),g_{n}h_{n}(c_{n}))<(\beta,\eta)$

fails. Recall that $d_{\mathit{FS}\text{-}\mathrm{fol}}$ is left
$G$-invariant, see Lemma 2.4 (ii). Hence we get from $c_{n}\in
h_{n}(c_{n})\cdot U_{\alpha,1/n}^{\mathrm{fol}}(c_{0})$ and $g_{n}c_{n}\in
h_{n}(g_{n}c_{n})\cdot U_{\alpha,1/n}^{\mathrm{fol}}(c_{0})$ that
$d_{\mathit{FS}}(c_{n},h_{n}(c_{n})c_{0})\leq(\alpha,1/n)$ and
$d_{\mathit{FS}}(c_{n},g_{n}^{-1}h_{n}(g_{n}c_{n})c_{0})\leq(\alpha,1/n)$
holds for $n\in I$. Put $a_{n}:=h_{n}(c_{n})^{-1}g_{n}^{-1}h_{n}(g_{n}c_{n})$.
We conclude from Lemma 2.5 that we can arrange by replacing $I$ by a possibly
smaller infinite subset that

$d_{\mathit{FS}\text{-}\mathrm{fol}}(a_{n}c_{0},c_{0})<(2\alpha,1/n)$

holds for $n\in I$. We conclude from Lemma 2.1 (i), that $a_{n}c_{0}$ stays in
the closed ball $K$ of radius $2\alpha+1$ around $c$. Since $\mathit{FS}$ is a
proper metric space, $K$ is compact. Since $\mathit{FS}$ is a proper $G$-space
by Lemma 2.3, we conclude from Lemma B.1 (iii) that $g\in G\mid g\cdot K\cap
K\not=\emptyset\\}$ is a compact subset of $G$ and contains the sequence
$(a_{n})_{n\geq 0}$. Hence we can arrange by passing to subsequences that
$\lim_{n\to\infty}a_{n}=a$ holds in $G$ for some $a\in G$. There are
$\tau_{n}\in[-2\alpha,2\alpha]$ with
$d_{\mathit{FS}}(a_{n}\Phi_{\tau_{n}}(c_{0}),c_{0})<1/n$. This implies
$\lim_{n\to\infty}a_{n}\Phi_{\tau_{n}}(c_{0})=c_{0}$. We can arrange by
passing to subsequences $\lim_{n\to\infty}\tau_{n}=\tau$ for some
$\tau\in[-2\alpha,2\alpha]$. We conclude
$\lim_{n\to\infty}a_{n}\Phi_{\tau_{n}}(c_{0})=a\Phi_{\tau}(c_{0})$ from Lemma
2.1 (ii). This shows $a\Phi_{\tau}(c_{0})=c_{0}$ and hence $a\in V_{c_{0}}$.
As the flow is of at most unit speed, see Lemma 2.1 (i), and $d_{\mathit{FS}}$
is left $G$-invariant, we get

$d_{\mathit{FS}}(ac_{0},c_{0})=d_{\mathit{FS}}(ac_{0},a\Phi_{\tau}(c_{0}))=d_{\mathit{FS}}(c_{0},\Phi_{\tau}(c_{0}))\leq|\tau|\leq
2\alpha.$

Hence $d_{G}(a,e)\leq\beta$ by 9.3. Since $\lim_{n\to\infty}a_{n}=a$ and $a\in
V_{c_{0}}$ hold, there is a natural number $N$ such that
$d_{V_{c_{0}}-{\mathrm{fol}}}(a_{n},e)<(\beta,\eta)$ holds for $n\in I$ with
$n\geq N$. As $d_{G}$ and hence $d_{V_{c_{0}}-{\mathrm{fol}}}$ are left
$G$-invariant, we get
$d_{V_{c_{0}}-{\mathrm{fol}}}(h_{n}(g_{n}c_{n}),g_{n}h_{n}(c_{n}))<(\beta,\eta)$
for $n\in I$ with $n\geq N$, a contradiction. ∎

The following addendum to Proposition 9.2 strengthens the conclusion in the
case where the period $\tau_{c_{0}}$ of $c_{0}$ defined in (2.6c) is large
relative to the given $\alpha$. Recall that we have defined the compact
subgroup $K_{c_{0}}\subset G$ to be isotropy group $G_{c_{0}}$ of
$c_{0}\in\mathit{FS}$ in (2.6a). The difference to Proposition 9.2 is that in
the two conclusions $d_{K_{c_{0}}-{\mathrm{fol}}}$ is used, not
$d_{V_{c_{0}}-{\mathrm{fol}}}$.

######  Addendum 9.4.

Let $\mathit{FS}_{0}\subseteq\mathit{FS}$ be compact. For all $\alpha>0$ there
are $\beta>0$, $\ell>0$ such that the following is true. For all $\eta>0$,
$c_{0}\in\mathit{FS}_{0}$ with $\tau_{c_{0}}>\ell$ there are $\delta>0$,
$h\colon G\cdot U^{{\mathrm{fol}}}_{\alpha,\delta}(c_{0})\to G$ satisfying the
following.

  1. (i)

for $c,c^{\prime}\in G\cdot U^{{\mathrm{fol}}}_{\alpha,\delta}(c_{0})$ we have

$d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime})<(\alpha,\delta)\quad\implies\quad
d_{K_{c_{0}}-{\mathrm{fol}}}(h(c),h(c^{\prime}))<(\beta,\eta);$

  2. (ii)

for $g\in G$, $c\in G\cdot U^{{\mathrm{fol}}}_{\alpha,\delta}(c_{0})$ we have

$d_{K_{c_{0}}-{\mathrm{fol}}}(h(gc),gh(c))<(\beta,\eta).$

###### Proof.

We can argue almost exactly as in the proof of Proposition 9.2. For the
element $a\in V_{c_{0}}$ produced in the proof of both conclusions there we
also proved $d_{\mathit{FS}}(ac_{0},c_{0})\leq 3\alpha$. Thus if
$\tau_{c_{0}}>\ell:=3\alpha$, then we must have $ac_{0}=c_{0}$, i.e., $a\in
K_{c_{0}}$ and the conclusions follow for $d_{K_{c_{0}}-{\mathrm{fol}}}$ in
place of $d_{V_{c_{0}}-{\mathrm{fol}}}$. ∎

######  Remark 9.5 (The role of $\mathit{FS}_{0}$).

The construction of maps $h\colon U\to G$ will depend on a choice of base
point for the orbit $Gc_{0}$, namely $c_{0}$. The same construction with
respect to a different base point $g_{0}c_{0}$ would also work, but with
respect to a different collection of subgroups and constant $\beta$. But the
subgroups and constant $\beta$ in Proposition 6.8 are required to be uniform
over all orbits. Therefore the base points for different orbits have to be
chosen somewhat consistently; in our argument we have done this by using only
base points from a fixed compact subset $\mathit{FS}_{0}$ of $\mathit{FS}$.

###  9.b. Proof of Proposition 6.8

######  Lemma 9.6.

Let $c_{0}\in\mathit{FS}$. Then there exists an open neighborhood $U$ of
$c_{0}$ in $\mathit{FS}$ and a compact open subgroup $K$ of $G$ such that
$K_{c}\subseteq K$ for all $c\in U$.

###### Proof.

Recall that $G$ acts cocompactly, isometrically, properly, and smoothly on
$X$. There is an open neighborhood $W\subseteq X$ of $c_{0}(0)$ such that
$G_{x}\subseteq G_{c_{0}(0)}$ for all $x\in W$, see Lemma B.1 (iv). Now
$K:=G_{c_{0}(0)}$ and $U:=\\{c\in\mathit{FS}\mid c(0)\in W\\}$ satisfy the
assertion. ∎

The following proof of Proposition 6.8 is the only place where we use
Assumption 2.7.

###### Proof.

Let $\mathit{FS}_{0}$ be the compact subset of $\mathit{FS}$ from Assumption
2.7. Given $\alpha>0$ Proposition 9.2 and Addendum 9.4 provide us with numbers
$\beta>0$, $\ell>0$.

Next we use (2.7b) and 9.6 to find
$\mathcal{V}\subseteq{\mathcal{C}\mathrm{vcy}}$ finite and a finite cover
$\mathcal{W}$ of $\mathit{FS}_{0}$ such that for any $W\in\mathcal{W}$ there
are $K_{W},V_{W}\in{\mathcal{C}\mathrm{vcy}}$ satisfying

  1. (9.7a)

for all $c\in W$ we have $K_{c}\subseteq K_{W}$;

  2. (9.7b)

for all $c\in W$ with $0<\tau_{c}\leq\ell$ we have $V_{c}\subseteq V_{W}$.

Let now $\eta>0$ and $c_{0}\in\mathit{FS}$ be given. According to (2.7a)
$\mathit{FS}_{0}$ is a fundamental domain for the $G$ action. This allows us
to choose $g_{0}\in G$, $W\in\mathcal{W}$ such that $g_{0}c_{0}\in
W\subseteq\mathit{FS}_{0}$. If $\tau_{c_{0}}=\tau_{g_{0}c_{0}}>\ell$, then we
set $V:=K_{W}$ and note that by (9.7a) $K_{g_{0}c_{0}}\subseteq V$. If
$\tau_{c_{0}}=\tau_{g_{0}c_{0}}\leq\ell$ then we set $V:=V_{W}$ and note that
by (9.7b) $V_{g_{0}c_{0}}\subseteq V$. Now Proposition 9.2 and Addendum 9.4
give us $\delta>0$ and $h\colon G\cdot
U^{\mathrm{fol}}_{\alpha,\delta}(c_{0})\to G$ satisfying for $c,c^{\prime}\in
G\cdot U^{{\mathrm{fol}}}_{\alpha,\delta}(c_{0})$, $g\in G$

$\displaystyle d_{V\text{-}\mathrm{fol}}(h(c),h(c^{\prime}))$ $\displaystyle<$
$\displaystyle(\beta,\eta)\quad\text{provided}\;d_{\mathit{FS}\text{-}\mathrm{fol}}(c,c^{\prime})<(\alpha,\delta);$
$\displaystyle d_{V\text{-}\mathrm{fol}}(h(gc),gh(c))$ $\displaystyle<$
$\displaystyle(\beta,\eta).$

Of course $U:=G\cdot U^{\mathrm{fol}}_{\alpha,\delta}(c_{0})$ is
$G$-invariant. ∎

This finishes the proof of our main Theorem 1.2.

##  Appendix A The Bruhat-tits building for reductive $p$-adic groups

Let $K$ be a non-Archimedian local field, i.e., a finite extension of the
field of $p$-adic numbers or the field of formal Laurent series $k((t))$ over
a finite field $k$. Consider an algebraic group $G$ over $K$ whose component
of the identity is reductive. Let $G(K)$ be its group of $K$-points. We will
simply say that $G(K)$ is a reductive $p$-adic group. We will need the action
of $G(K)$ on the associated (extended) Bruhat-Tits building. The original
reference for the Bruhat-Tits building is [10, 11]. Summaries of the
construction can be found in [21] and in [20, Sec. I.1].

To set up notation we briefly review aspects of the construction. Let $A$ be
the real affine space constructed in [21, 1.2, p.31,32]. It comes equipped
with an action of a subgroup $N(K)$ of $G(K)$. The affine space is finite
dimensional and the action of $N(K)$ is cocompact666This is not explicitly
mentioned [21, 1.2, p.31,32] but follows from the construction.. There is also
a collection $\Phi_{\text{af}}$ of affine linear function $\alpha\colon
A\to\mathbb{R}$, these are the affine roots [21, 1.6, p.33]. This set is
symmetric, i.e., if $\alpha\in\Phi_{\text{af}}$ then
$-\alpha\in\Phi_{\text{af}}$. After identifying $A$ with the associated linear
space $V$ the affine roots can be described as follows: there are finitely
many linear function $a\colon V\to\mathbb{R}$ (the roots) and for each $a$
there is a discrete set $\Gamma_{a}\subseteq\mathbb{R}$ such that the affine
roots are the maps $v\mapsto a(v)+l$ where $l\in\Gamma_{a}$. (This follows
from the discussion in [21, 1.6, p.33], see also [20, p. 103]). Associated to
$\alpha\in\Phi_{\text{af}}$ is the half-apartment $A_{\alpha}=\\{x\in
A\mid\alpha(x)\geq 0\\}$ and the wall $\partial A_{\alpha}=\\{x\in
A\mid\alpha(a)=0\\}$. Chambers of $A$ are the connected components of the
complements of the walls 777In the quasi-simple case the facets are simplices;
in the semi-simple case the facets are poly-simplices (i.e., finite products
of simplices); in general the facets are products of affine spaces with poly-
simplices [21, 1.7, p.33].. The facets of the chambers are called the facets
of $A$. The building $X$ is constructed as a quotient of $G\times A$ [21, 2.1,
p.43]. The quotient map $G(K)\times A\to X$ is $G(K)$-equivariant and the
$G(K)$-action on $X$ extends the $N(K)$-action on $A$. The translates of $A$
under $G(K)$ are the apartments of $X$. As $N(K)$ acts cocompactly on $A$ and
since $X$ is the union of its apartments the action of $G(K)$ on $X$ is
cocompact as well. The apartments $gA$ inherit an affine structure and a
partition into facets from $A$; these structures agree on intersections of
apartments; any two points (in fact any two facets) of $X$ are contained in a
common apartment [21, 2.2.1, p.44]. Given two apartments $A^{\prime}$ and
$A^{\prime\prime}$ there is $g\in G$ with $gA^{\prime}=A^{\prime\prime}$ such
that $g$ fixes $A^{\prime}\cap A^{\prime\prime}$ pointwise [21, 2.2.1]. This
can be used to construct a $G(K)$-invariant $\operatorname{CAT}(0)$-metric
$d_{X}$ on $X$ [21, 2.3, p.45]888In [21] the terminology of
$\operatorname{CAT}(0)$-spaces is not used, but the inequality given there is
equivalent to the $\operatorname{CAT}(0)$-condition, see [9, p. 163]..
Apartments are then flat subspace of $X$. The action of $G(K)$ on $X$ is also
proper [21, p.45]. By our assumption on $K$ its residue field (denoted
$\overline{K}$ in [21]) is finite. This assumption is used in some of the
following results from [21]. The stabilizer groups of chambers (and therefore
of facets) contain the Iwahori subgroups [21, p.54] and these subgroups are
open [21, p.55]. In particular, all stabilizer groups for facets are open and
the action of $G(K)$ on $X$ is smooth. The chambers of $X$ can be subdivided
to give $X$ the structure of a locally finite simplicial complex where the
action of $G(K)$ is simplicial [21, 2.3.1, p.45]. Altogether $X$ is a finite
dimensional $\operatorname{CAT}(0)$-space with a a proper, continuous,
isometric, smooth, cocompact $G(K)$-action. Assumption 2.7 for the action of
$G(K)$ on $X$ is verified in Proposition A.7 below.

######  Lemma A.1.

Any generalized geodesic $c\colon\mathbb{R}\to X$ is contained in a translate
of $A$.

###### Proof.

We may assume $c(0)\in A$. For $n\in\mathbb{N}$ we find an apartment
$A^{\prime}$ that contains $c(\pm n)$. By the construction of the metric
$A^{\prime}$ will then contain $c([-n,n])$. Now choose $g_{n}\in G$ such that
$g_{n}A=A^{\prime}$ and $g_{n}$ fixes $A\cap A^{\prime}$. Then
$(g_{n})_{n\in\mathbb{N}}$ is a sequence in the compact subgroup of $G(K)$
that fixes $c(0)$ and has an accumulation point $g$. As the action of $G(K)$
on $X$ is continuous the $g_{n}A$ must agree with $gA$ on larger and larger
neighborhoods of $c(0)$. It follows that the apartment $gA$ contains the image
of $c$. ∎

In the following we write $\mathit{FS}_{\infty}$ for the subspace of
$\mathit{FS}$ consisting of all (bi-infinite) geodesics $c\colon\mathbb{R}\to
X$ and for $Y\subseteq X$ we set
$\mathit{FS}_{\infty}(Y):=\mathit{FS}(Y)\cap\mathit{FS}_{\infty}$.

######  Lemma A.2.

Let $c_{0}\in\mathit{FS}_{\infty}(A)$. Then there is $\epsilon>0$ such that
for all $\alpha\in\Phi_{\text{af}}$, with
$c_{0}\not\in\mathit{FS}_{\infty}(A_{\alpha})$ we have

$d_{\mathit{FS}}\big{(}c_{0},\mathit{FS}_{\infty}(A_{\alpha})\big{)}\geq\epsilon.$

###### Proof.

As there are only finitely many roots it suffices to consider the affine roots
associated to a fixed root $a$. The walls associated to these affine roots are
then all parallel and the half-apartments $A_{\alpha}$ are linearly ordered by
inclusion (because $\Gamma_{a}$ is discrete). If $c_{0}$ is not parallel to
these walls, then no half-apartment $A_{\alpha}$ contains $c_{0}$ (or any
geodesic parallel to $c_{0}$) and $c_{0}$ intersects all the walls $\partial
A_{\alpha}$ in the same angle. It is then not difficult to bound
$d_{\mathit{FS}}(c_{0},\mathit{FS}_{\infty}(A_{\alpha}))$ in terms of this
angle. If $c_{0}$ is parallel to the walls $\partial A_{\alpha}$, then among
the $A_{\alpha}$ not containing $c_{0}$ there is a maximal half-apartment
$A_{\alpha_{0}}$ and we can use
$\epsilon:=d_{\mathit{FS}}(c_{0},\mathit{FS}_{\infty}(A_{\alpha_{0}}))$. ∎

######  Lemma A.3.

Let $c_{0}\in\mathit{FS}_{\infty}(A)$. Then there is $\epsilon>0$ such that
for all $g\in G$ we have

$d_{\mathit{FS}}\big{(}c_{0},\mathit{FS}_{\infty}(A\cap
gA)\big{)}\in\\{0\\}\cup(\epsilon,\infty).$

###### Proof.

The intersection $gA\cap A$ is a union of facets of $A$ and convex. It follows
that if $c_{0}$ is not contained in $gA\cap A$, then $gA\cap A$ is contained
in a half-apartment $A_{\alpha}$ that does not contain $c_{0}$. The assertion
follows now from Lemma A.2. ∎

######  Lemma A.4.

Let $c_{0}\in\mathit{FS}_{\infty}(A)$. Then
$Gc_{0}\cap\mathit{FS}_{\infty}(A)$ is discrete.

###### Proof.

Choose real numbers $t_{-}<t_{+}$. As geodesics in $A$ have unique extensions
in $A$ we observe the following: if $gc_{0}\in\mathit{FS}_{\infty}(A)$ and
$c_{0}(t_{\pm})=gc_{0}(t_{\pm})$, then $c_{0}=gc_{0}$.

Let now $g_{n}\in G$ with $g_{n}c_{0}\in\mathit{FS}_{\infty}(A)$ and
$g_{n}c_{0}\to c_{1}\in\mathit{FS}_{\infty}(A)$ as $n\to\infty$. As the action
of $G(K)$ on $X$ is smooth all orbits for this action are discrete. Thus
$g_{n}c_{0}(t_{\pm})=c_{1}(t_{\pm})$ for almost all $n$. The above observation
now implies that $g_{n}c_{0}$ is eventually constant. Thus
$Gc_{0}\cap\mathit{FS}(A)$ is discrete, as asserted. ∎

Recall that an element $c$ in $\mathit{FS}$ is called periodic if there exists
$g\in G$ and $t\in\mathbb{R}$ with $t>0$, and $gc=\Phi_{t}(c)$. If $c$ is
periodic, then necessarily $c\in\mathit{FS}_{\infty}$.

######  Lemma A.5.

Let $c_{0}\in\mathit{FS}_{\infty}(A)$. Let $\beta>0$. Then there is
$\epsilon>0$ such that the following holds. Let $c\in\mathit{FS}_{\infty}(A)$
with $d_{\mathit{FS}}(c,c_{0})<\epsilon$, $g\in G$, $t\in[-\beta,\beta]$ with
$gc=\Phi_{t}c$. Then $g\in V_{c_{0}}$.

###### Proof.

Assume this fails. Then there are sequences $(c_{n})_{n\geq 0}$ in
$\mathit{FS}_{\infty}(A)$, $(t_{n})_{n\geq 0}$ in $[-\beta,\beta]$, and
$(g_{n})_{n\geq 0}$ in $G(K)$ such that $\lim_{n\to\infty}c_{n}=c_{0}$,
$g_{n}c_{n}=\Phi_{t_{n}}c_{n}$, but $g_{n}\not\in V_{c_{0}}$. By passing to
subsequences we can arrange $\lim_{n\to\infty}t_{n}=t$ for some
$t\in[-\beta,\beta]$. Then we get
$\lim_{n\to\infty}g_{n}c_{n}=\lim_{n\to\infty}\Phi_{t_{n}}(c_{n})=\Phi_{t}c_{0}$
from Lemma 2.1 (ii). As $G\curvearrowright\mathit{FS}$ is proper, see Lemma
2.3, the $g_{n}$ vary over a relatively compact set. Thus we can pass to a
further subsequence and assume that $\lim_{n\to\infty}g_{n}=g$ for some $g\in
G$. Then $\lim_{n\to\infty}g_{n}c_{0}=gc_{0}$. As
$G\curvearrowright\mathit{FS}$ is isometric we also have
$\lim_{n\to\infty}g_{n}c_{n}=gc_{0}$. Thus $gc_{0}=\Phi_{t}(c_{0})$. We have
$g_{n}c_{n}=\Phi_{t_{n}}c_{n}\in\mathit{FS}(A)$. Thus
$c_{n}\in\mathit{FS}_{\infty}((g_{n})^{-1}A)$. Lemma A.3 implies that
$c_{0}\in\mathit{FS}_{\infty}((g_{n})^{-1}A)$ for almost all $n$. Thus
$g_{n}c_{0}\in\mathit{FS}(A)$ for almost all $n$. Now Lemma A.4 implies that
$g_{n}c_{0}=gc_{0}$ for almost all $n$. Thus
$g_{n}c_{0}=gc_{0}=\Phi_{t}(c_{0})$ for almost all $n$, contradicting
$g_{n}\not\in V_{c_{0}}$. ∎

######  Lemma A.6.

Let $c_{0}\in\mathit{FS}_{\infty}(A)$. Let $\ell>0$. Then there is
$\epsilon>0$ such that for all $c\in\mathit{FS}_{\infty}(A)$ with
$d_{\mathit{FS}}(c,c_{0})<\epsilon$ and $0<\tau_{c}<\ell$ we have
$V_{c}\subseteq V_{c_{0}}$.

Recall that have defined the group $K_{c}$ to be the $G(K)$-isotropy group of
$c$ in (2.6a).

###### Proof of Lemma A.6.

Let $c\in\mathit{FS}(A)$ with $\tau_{c}>0$, i.e., $c$ is periodic. Lemma 2.8
tells us that there is $v\in V_{c}$ such that $\Phi_{\tau_{c}}(c)=vc$ and that
$v$ together with $K_{c}$ generates $V_{c}$. The result follows therefore from
Lemma A.5 ∎

######  Proposition A.7.

The action of $G(K)$ on $\mathit{FS}(X)$ satisfies Assumption 2.7.

###### Proof.

As discussed above the action of the subgroup $N(K)$ on $A$ is cocompact. This
implies that the action of $N(K)$ on $\mathit{FS}(A)$ is cocompact as well,
see Lemma 2.3. Thus we find $\mathit{FS}_{0}\subseteq\mathit{FS}(A)$ compact
with $N(K)\cdot\mathit{FS}_{0}=\mathit{FS}(A)$. By Lemma A.1 we have
$G(K)\cdot\mathit{FS}(A)=\mathit{FS}$. So $G\cdot\mathit{FS}_{0}=\mathit{FS}$,
i.e., (2.7a) is satisfied.

Towards (2.7b), we first observe that $\tau_{c}<\infty$ implies
$c\in\mathit{FS}_{\infty}$, see Lemma 2.8. Let now
$c_{0}\in\mathit{FS}_{0}\subseteq\mathit{FS}(A)$ and $\ell>0$ be given. We
need to find an open neighborhood $U$ of $c_{0}$ in $\mathit{FS}_{0}$ such
that for all $c\in U$ with $\tau_{c}<\ell$ we have $V_{c}\subseteq V_{c_{0}}$.
As $\mathit{FS}_{\infty}\subseteq\mathit{FS}$ is closed we can take
$\mathit{FS}_{0}\setminus\mathit{FS}_{\infty}$ if
$c_{0}\not\in\mathit{FS}_{\infty}$. If $c_{0}\in\mathit{FS}_{\infty}$, then
Lemma A.6 provides a suitable $\epsilon$-neighborhood. Thus (2.7b) is
satisfied as well. ∎

##  Appendix B Basics about group actions

A (continuous) map $f\colon X\to Y$ of (compactly generated topological)
spaces is called _proper_ if preimages of compact subsets are compact again. A
$G$-space $X$ is called _proper_ if the map $\Theta^{G}_{X}\colon G\times X\to
X\times X,\;(g,x)\mapsto(x,gx)$ is proper. It is called _smooth_ if all
isotropy group are open. It is called _cocompact_ if the quotient space $X/G$
is compact.

######  Lemma B.1.

Let $G$ be a locally compact Hausdorff group and let $X$ be a $G$-space.

  1. (i)

The $G$-space $X$ is proper if and only if for any $x\in X$ there is an open
neighborhood $U$ such that the subset $\\{g\in G\mid g\cdot U\cap
U\neq\emptyset\\}$ of $G$ is relatively compact, i.e., its closure in $G$ is
compact;

  2. (ii)

The isotropy groups of a proper $G$-space are all compact. A $G$-$CW$-complex
is proper if and only if all its isotropy groups are compact;

  3. (iii)

If the $G$-space $X$ is proper, then for every compact subset $K\subseteq X$
the subset $\\{g\in G\mid gK\cap K\not=\emptyset\\}$ of $G$ is compact. The
converse is true if $X$ is locally compact;

  4. (iv)

Let $X$ be a metric space with isometric proper smooth $G$-action. Then for
any $x\in X$ there exists $\epsilon>0$ satisfying:

    1. (a)

$G_{x}=\\{g\in G\mid g\cdot B_{\epsilon}(x)\cap
B_{\epsilon}(x)\not=\emptyset\\}$;

    2. (b)

The map

$\alpha\colon G\times_{G_{x}}B_{\epsilon}(x)\xrightarrow{\cong}G\cdot
B_{\epsilon}(x),\quad(g,y)\mapsto gy$

is a $G$-homeomorphism;

    3. (c)

We have $G_{y}\subseteq G_{x}$ for every $y\in B_{\epsilon}(x)$.

  5. (v)

Let $X$ be a proper $G$-space. Let $A\subseteq X$ be a closed subspace. Let
$G_{A}=\\{g\in G\mid gA=A\\}$. Then the $G_{A}$-space $A$ is proper;

  6. (vi)

Let $H$ be a (closed) subgroup of $G$. If the $G$-space $X$ is proper, then
its restriction to an $H$-space is proper.

  7. (vii)

Let $f\colon X\to Y$ be a proper $G$-map. If $Y$ is proper, then $X$ is
proper;

  8. (viii)

Let $X$ be a metric space on which $G$ acts isometrically and properly. Then
$X$ is smooth if and only if each orbit $Gx$ (equipped with the subspace
topology from $X$) is discrete;

  9. (ix)

Suppose that $X$ is a locally compact metric space on which $G$ acts
isometrically. Then $X$ is proper and smooth if and only if each orbit $Gx$ is
discrete and each isotropy group $G_{x}$ is compact.

  10. (x)

If the $G$-space $X$ contains a compact subset $C$ with $G\cdot C=X$, then $X$
is cocompact.

If the $G$-space $X$ is locally compact and cocompact, then there is a compact
subset $C\subseteq X$ satisfying $G\cdot C=X$;

  11. (xi)

Let $f\colon X\to Y$ be a proper $G$-map. If $Y$ is locally compact and
cocompact, then $X$ is cocompact.

###### Proof.

(i) See [22, Proposition 3.21 in Chapter I on page 28].
(ii) Obviously the isotropy groups of a proper $G$-space are all compact. The
claim about $G$-$CW$-complexes is proved in [17, Theorem 1.23 on page 18].
(iii) The set $\\{g\in G\mid g\cdot K\cap K\not=\emptyset\\}$ is the image of
$(\Theta^{G}_{X})^{-1}(K\times K)$ under the projection $G\times X\to G$.
Hence it is compact if $X$ is proper. The converse follows from assertion (i).
(iv) Suppose assertion (iv)a is not true. Then there is an $x\in X$ such that
for every $\epsilon>0$ the isotropy group $G_{x}$ is not equal to $\\{g\in
G\mid g\cdot B_{\epsilon}(x)\cap B_{\epsilon}(x)\not=\emptyset$. Hence we can
choose a sequence of elements $(x_{n})_{n\geq 0}$ in $X$ and a sequence of
elements $(g_{n})_{n\geq 0}$ in $G$ such that $x_{n}$ and $g_{n}x_{n}$ belong
to $B_{1/n}(x)$ and $g_{n}\notin G_{x}$ holds. Because of assertion (i) there
is an open neighborhood $U$ such that the subset $\overline{\\{g\in G\mid
g\cdot U\cap U\neq\emptyset\\}}$ of $G$ is compact. By passing to subsequences
we can arrange $x_{n}\in U$ for $n\geq 0$. Then $g_{n}$ belongs to the compact
subset $\overline{\\{g\in G\mid g\cdot U\cap U\neq\emptyset\\}}$ of $G$. By
passing to subsequences, we can arrange that there is an element $g\in G$ with
$\lim_{n\to\infty}g_{n}=g$. Since $\lim_{n\to\infty}x_{n}=x$ and
$\lim_{n\to\infty}g_{n}x_{n}=x$ holds and $G$ acts isometrically, we conclude
$\lim_{n\to\infty}g_{n}x=x$. This implies $g\in G_{x}$ because of
$\lim_{n\to\infty}g_{n}=g$. Since $G_{x}$ is open in $G$ and
$\lim_{n\to\infty}g_{n}=g$, we get for almost all $g_{n}$ that $g_{n}=g$ and
hence $g_{n}\in G_{x}$, a contradiction. This proves assertion (iv)a.

Next we show assertion (iv)b for the $\epsilon$ for which assertion (iv)a is
true. Let $s\colon G/G_{x}\to G$ be a map of sets such that its composition
with the projection $p\colon G\to G/G_{x}$ the identity on $G/G_{x}$. Since
$G/G_{x}$ is discrete, we obtain a homeomorphism

$\beta\colon G/G_{x}\times
B_{\epsilon}(x)\xrightarrow{\cong}G\times_{G_{x}}B_{\epsilon}(x),\quad(gG_{x},y)\mapsto(s(gG_{x}),y).$

Its inverse is given by $(g,y)\mapsto(gG_{x},s(gG_{x})^{-1}gy)$. The composite
$\beta\circ\alpha$ is the map $G/G_{x}\times B_{\epsilon}(x)\to G\cdot
B_{\epsilon}(x),(gG_{x},y)\mapsto s(gG_{x})\cdot y$ which is a homeomorphism
since $G\cdot B_{\epsilon}(x)$ is the disjoint union of open subsets
$\coprod_{gG_{x}\in G/G_{x}}s(gG_{x})\cdot B_{\epsilon}(x)$ by assertion
(iv)a. Hence assertion (iv)b is true.

Assertion (iv)c follows directly from assertion (iv)a.
(v) Let $f\colon X\to Y$ be a proper map and $B\subseteq Y$ closed. Consider
any closed subspace $A\subseteq f^{-1}(B)$. Then the induced map $f|_{A}\colon
A\to B$ is obviously proper. Now the claim follows from the commutativity of
the following diagram

$\textstyle{G_{B}\times
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta^{G_{B}}_{A}}$$\textstyle{B\times
B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta^{G}_{X}}$$\textstyle{X\times
X}$

whose vertical arrows are the obvious inclusions.
(vi) This follows from the fact that the restriction of a proper map to a
closed subspace is again proper.
(vii) Suppose that $Y$ is proper. The following diagram

$\textstyle{G\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta^{G}_{X}}$$\scriptstyle{\operatorname{id}_{G}\times
f}$$\textstyle{X\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f\times
f}$$\textstyle{G\times
Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Theta^{G}_{Y}}$$\textstyle{Y\times
Y}$

commutes. Since the lower horizontal arrow and the vertical arrows are proper,
the upper arrow is proper, see [17, Lemma 1.16 on page 14], in other words,
$X$ is proper.
(viii) This follows from the fact that for a proper $G$-space the canonical
map $G/G_{x}\to Gx$ is a homeomorphism for every $x\in X$, see [22,
Proposition 3.19 (ii) in Chapter I on page 28] or [17, Lemma 1.19 (iii) on
page 16].
(ix) Suppose that $X$ is proper and smooth. Then each isotropy group is
compact and each $G$-orbit is discrete by assertions (ii) and (viii).

Suppose that each each orbit $Gx$ is discrete and each isotropy group $G_{x}$
is compact. By assertion (viii) it suffices to show that the $G$-action is
proper. Consider $x\in X$. Since $Gx$ is discret and $X$ locally compact, we
can choose a compact neighborhood $U$ of $x$ in $X$ satisfying $U\cap
Gx=\\{x\\}$. It suffices to show that the subset $\\{g\in G\mid gU\cap
U\not=\emptyset\\}$ of $G$ is relative compact because of assertion (i). We
can equip $G$ with a left invariant proper metric, see [14]. It suffices to
show that any sequence $(g_{n})_{n\in\mathbb{N}}$ of element in $G$ satisfying
$g_{n}\cdot U\cap U\not=\emptyset$ has a convergent subsequence. Choose for
each $n\geq 0$ elements $u_{n}$ and $u_{n}^{\prime}$ in $U$ such that
$g_{n}u_{n}=u_{n}^{\prime}$ holds. Since $U$ is compact, we can arrange after
passing to subsequences that $\lim_{n\to\infty}u_{n}=u$ and
$\lim_{n\to\infty}u_{n}^{\prime}=u^{\prime}$ holds for appropriate elements
$u,u^{\prime}\in U$. Since $G$ acts by isometries on $X$, we get
$\lim_{n\to\infty}g_{n}u=u^{\prime}$. As $Gu$ is a discrete subspace of $X$,
we can arrange after passing to subsequences that $g_{n}u=u^{\prime}$ holds
for $n\geq 0$. Hence $g_{0}^{-1}g_{n}u=u$ for all $n\geq 0$. Since $G_{u}$ is
compact, we can arrange after passing to subsequences that $g_{0}^{-1}g_{n}$
is a convergent sequence in $G$. This implies that $g_{n}$ is a convergent
sequence in $G$.
(x) Let $p\colon X\to X/G$ be the projection. If $C\subseteq X$ is a compact
subset with $G\cdot C=X$, then $p(C)=X/G$ and hence $X/G$ is compact.

Suppose that $X$ is locally compact and $X/G$ is compact. We can find for
every $x\in X$ an open neighborhood $U(x)$ such that $\overline{U(x)}$ is
compact. We obtain an open covering $\\{p(U(x))\mid x\in X\\}$ of $X/G$. Since
$X/G$ is compact, we can find a finite subset $I\subseteq X$ with
$\bigcup_{x\in I}p(U(x))=X/G$. Then $C=\bigcup_{x\in I}\overline{U(x)}$ is a
compact subset of $X$ with $G\cdot C=X$.
(xi) Since $Y$ is locally compact and cocompact, we can choose by assertion
(x) a compact subset $C\subseteq Y$ with $G\cdot C=Y$. Since $f$ is proper,
$D:=f^{-1}(C)$ is a compact subset of $X$. Since $G\cdot D=X$ holds, $X/G$ is
compact by assertion (x), in other words, $X$ is cocompact. This finishes the
proof of Lemma B.1. ∎

######  Example B.2 (Non-proper action).

The action of $\mathbb{Z}$ on $S^{1}$ by rotating through an irrational angle
is free, isometric, and smooth, but not proper. All $\mathbb{Z}$-orbits are
dense and not discrete. The canonical $G$-map map $G/G_{x}\to Gx$ is
continuous and bijective, but not a homeomorphism.

## References

  * [1] H. Abels, A. Manoussos, and G. Noskov.  Proper actions and proper invariant metrics.  J. Lond. Math. Soc. (2), 83(3):619–636, 2011.
  * [2] A. Bartels.  $K$-theory and actions on Euclidean retracts.  In Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures, pages 1041–1062. World Sci. Publ., Hackensack, NJ, 2018.
  * [3] A. Bartels and W. Lück.  Geodesic flow for CAT(0)-groups.  Geom. Topol., 16:1345–1391, 2012.
  * [4] A. Bartels and W. Lück.  The Borel conjecture for hyperbolic and CAT(0)-groups.  Ann. of Math. (2), 175:631–689, 2012.
  * [5] A. Bartels and W. Lück.  Algebraic $K$-theory of reductive $p$-adic groups.  Preprint, arXiv:2306.03452 [math.KT], 2023.
  * [6] A. Bartels, W. Lück, and H. Reich.  Equivariant covers for hyperbolic groups.  Geom. Topol., 12(3):1799–1882, 2008.
  * [7] A. Bartels, W. Lück, and H. Reich.  The $K$-theoretic Farrell-Jones conjecture for hyperbolic groups.  Invent. Math., 172(1):29–70, 2008.
  * [8] P. Baum, A. Connes, and N. Higson.  Classifying space for proper actions and ${K}$-theory of group ${C}^{\ast}$-algebras.  In $C^{\ast}$-algebras: 1943–1993 (San Antonio, TX, 1993), pages 240–291. Amer. Math. Soc., Providence, RI, 1994.
  * [9] M. R. Bridson and A. Haefliger.  Metric spaces of non-positive curvature.  Springer-Verlag, Berlin, 1999.  Die Grundlehren der mathematischen Wissenschaften, Band 319.
  * [10] F. Bruhat and J. Tits.  Groupes réductifs sur un corps local.  Inst. Hautes Études Sci. Publ. Math., 41:5–251, 1972.
  * [11] F. Bruhat and J. Tits.  Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée.  Inst. Hautes Études Sci. Publ. Math., 60:197–376, 1984.
  * [12] R. Engelking.  Dimension theory.  North-Holland Publishing Co., Amsterdam-Oxford-New York; PWN—Polish Scientific Publishers, Warsaw, 1978.  Translated from the Polish and revised by the author, North-Holland Mathematical Library, 19.
  * [13] F. T. Farrell and L. E. Jones.  Isomorphism conjectures in algebraic ${K}$-theory.  J. Amer. Math. Soc., 6(2):249–297, 1993.
  * [14] U. Haagerup and A. Przybyszewska.  Proper metrics on locally compact groups, and proper affine isometric actions on Banach spaces.  Preprint, arXiv:0606794 [math.OA], 2006.
  * [15] D. Kasprowski and H. Rüping.  Long and thin covers for flow spaces.  Groups Geom. Dyn., 11(4):1201–1229, 2017.
  * [16] D. Kasprowski and H. Rüping.  Long and thin covers for flow spaces.  Groups Geom. Dyn., 11(4):1201–1229, 2017.
  * [17] W. Lück.  Transformation groups and algebraic ${K}$-theory, volume 1408 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 1989.
  * [18] W. Lück.  Survey on classifying spaces for families of subgroups.  In Infinite groups: geometric, combinatorial and dynamical aspects, volume 248 of Progr. Math., pages 269–322. Birkhäuser, Basel, 2005.
  * [19] W. Lück and H. Reich.  The Baum-Connes and the Farrell-Jones conjectures in $K$\- and $L$-theory.  In Handbook of $K$-theory. Vol. 1, 2, pages 703–842. Springer, Berlin, 2005.
  * [20] P. Schneider and U. Stuhler.  Representation theory and sheaves on the Bruhat-Tits building.  Inst. Hautes Études Sci. Publ. Math., 85:97–191, 1997.
  * [21] J. Tits.  Reductive groups over local fields.  In Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, pages 29–69. Amer. Math. Soc., Providence, R.I., 1979.
  * [22] T. tom Dieck.  Transformation groups.  Walter de Gruyter & Co., Berlin, 1987.
  * [23] C. Wegner.  The $K$-theoretic Farrell-Jones conjecture for CAT(0)-groups.  Proc. Amer. Math. Soc., 140(3):779–793, 2012.