Convergence actions and Specker compactifications
Clément Toromanoff

arXiv:1604.04488v2 [] 28 Mar 2018

March 29, 2018
Abstract
The aim of this paper is to make links between the Specker compactifications of a locally compact
group and its convergence actions on compacta. If G is a convergence group acting on a compactum T
we prove that the closure of G in the attractor sum G⊔T [14] is a quasi-Specker compactification of G.
Together with a theorem due to H. Abels [2], this implies that for any convergence action such that the
limit set Λ is totally disconnected there exists a surjective G-equivariant continuous map Ends G → Λ.
Conversely, when the group is compactly generated we show that any Specker compactification gives
rise to a convergence action. Given two minimal convergence actions of a compactly generated group
on totally disconnected compacta M and N, we can then prove the existence of a minimal compactum
T admitting a convergence action of G and G-equivariant continuous maps T → M and T → N. We
end the paper giving an interpretation of the set of ends of a compactly generated group as the
completion by a “visibility” uniformity [14].

1

Introduction

Convergence groups were introduced by F.W. Gehring and G.J. Martin [12]. The convergence property that defines these groups reflects an essential dynamical behaviour of a kleinian group acting
on the boundary of a hyperbolic space. Wider families of groups satisfy this property, for instance
discrete hyperbolic (resp. relatively hyperbolic) groups can be defined as uniform (resp. expansive)
convergence groups (see [5, 13, 21] and references therein).
We define here the convergence property for a locally compact group acting on a compact Hausdorff space (compactum for short). Locally compact hyperbolic groups can also be characterized as
uniform convergence groups (see [7]). It seems to be interesting to ask which definition of a relatively
hyperbolic group would fit the best to the locally compact case and whether one can extend results
such as in [13, 21, 22].
We refer to the introduction of [13] for a history of convergence groups and motivations for the
locally compact case. More results about discrete convergence groups can be found in [6, 9, 12, 20].
Let us fix a (always Hausdorff) locally compact non compact topological group G acting continuously on a compactum T . We do not assume that T is metrizable so that we need to work with
nets rather than sequences, see section 2 for precise definitions. We identify G with a subgroup of
homeomorphisms of T and we assume that ∣T ∣ > 2.
A net on G is wandering if it has no subnet converging to an element of G. A net (gn )n on G
collapses (to a except at b, where a, b ∈ T ) if for every neighborhood V of a in T and every compact
subset K ⊆ T ∖ {b} we have gn ⋅ K ⊆ V for n large enough. In other words, a net is collapsing if it
converges to a constant map uniformly on compact subsets of T minus one point. Note that a and b
need not be distinct.
Following [6], we say that the action of G on T is convergence (or that G is a convergence group)
if every wandering net on G has a collapsing subnet. The action is convergence if and only if the
1

action on the space on distinct triples is proper, see section 3. The limit set Λ of a convergence action
is the set of all points a ∈ T such that there exists a net on G collapsing to a, it is a G-invariant
closed subset of T .
We show in this paper that the convergence actions of G are closely related to its Specker compactifications in the sense of H. Abels [2]. A quasi-Specker compactification of G is a compactum X
satisfying the following properties:
• X contains G as a dense open subset,
R̂ ∶ G × X → X

is continuous (it extends the right multiplix ⋅ g if x ∈ G
x if x ∈ X ∖ G
cations by elements of G by the identity on X ∖ G),

• the map

(g, x) ↦ {

• for all g ∈ G, the left multiplication

Lg ∶ G → G
extends to a continuous map L̂g ∶ X → X.
h ↦ g⋅h

When in addition the “boundary” X ∖ G is totally disconnected (i.e. its only connected subsets
are the singletons) we say that X is a Specker compactification of G.
Note that this definition does not depend on the action of G on T , note also that the map
L̂ ∶ G × X → X
is an action of G on X extending the left multiplications by elements of G.
(g, x) ↦ L̂g (x)
In the cases we will be interested in this action will be continuous.
Assume that G is a convergence group acting on a compactum T , we denote by ⋅ both the
multiplication of G and the action of G on T . V. Gerasimov introduced in [14] a topology on the
disjoint union X = G ⊔ T , called attractor sum, such that X is a compactum and to which there is a
natural extension of the convergence action of G. Our first result is:
Theorem A (Theorem 11). Let G be a locally compact topological group acting on a compactum
T with convergence property and let X = G ⊔ T be the attractor sum. Assume that ∣T ∣ ≠ 2.
L̂ ∶ G × X → X
The map
is continuous, and the closure of G in X is a quasi-Specker
(g, x) ↦ g ⋅ x
compactification of G (with L̂g the restriction of L̂(g, ⋅) for all g ∈ G).
As a set, the closure of G in the attractor sum is the disjoint union of G and the limit set
Λ (see above). When the limit set is totally disconnected, we obtain then a (complete) Specker
compactification of G.
Let G be an (abstract) locally compact topological group. If G acts on two compacta M and N,
we call G-morphism a G-equivariant continuous map M → N.
By [2, Satz 1.8] the end compactification XE of G (in the sense of [18]), for which the boundary
E = XE ∖ G is the space of ends of G, is a Specker compactification. It is universal, i.e. for any other
Specker compactification X of G there exists a surjective G-morphism (for the L̂ actions) f ∶ XE → X
extending the identity on G. Together with Theorem A, this implies the following:
Corollary B (Proposition 20). For every convergence action of G on a compactum T such that the
limit set Λ is totally disconnected, there exists a surjective G-morphism E → Λ.

Let us fix a convergence action of G on T with limit set Λ. We consider the equivalence relation on
Λ “belong to the same connected component” and denote with Λ′ the associated quotient space, it is
a totally disconnected compactum. We show that the natural induced action of G on T ′ = (T ∖Λ)⊔Λ′
is convergence with limit set Λ′ (Lemma 7). Corollary B implies then the following generalization
of a result of R. S. Kulkarni [17] to the locally compact case (note that we do not require here the
group to be finitely or compactly generated):
2

Corollary C (Proposition 20). If a locally compact topological group admits a convergence action
such that the limit set has more than two components then the group has infinitely many ends.
Our next result deals with compactly generated groups:
Theorem D (Theorem 26). Let G be a locally compact compactly generated topological group. If
X is a Specker compactification of G then the action L̂ of G on X is continuous, convergence and its
limit set is X ∖ G.
Let us consider the family of all totally disconnected spaces which are limit sets of a convergence
action of G. When G is compactly generated Theorem D strengthens Corollary B, stating that this
family has a maximal element, the set of ends of G.
Let G act with convergence property on two compacta M and N. Following V. Gerasimov and
L. Potyagailo [15], we call pullback of these actions a triple (T, πM , πN ) such that T is a compactum
on which G acts with convergence property and πM ∶ ΛT → ΛM and πN ∶ ΛT → ΛN are G-morphisms.
′
′
A pullback (T, πM , πN ) is minimal if for any other pullback (T ′ , πM
, πN
) there exists a G-morphism
′
′
φ ∶ T ′ → T such that πM ○ φ = πM and πN ○ φ = πN .
Using our previous results we show the following (recall that a convergence action of G on a
compactum T is minimal if the limit set is the whole T ):
Corollary E (Theorem 28). Any two minimal convergence actions of a locally compact compactly
generated topological group on totally disconnected compacta containing more than two points admit
a minimal pullback.
Note that Theorem D and Corollary E are not true if we omit the totally disconnectedness
property (or if we replace “Specker” by “quasi-Specker”). We also need to assume that the group
is compactly generated. Counterexamples are due to [3] and [15], see the discussion at the end of
section 6.
We conclude our paper giving an interpretation of the space of ends of a locally compact compactly
generated group as a “visibility completion” .
Recall that the notion of the ends of a topological space or group is originally due to H. Freudenthal
[10, 11]. J. R. Stallings gives in [19] a construction of the space of ends of a finitely generated group
G (or of a connected locally finite graph) as the set of maximal ideals of a Boolean subalgebra of the
power set P(G). H. Abels extends in [2] this construction to any locally compact space or group.
Any end of X in the sense of H. Freudenthal naturally gives an end in the sense of H. Abels.
When X is locally compact, connected and locally connected, this induces in fact a homeomorphism
so that the two definitions coincide (it is a consequence of [4, I,p.117,ex.19]).

Let Γ = (V, E) be a locally finite connected graph. Following V. Gerasimov [14, 4.1], we consider
the uniformity on V generated by the sets of couples of vertices that can be connected by a path
avoiding a fixed finite set of edges. This uniformity is a “visibility”, see [14].
The boundary of the completion by this uniformity can naturally be seen as the set of ends of Γ
in the sense of H. Freudenthal, and V. Gerasimov points out the fact that this is homeomorphic to
the set of ends of Γ in the sense of J. R. Stallings. In particular the two definitions also coincide in
this context.
We show that the same holds for a compactly generated locally compact group (see section 7 for
precise definitions):
Proposition F (Theorem 32). Let G be a compactly generated locally compact topological group.
The space of ends of G (in the sense of H. Abels) is homeomorphic to the boundary of the completion
of G by a “visibility” uniformity.
3

Structure of the paper
Section 2 of our paper provides topological preliminaries, especially the definition and basic properties
of the nets. In section 3 we establish some properties concerning locally compact convergence groups.
In section 4 we give the construction of the attractor sum and prove Theorem A. In section 5 we
recall Abels’ classification of the Specker compactifications of a group, we can then define the space
of ends and prove Corollaries B and C. In section 6 we give some properties of the Cayley graph of
a compactly generated group and we prove Theorem D and Corollary E. Section 7 is devoted to the
proof of Proposition F.
Acknowledgements
This paper was written during my doctoral thesis, I am grateful to my director Leonid Potyagailo
for patiently listening to me, reading preliminaries versions of the paper and providing numerous
corrections. I also want to thank Professor Herbert Abels for inviting me to Bielefeld and the
interesting discussions we had. This work was supported in part by the Labex CEMPI (ANR-11LABX-0007-01).

2

Preliminaries

In the whole paper ∣A∣ denotes the cardinality of a set A. A subset of a topological space is bounded
if it is contained in a compact subspace.
Let X be a topological space. For all x ∈ X, we denote with NX (x) the family of all (not
necessarily open) neighborhoods of x in X. Recall that a local basis at a point x ∈ X is a subfamily
B of NX (x) such that every neighborhood of x contains a member of B. A compactum is a compact
Hausdorff space. By convention, all locally compact spaces (or groups) that we consider are assumed
to be Hausdorff.
It is convenient for us to work with “nets”, which we define below. This notion is equivalent to
the notion of filter (to every net we can associate a filter with the same convergence properties and
conversely), we believe it is easier to describe the dynamical properties of convergence using nets.
We fix a topological space X.
Definitions 1.
• A set D is directed by a transitive reflexive relation ⩾ if for all n, m ∈ D there exists p ∈ D
such that p ⩾ n and p ⩾ m.
• A property P defined of a directed set (D, ⩾) is true for n large enough if there exists m ∈ D
such that whenever n ∈ D satisfies n ⩾ m we have P (n).

• A net on X is a map z ∶ D → X where D is a directed set, we will write it (zn )n∈D or simply
(zn )n . We also say that the net is on a subset A of X if zn ∈ A for all n.
• A net (zn )n on X converges to a point x ∈ X (we write then zn Ð→ x) if for all U ∈ NX (x),
n→∞
zn ∈ U for n large enough.

• A net (ye )e∈E is a subnet of a net (zd )d∈D if there exists a map N ∶ E → D such that y = z ○ N
(i.e. ∀e ∈ E, ye = zN (e) ) and such that for all d ∈ D we have N(e) ⩾ d for e large enough.
We will often not need to explicit the map N and we will write the subnet (ze )e rather than
(zN (e) )e∈E .

• Let (zn )n be a net on X. A point x ∈ X is a cluster point of (zn )n if there exists a subnet of
(zn )n converging to x. The net (zn )n is wandering if it has no cluster point.
4

Note that if (zn )n is a net on X and if P is a property on elements of X, the assertion “P (zn )
for n large enough” is false if and only if there exists a subnet (zi )i such that P (zi ) is false for all i.

Recall the following characterizations (widely used in the langage of sequences when the spaces
are assumed metrizable for instance), see [16] for instance:

• A map f ∶ X → Y (where Y is a topological space) is at a point x ∈ X if and only if for every
net on X converging to x, the net (f (xn ))n converges to f (x).
• A subset A ⊆ X is compact if and only if every net on A has a cluster point in A.
This justify that we repeatedly use formulations such as “up to passing to a subnet” in our proofs.
Recall that X is totally disconnected if its only connected subsets are its singletons, by convention the empty space is not totally disconnected.

3

Locally compact convergence groups

We establish in this section basic facts about locally compact convergence groups acting on a (non
necessarily metrizable) compactum.
We first give the definition of convergence group that we will use. We show then that standard
statements concerning the limit set of such an action extend to the locally compact case. We conclude
with a lemma that provides convergence actions with totally disconnected limit sets.
Let us fix a locally compact group G acting continuously on a compactum T .
Definitions 2.
• A net (gn )n on G is collapsing if there exists a, b ∈ T such that for every compact subset
K ⊆ T ∖ {b} and every neighborhood U ∈ NT (a) we have gn (K) ⊆ U for n large enough. In this
case we write gn ∣(T ∖{b}) −−↠ a.
n→∞

• The group G is a convergence group, or the action of G on T is convergence, if every
wandering net on G has a collapsing subnet.
Note that if G is compact or if ∣T ∣ ≤ 2 the action is always convergence. When ∣T ∣ > 2, any
collapsing net (gn )n is wandering and such that the set {gn ∣ n} is not bounded.
A more standard definition of convergence is the 3-properness, that we define now. Let T be the
locally compact subspace {(x, y, z) ∈ T 3 ∣ x ≠ y ≠ z ≠ x}e it is the space of distinct triples of T .
Definition 3. The action of G on T is proper on triples, or 3-proper, if for all compact subsets
K and L of T the set {g ∈ G ∣ g ⋅ K ∩ L ≠ ∅} is bounded.
When G is a discrete group, the usual terminology is “properly discontinuous” instead of “proper”.
B. Bowditch proves in [6, §1] that if G is discrete then G is a convergence group if and only if its
action on T is 3-properly discontinuous. His arguments extend without difficulty to our situation,
we have then:
Proposition 4. The action of G on T is convergence if and only if it is 3-proper.
In the following we assume that G is a convergence group.
Definitions 5.
• The action is proper at t ∈ T if there exists V ∈ NT (t) such that the set {g ∈ G ∣ g ⋅ V ∩ V ≠ ∅}
is bounded. The ordinary set Ω is the set of all points where the action is proper.
5

• The limit set Λ is the set of attractive points, i.e.:
Λ = {u ∈ T ∣ ∃v ∈ T, ∃(gn )n net on G such that gn ∣(T ∖{v}) −−↠ u}.
n→∞

• The action is minimal if Λ = T .
Proposition 6.
• The limit set Λ is a closed G-invariant subset of T and we have T = Λ ⊔ Ω.
• The restriction of the action of G on Λ is convergence and minimal.
• If ∣Λ∣ ⩾ 3 then Λ is perfect (i.e. every point is an accumulation point), hence infinite.

Proof. For all t ∈ T , a subset K ⊆ T ∖ {t} is compact if and only if the set T ∖ K is an (open)
neighborhood of t in T . As the action is continuous, for all u, v ∈ T , for all g ∈ G and for every
net (gn )n on G, we have gn ∣(T ∖{v}) −−↠ u if and only if (g ⋅ gn )∣(T ∖{v}) −−↠ g ⋅ u, we deduce that Λ is
n→∞
n→∞
G-invariant.
Let us now show that Λ = T ∖ Ω, as Ω is open by construction, Λ is closed. If ∣T ∣ ≤ 2 we easily get
Λ = T , we assume then that ∣T ∣ ≥ 3.
First, let λ ∈ Λ: there exists µ ∈ T and a wandering net (gn )n on G such that gn ∣(T ∖{µ}) −−↠ λ.
n→∞
We can then pick two distinct points x, y ∈ T ∖ {µ} and we have gn ⋅ x, gn ⋅ y Ð→ λ so that λ is non
n→∞
isolated. Under the hypothesis ∣Λ∣ ⩾ 3 we can assume x, y ∈ Λ, as gn ⋅ y, gn ⋅ y ∈ Λ for all n we have the
last point.
Let V ∈ NT (λ), there exists a point v ∈ V ∖ {λ, µ}, and as gn ⋅ v Ð→ λ, we have gn ⋅ V ∩ V ≠ ∅ for
n→∞
n large enough. The set {g ∈ G ∣ g ⋅ V ∩ V ≠ ∅} being not bounded we have λ ∈ T ∖ Ω.
Let then λ ∈ T ∖ Ω. For every V ∈ NT (λ) and every compact subset K ⊆ G there exists then
v = v(V, K) ∈ V and gv ∈ G ∖ K such that gv ⋅ v ∈ V . By construction the net (gv )(V,K) is wandering so
that there exists u, v ∈ T and a subnet (gi )i satisfying gi ∣(T ∖{v}) −−↠ u, and as vi Ð→ v and gi ⋅ vi Ð→ v

we deduce that λ ∈ {u, v} ⊆ Λ. We have the first point, the second is then easy.
i→∞

i→∞

i→∞

Our next lemma will be used to get convergence actions with totally disconnected limit sets.
Let Λ′ be the quotient space of Λ by the equivalence relation “belong to the same connected
component”. We extend this relation to T with the equality on Ω, we write T ′ the quotient space
and π ∶ T → T ′ the canonical projection. Note that T ′ = Ω ⊔ Λ′ .
Lemma 7. The space T ′ is a compactum and Λ′ is a totally disconnected subspace. The formula
g ⋅ π(t) = π(g ⋅ t) induces a well-defined action of G on T ′ . This action is continuous and convergence,
with limit set Λ′ .
Note that π ∶ T → T̃ is continuous, surjective and G-equivariant by construction.
Proof. It is standard that Λ′ is a totally disconnected compactum (see [4] for instance), it is a subspace
of T ′ by construction. The space T ′ is compact as the quotient of a compact space, we show that it
is Hausdorff: let t1 , t2 be distinct points in T ′ .
Assume first that t1 ∈ Ω. Note that A = π −1 (t2 ) is a compact subset of T not containing t1 . As T
is a compactum and as Ω is open in T , there exists a compact neighborhood K of t1 in T contained
in Ω and such that K ∩ A = ∅. Note then that K is a neighborhood of t1 in T ′ and that T ′ ∖ K is a
neighborhood of t2 in T ′ , disjoint from K.
Assume now that t1 , t2 ∈ Λ′ . As Λ′ is a totally disconnected compactum there exists a closed and
open subset U of Λ′ such that t1 ∈ U and t2 ∉ U (see for instance [8]), we write V = π −1 (U), it is a
closed and open subset of Λ. As T is a compactum, there exists disjoint open subsets O1 and O2 of
6

T such that V ⊆ O1 and Λ ∖ V ⊆ O2 . We check then that π −1 (π(Oi )) = Oi for i ∈ {1, 2}, so that π(O1 )
and π(O2 ) are disjoint neighborhood of t1 and t2 in T ′ .

The map (g, π(t)) ↦ π(g ⋅ t) is well-defined because Λ is G-invariant and because G acts by
homeomorphisms so that it acts on the set of components of Λ. We easily check that it is an action
of G on T̃ .
Let g ∈ G and t′ ∈ T ′ and let (gn )n (resp. (t′n )n ) be a net on G (resp. on T ′ ) such that gn Ð→ g
n→∞
in G (resp. t′n Ð→ t′ in T ′ ). We show that the only cluster point of the net (gn ⋅ t′n )n is g ⋅ t′ , as T ′
n→∞
is compact we have then gn ⋅ t′n Ð→ g ⋅ t′ in T ′ . Let then assume that some subnet (gi ⋅ t′i )i converges
n→∞
in T ′ . There exists a net (ti )i on T such that π(ti ) = t′i for all i, up to passing to a subnet we can
assume that ti Ð→ t where t ∈ T . As π is continuous we have t′ = π(t), and as the action of G on T
is continuous, we have gi ⋅ ti Ð→ g ⋅ t so that gi ⋅ t′i = gi ⋅ π(ti ) = π(gi ⋅ ti ) Ð→ π(g ⋅ t) = g ⋅ π(t) = g ⋅ t′ .
i→∞

Let now (gn )n be a net on G, we assume that there exists a, b ∈ T satisfying gn ∣(T ∖{b}) −−↠ a and
n→∞
we show that gn ∣(T ′ ∖{π(b)}) −−↠ π(a). Let V ∈ NT ′ (π(a)) and let K be a compact subset of T ′ ∖ {π(b)}.
n→∞
We have π −1 (V ) ∈ NT (a) and π −1 (K) is closed hence compact and contained in T ∖ {b} so that
gn ⋅ (π −1 (K)) ⊆ π −1 (V ) for n large enough. As π is surjective we deduce that gn ⋅ K = gn ⋅ π(π −1 (K)) =
π(gn ⋅ π −1 (K)) ⊆ π(π −1 (V )) ⊆ V for n large enough.
In other words, collapsing nets “relatively to T ” are also collapsing “relatively to T ′ ” so that the
convergence property for the action on T gives the convergence property for the action on T ′ . We
also deduce that the limit set of the convergence action of G on T ′ is Λ′ , we have the lemma.
i→∞

4

i→∞

Attractor sum

We show in this section that any convergence action induces a quasi-Specker compactification. The
main tool that we use in this prospect is the attractor sum construction of V. Gerasimov [14, §8].
Let us fix again a locally compact group G acting with convergence property on a compactum T .
We first define a topology on the union X = G ⊔ T which extends the original topologies of G
and T , it is the attractor sum. We then show that the space X is a compactum (Proposition 9) and
that the right and left translations extend on X as required (Theorem 10), this yields Theorem A.
We conclude the section with a short proof that the action of G on the attractor sum is convergence
(Proposition 13) and a lemma allowing to extend G-equivariant maps on attractor sums.
The fact that the action of G on the compactum X is convergence is already proven in [14],
we believe that our methods using (collapsing) nets give an other interesting point of view on the
attractor sum.
We denote with ⋅ both the multiplication in G and the action of G on T , so that g ⋅ x makes sense
for x ∈ G or x ∈ T (where g ∈ G).
For any net (gn )n on G and any subset H ⊆ G, let
and

A(gn )n = {u ∈ T ∣ ∃v ∈ T, ∃(gi )i subnet of (gn )n ∶ gi ∣(T ∖{v}) −−↠ u}
A(H)

= {u ∈ T ∣ ∃v ∈ T,

∃(gi )i net on H ∶

i→∞

gi ∣(T ∖{v}) −−↠ u}
i→∞

be the sets of attractor points of (gn )n and H in T .

Definition-Proposition 8. Let X be the formal disjoint union X = G ⊔ T . We say that a subset
P ⊆ X is closed if P ∩ G is closed in G, P ∩ T is closed in T and A(P ∩ G) ⊆ P ∩ T .
The family of closed subsets induces the structure of a topological space on X. We call this space
the attractor sum of G and T (relatively to the action). In this space G is an open subset and T
is a closed subset, moreover the topologies of G and T as subspaces coincide with their initial ones.
7

The proof is straightforward. Note that the case ∣T ∣ = 1 gives the Alexandroff one-point compactification of G. In the case ∣T ∣ = 2, for all H ⊆ G we have A(H) ∈ {∅, T } and the attractor sum is the
Alexandroff compactification with a “doubled” point at infinity. These two points are closed but not
separated in X.
In the following, we assume then that ∣T ∣ ≥ 3.
Proposition 9.
• The attractor sum X = G ⊔ T is a compactum.
• For every net (gn )n on G and for all t ∈ T , we have gn Ð→ t in X if and only if (gn )n is
n→∞
wandering and A(gn )n = {t}.
The second point gives the two following facts:
- if a net (gn )n on G and if t, u ∈ T satisfy gn ∣(T ∖{u}) −−↠ t then gn Ð→ t and gn−1 Ð→ u in X,
n→∞

n→∞

n→∞

- the set of accumulation points of G in T is exactly the limit set of the action.
Proof. We first show that X is Hausdorff, using the same proof as V. Gerasimov [14, Prop. 8.2.4].
Let p and q be distinct points of X, we distinguish three cases:
• Case 1: p, q ∈ G. As G is Hausdorff and open in X we can find disjoint neighborhoods of p and
q in G hence in X.
• Case 2: p ∈ G and q ∈ T . Let us pick a compact K ∈ NG (p), then K is a neighborhood of p in
X, closed in X because A(K) = ∅, hence X ∖ K is a neighborhood of q in X disjoint to K.
• Case 3: p, q ∈ T . We need some preliminaries.

Let us fix a subset ξ ⊆ T of cardinality 3. For every subset A ⊆ T let ξA = {g ∈ G ∣ ∣A ∩ g ⋅ ξ∣ ⩾ 2}
and ξ + A = A ∪ ξA. We easily check that ξ and ξ + commutes with the complement (in G or in
X respectively) and that for all disjoint subsets A and B of G the subsets ξ + A and ξ +B of X
are still disjoint. Let A be a closed subset of T , then T ∩ ξ + A = A is closed in T . Let (gn )n be
a net on G ∩ ξ + A = ξA converging to a point g ∈ G, we show that g ∈ ξA, as a consequence ξA
is closed in G. For all n we have ∣A ∩ gn ⋅ ξ∣ ⩾ 2, as ξ is finite and A is compact, up to passing
to a subnet we can assume that there exists distinct points x1 and x2 in ξ such that gn ⋅ x1 ∈ A
and gn ⋅ x2 ∈ A for all n, and that there exists points a1 , a2 ∈ A such that gn ⋅ x1 Ð→ a1 and
n→∞
gn ⋅ x2 Ð→ a2 . The action of G on T is continuous so that g ⋅ x1 = a1 ∈ A and g ⋅ x2 = a2 ∈ A, hence
n→∞
g ∈ ξA. Finally, if there exists u ∈ A(ξA) ∖ A, there exists v ∈ T and a net (gn )n on ξA such
that gn ∣(T ∖{v}) −−↠ u, and as X ∖ A ∈ NX (u), we have gn (ξ ∖ {v}) ⊆ X ∖ A for n large enough,
n→∞
but ∣ξ ∖ {v}∣ ⩾ 2 and gn ∈ ξA for all n gives a contradiction. We deduce that A(ξA) ⊆ A and we
conclude that ξ + A is closed in X.
Now, if Op and Oq are disjoint open neighborhoods of p and q in T then ξ +Op and ξ +Oq are
disjoint open subsets of X containing p and q.

We show now the second point: let (gn )n be a net on G and let t ∈ T .
We first assume that gn Ð→ t in X. The net (gn )n (seen as a net on G) is wandering, by the
n→∞
convergence property we have A(gn )n ≠ ∅. For r ∈ T ∖ {t}, there exists Or ∈ NX (r) open and such
that Fr = X ∖ Or belongs to NX (t). On one hand Fr is closed in X so that A(G ∩ Fr ) ⊆ T ∩ Fr (which
does not contain r), one the other hand gn Ð→ t in X so that gn ∈ G ∩ Fr for n large enough. We
n→∞
deduce that r ∉ A(gn )n , and necessarily A(gn )n = {t}.
8

Conversely, assume that (gn )n is wandering and that A(gn )n = {t}. For every O ∈ NX (t) open,
the complement F = X ∖ O is closed in X so that A(G ∩ F ) ⊆ T ∩ F (which does not contain t), hence
there is no subnet of (gn )n on G ∩ F . This means exactly that gn ∈ G ∖ F ⊆ O for n large enough, so
that gn Ð→ t in X.
n→∞

We can now show that X is compact. Let (xn )n be a net on X, up to passing to a subnet, we
limit the proof to the three following cases. First case: xn ∈ T for all n. As T is compact the net has
a cluster point in T and it is also a cluster point in X. Second case: xn ∈ G for all n and the net has
a cluster point in G. Again, this is a cluster point in X. Third case: xn ∈ G for all n and the net
is wandering. By the convergence property there exists a subnet (xi )i and points u, v ∈ T such that
xi∣(T ∖{v}) −−↠ u, by the previous point we have xi Ð→ u in X. We have shown that every net on X
i→∞
i→∞
has a cluster point: X is compact.
Theorem 10. Assume that G is a topological group.
R̂ ∶ G × X → X
L̂ ∶ G × X → X
The actions
are continuous.
x ⋅ g if x ∈ G and
(g, x) ↦ g ⋅ x
(g, x) ↦ {
x if x ∈ T
Proof. The two maps are actions of G on X by construction. We only prove the continuity of R̂, the
case of L̂ is similar.
It is enough to show the continuity at a point (g, t) ∈ G × T . Let (gn )n be a net on G and (xn )n be
a net on X such that gn Ð→ g in G and xn Ð→ t in X, we show that R̂(gn , xn ) = xn ⋅gn Ð→ R̂(g, t) = t
n→∞
n→∞
n→∞
in X.
Using the partition {n ∣ xn ∈ T } ⊔ {n ∣ xn ∈ G} and noting that R̂∣(G×T ) is continuous, it is enough
to consider the case where xn ∈ G for all n. By the previous proposition the net (xn )n is wandering
and satisfies A(xn )n = {t}, we want to show that the net (xn ⋅ gn )n is wandering and has only one
attractor point, namely t.

By contradiction, assume that there exists a subnet (xi ⋅ gi )i converging in G to a point h ∈ G. As
gi Ð→ g in G we have gi−1 Ð→ g −1 in G so that xi = (xi ⋅ gi ) ⋅ gi−1 Ð→ h ⋅ g −1 in G hence in X, this is in
contradiction with xi Ð→ t ∈ T . We deduce that the net (xn ⋅ gn )n is wandering.
i→∞

i→∞

i→∞

Let then u, v ∈ T and (xi ⋅ gi )i∈I be a subnet such that (xi ⋅ gi )∣(T ∖{v}) −−↠ u, we show that
i→∞

xi∣(T ∖{g⋅v}) −−↠ u so that necessarily u = t. Let then K be a compact subset of T ∖ {g ⋅ v} and
i→∞
V ∈ NT u.
We first claim that there exists a neighborhood W0 ∈ NT (v) such that gi−1 ⋅ K ⊆ T ∖ W0 for i
large enough. Assume that this is not true: for all W ∈ NT (v) and for all i ∈ I, there exists then
j = j(i, W ) ⩾ i and kj ∈ K such that gj−1 ⋅kj ∈ W . The map (gj−1 ⋅kj )(i,W ) is then a net on T , converging
to v by construction. As K is compact, some subnet (kj ′ )j ′ of (kj )(i,W ) converges to a point k ∈ K
distinct from g ⋅ v. But the action of G on T is continuous so that the net (gj−1′ ⋅ kj ′ )j ′ converges to
g −1 ⋅ k ≠ v, a contradiction.
Without loss of generality, we can assume that T ∖ W0 is a compact subset of T ∖ {v}. As
(xi ⋅ gi )∣(T ∖{v}) −−↠ u, for i large enough we have xi ⋅ K = (xi ⋅ gi ) ⋅ (gi−1 ⋅ K) ⊆ (xi ⋅ gi ) ⋅ (T ∖ W0 ) ⊆ V .
i→∞

We have the stated result, we deduce that A(xn ⋅ gn )n = {t}.
i→∞

We can now prove Theorem A:
Theorem 11. If G is a locally compact convergence group acting on a compactum T containing at
least three points then the closure of G in the attractor sum of G and T , endowed with the restriction
of the L̂-action defined previously, is a quasi-Specker compactification of G (see the introduction or
Definitions 15).
9

We will use the following corollary:
Corollary 12. Let G be a locally convergence group acting on a compactum T containing at least
three points with a totally disconnected limit set. The closure of G in the attractor sum of G and T
is a (complete) Specker compactification of G.
Proof. Let us write Ḡ the closure of G in the attractor sum X of G and T . As X is a compactum so
is Ḡ, and it contains G as a dense subset by definition. As a set Ḡ is the disjoint union of G and the
limit set Λ by proposition 9. As Λ is G-equivariant the maps L̂ and R̂ restrict to maps G × Ḡ → Ḡ
which are continuous and extend the multiplications of G.
As the third point of Definitions 15 only depends on the topology of the boundary, we get the
corollary.
Note that by Lemma 7 any convergence action induces a convergence action with a totally disconnected limit set, hence a Specker compactification
Note also that Theorems 10 and 11 are trivially true if ∣T ∣ = 1, if ∣T ∣ = 2 we still have Theorem 10
(in the attractor sum, every wandering net on G converges to any point in T , and any non wandering
net has all its cluster points in G).
We now give a short proof of [14, 8.3]:
Proposition 13. The action of G on the attractor sum X of G and T is convergence.
Proof. Let (gn )n be a wandering net on G, we can assume that there exists u, v ∈ T such that
gn ∣(T ∖{v}) −−↠ u (we have then gn Ð→ u in X, see Proposition 9). We claim that gn ∣(X∖{v}) −−↠ u.
n→∞
n→∞
n→∞
Arguing by contradiction, assume that this is not the case. There exists then a compact subset
K ⊆ X ∖ {v}, a neighborhood U ∈ NX (u), a subnet (gi )i and a net (xi )i on K such that gi ⋅ xi ∉ U
for all i. We can assume that there exists a point x ∈ K ⊆ X ∖ {v} and a point y ∈ X ∖ {u} such that
xi Ð→ x and gi ⋅ xi Ð→ y in X.
i→∞
i→∞
Assume first that x ∈ G. We have then xi ∈ G for i large enough hence xi Ð→ x in G, but gi Ð→ u
i→∞

i→∞

in X and the map R̂ is continuous so that gi ⋅ xi = R̂(xi , gi ) Ð→ R̂(x, u) = u ≠ y in X, a contradiction.

We deduce that x ∈ T , and as gi−1 Ð→ v in X the same argument applied to xi = gi−1 ⋅ (gi ⋅ xi ) gives
i→∞
y ∈ T.
Up to passing to a subnet, we can assume either that xi ∈ T for all i or that xi ∈ G for all i. The
first case contradicts the assumption gi ∣(T ∖{v}) −−↠ u, and in the second case, we can assume that there
i→∞

exists λ, µ ∈ T such that xi∣(T ∖{λ}) −−↠ x and gi ⋅ xi ∣(T ∖{µ}) −−↠ y. Let us pick a point p ∈ T ∖ {λ, µ}, on
i→∞

one hand we have (gi ⋅ xi ) ⋅ p Ð→ y because p ≠ µ and on a other hand we have xi ⋅ p Ð→ x ≠ v so that
i→∞

i→∞

gi ⋅ (xi ⋅ p) Ð→ u, once more a contradiction.
i→∞

i→∞

i→∞

We will need the following result in section 6, it has already been proved in the case of a discrete
group [15, Lemma 5.3].
Lemma 14. Let G act with convergence property on two compacta M and N and let f ∶ M → N be
a non-constant continuous G-equivariant map. If ∣M∣ ≥ 3 then the map f˜ ∶ G ⊔ M → G ⊔ N extending
f on the attractor sums by the identity on G is continuous.
Note that necessarily ∣N∣ ≥ 3.
Proof. Under the hypotheses of the lemma we first show the following: if ma , mr ∈ M, if na , nr ∈ N and
if (gn )n is a wandering net on G such that gn ∣(M ∖{mr }) −−↠ ma and gn ∣(N ∖{nr }) −−↠ na then na = f (ma )
n→∞
n→∞
and nr = f (mr ).
10

Note first that mr is not isolated in M. Indeed, it follows from the definition of a collapsing
net that gn−1∣(M ∖{ma }) −−↠ mr , as ∣M∣ ≥ 3 we can find x, y ∈ M such that ∣{x, y, a}∣ = 3, we have then
n→∞
gn−1 ⋅ x, gn−1 ⋅ y Ð→ ma , and gn−1 ⋅ x ≠ gn−1 ⋅ y for all n.
n→∞
Note then that f −1 (nr ) ∪ {mr } ⊊ M. If not, as mr is not isolated in M we can find a net (mn )n
on M converging to mr and such that mn ≠ mr for all n, by assumption f (mn ) = nr for all n, the
continuity of f gives then f (mr ) = nr so that f is constant, a contradiction.
We can then take m ∈ M ∖ (f −1 (nr ) ∪ {mr }), on one hand m ≠ mr so that gn ⋅ m Ð→ ma and
n→∞
f (gn ⋅ m) Ð→ f (ma ), on an other hand f (m) ≠ nr so that gn ⋅ f (m) Ð→ na , but gn ⋅ f (m) = f (gn ⋅ m)
n→∞
n→∞
for all n so that na = f (ma ).
As gn−1 ∣(M ∖{ma }) −−↠ mr and gn−1 ∣(N ∖{na }) −−↠ nr the same proof gives nr = f (mr ).
n→∞
n→∞
We can now prove the lemma. Easily, f˜ is continuous on the open subsets G and ΩM of M. Using
the same techniques as in the proofs of Theorem 10 and Proposition 9, we prove that if (xn )n is a net
on G ⊔ M converging to a point λ ∈ ΛM , then f˜(xn ) Ð→ f˜(λ) = f (λ). It is enough to consider the
n→∞
case when xn ∈ G for all n, the net (xn )n is then wandering and satisfies AM (xn )n = {λ}. If na , nr ∈ N
and if (xi )i is a subnet satisfying xi ∣(N ∖{nr }) −−↠ na , we can assume that there exists ma , mr ∈ M such
that xi∣(M ∖{mr }) −−↠ ma , necessarily ma = λ and the previous result gives then na = f (λ). We deduce
i→∞

that AN (xn )n = {f (λ)} and the lemma.
i→∞

5

Specker Compactifications

The aim of this section is to recall Abels’ classification of the Specker compactifications of a given
group [2, Satz 1.6 and 1.8] (our Proposition 17), this yields the definition of the space of ends. We
conclude the section with the proof of Corollaries B and C.
Our methods in the following sections use a similar machinery, in order the paper to be selfcontained we provide some details here.
In the whole section we fix a locally compact topological group G.
Definitions 15. A topological space X is a Specker compactification of G if it satisfies the
following properties:
1. X is a compactum,
2. G is homeomorphic to a dense open subset of X (we will identify G and this subset),
3. the “boundary” Λ = X ∖G is totally disconnected (its only connected subsets are the singletons),
4. the map

R̂ ∶ G × X → X
(g, x) ↦ {

5. for all g ∈ G, the map

x ⋅ g if x ∈ G
x if x ∈ X ∖ G

is continuous,

Lg ∶ G → G
extends to a continuous map Lˆg ∶ X → X.
h ↦ g⋅h

A topological space X satisfying the three first points (related only to the topological structure
of G) is a zero-dimensional compactification of G. A topological space X satisfying all points
but the third is a quasi-Specker compactification of G.
Two compactifications X and Y of G are equivalent if the identity map G → G extends to a
homeomorphism X → Y .
Notations 16.
11

• For all A, B ⊆ G, let A + B = (A ∖ B) ∪ (B ∖ A) be the symmetric difference of A and B.
• If A is a subset of a topological space X, we write ∂X A the boundary of A for the topology on
X.
• Let B (resp. K) be the family of all bounded (resp. compact) subsets of G.
• Let U + = {A ⊆ G ∣ ∂G A ∈ K}. Endowed with symmetric difference and intersection U + is a
Boolean algebra (with identity elements 0 = ∅ for + and 1 = G for ∩) in which B is an ideal.
• Let U G = {A ∈ U + ∣ ∀K ∈ K ∖ {∅}, (A ⋅ K) + A ∈ B}, this is a subalgebra of U + containing B and
a G-module for the left action of G.
Proposition 17. The map X ↦ U (X) = {B ∩ G ∣ B ⊆ X and ∂X B ⊆ G} induces a bijection between
equivalence classes of zero-dimensional compactifications of G and subalgebras of U + containing B.
In this correspondance the (equivalence classes of) Specker compactifications of G correspond exactly
to the subalgebras of U G containing B which are also G-submodules of U G .
Sketch of the proposition’s proof. Let us first fix a zero-dimensional compactification X of G, we write
Λ = X ∖ G. The key fact to get the classifications is the following lemma:

Lemma 18. For all x ∈ X, the family Wx = {B ⊆ X ∣ x ∈ B and ∂X B ⊆ G} is a local basis at x.

As G is locally compact, the case x ∈ G is easy. Assume x ∈ Λ. As Λ is compact and totally
disconnected it has a base of closed and open subsets, and as X is a compactum it is normal (i.e.
disjoint closed subsets have disjoint neighborhoods), see for instance [8]. It is then an exercise to
show the lemma.

Let U (X) be as in the statement of the proposition, we easily check that it is a subalgebra of U +
containing B.
A subfamily F of U (X) is an unbounded ultrafilter in U (X) if it satisfies the following
properties :
(F0) X ∈ F (or F ≠ ∅),
(F1) for all A, B ∈ F we have A ∩ B ∈ F ,
(F2) for all A, B ∈ U (X), if A ∈ F and A ⊆ B then B ∈ F ,
(U) for all A ∈ U (X), either A ∈ F or G ∖ A ∈ F ,

(NB) F ∩ B = ∅.
Using the previous lemma we can show that for all λ ∈ Λ, the subfamily Fλ = {B ∩ G ∣ B ∈ Wλ } of
U (X) is an unbounded ultrafilter in U (X). Moreover, we can show that the map λ ↦ Fλ is a bijection
between Λ and the set of unbounded ultrafilters in U (X). We have then characterized points of Λ
only in terms of the subalgebra U (X) ⊆ U + .
Up to taking complement in P(U (X)) it is equivalent to consider prime ideals instead of ultrafilters (see [2]). Our choice is more consistant with section 7.
Let now U be any subalgebra of U + containing B. Following the previous results, let ∂U G be the
set of unbounded ultrafilters in U and let X(U ) = G ⊔ ∂U G. Let O denote the set of all open subsets
of G, for A ∈ U ∩ O we define A+ = A ∪ {F ∈ ∂U G ∣ A ∈ F }, let then U+O be the set of all A+ for
A ∈ U ∩ O. The family O ∪ U+O is then a base for a topology on X(U ) satisfying all properties of a
zero-dimensional compactification of G (see [2, Satz 1.6] for more details).
It is then enough to check that U (X(U )) = U and that X and X(U (X)) are equivalent for any
zero-dimensional compactification X of G in order to get the first part of the proposition.
12

Now, let us fix a Specker compactification X of G, we set again Λ = X ∖ G.
If A = B ∩ G where B ⊆ X satisfies ∂X B ⊆ G and if g ∈ G, we have g ⋅ A = L̂g (B) ∩ G and L̂g is a
homeomorphism so that ∂X (Lˆg (B)) = Lˆg (∂X B) ⊆ G. We deduce that U (X) is a G-module for the
left action. Using topological arguments, one can also show that U (X) ⊆ U G .

Conversely, let U be a subalgebra and a G-submodule of U G containing B. Let X be the corresponding zero-dimensional compactification of G (first part of the proposition), we have then
X ∖ G = ∂U G.
For all g ∈ G, we check that the map Lˆg such that Lˆg (h) = g ⋅h for h ∈ G and Lˆg (F ) = {g ⋅A ∣ A ∈ F }
for F ∈ ∂U G is continuous. We can also show that the map R̂ ∶ G × Y ↦ Y (see Definitions 15) is
continuous. In conclusion X is a Specker compactification, we have the proposition.
Proposition 19. The Specker compactification XE (G) = X(U G ) satisfies the following “universal”
property: for every Specker compactification X of G there exists a (unique) surjective G-equivariant
continuous map XE (G) → X extending the identity on G.

It follows that the space XE (G) is the ends compactification in the sense of E. Specker [18].
The space E(G) = XE (G) ∖ G is the space of ends of G, it is a totally disconnected compactum on
which G acts with the L̂-action. We will see in section 7 that when G is compactly generated, the
space of ends XE of G can be seen as the boundary of a completion of G by a “visibility” uniformity
on G.
Note that the map XE (G) → X given by the proposition restricts to a surjective G-equivariant
continuous map E(G) → X ∖ G.
Proof. We use the same terminology as in the previous proof.
First note that if U1 and U2 are subalgebras and submodules of U G such that U1 ⊆ U2 , then the
X(U2 ) → X(U1 )
x
if x ∈ G
map
is surjective, G-equivariant, continuous, and extends the
x ↦ {
x ∩ U1 if x ∈ ∂U2 G
identity on G.
Let then X be a Specker compactification of G. Taking U1 = U (X) and U2 = U G in the previous
point we get a map f ∶ XE (G) → X(U (X)), by the previous proof there exists a homeomorphism
ψ ∶ X(U (X)) → X, the map ψ ○ f is then the desired map XE (G) → X.
We can now prove Corollary B et C of the introduction:
Proposition 20. Let G be a locally compact convergence group acting on a compactum T .
• If the limit set Λ is totally disconnected and if ∣T ∣ ≠ 2 then there exists a G-equivariant surjective
continuous map E(G) → Λ.

• If the limit set has more than two components then G has infinitely many ends (i.e. ∣E(G)∣ = ∞).

Proof. Assume first that Λ is totally disconnected (then Λ ≠ ∅ by convention). The case ∣T ∣ = 1 is
trivial (we have then Λ = T ), and if ∣T ∣ ≥ 3 by Corollary 12 the closure G∪Λ of G in the attractor sum
G ∪ T is a Specker compactification of G, by the previous proposition there exists then a surjective
map E(G) → Λ.
Assume now that Λ has more than two components. Let Λ′ be the totally disconnected quotient
space of Λ where we collapsed connected components to points and T ′ = (T ∖Λ)∪Λ′ , then G acts on T ′
with convergence property and with limit set Λ′ (see Lemma 7). By propositon 6 Λ′ is infinite, by the
previous point there exists a surjective map E(G) → Λ′ , we conclude that E(G) is also infinite.

13

6

Case of a compactly generated group

We want now to prove Theorem D, from which we will deduce Corollary E. The tools that we use
are adapted from [1] and [2, §2 and §3].
We start with the construction of the Cayley graph of a compactly generated group G, we fix then
a Specker compactification X and show the convergence property for the action of G on X (Theorem
26), this yields Theorem D. We conclude with the proof of Corollary E and some considerations
about the hypotheses in Theorem D.
In the whole section, G is a compactly generated locally compact topological group, we write
multiplicatively (with ⋅) the group operation.
Lemma 21. There exists a compact symmetric neighborhood F ∈ NG (1) that generates G.
Proof. If K is a compact subset generating G (i.e. G = ⋃n∈N K n ) by Baire’s lemma some K n has a
non-empty interior, we then check that F = (K ∪ K −1 )2n is as in the statement of the lemma.
Definitions 22. The Cayley graph of G (relatively to the generating set F ) is the set (of edges)
Γ = {(x, x ⋅ f ) ∣ x ∈ G, f ∈ F }. A Γ-path connecting two points g, h ∈ G is a finite sequence
g = x0 , x1 , . . . , xn = h (where n ∈ N) of points in G such that (x0 , x1 ), . . . , (xn−1 , xn ) ∈ Γ, we say
that the path is in a subset A ⊆ G if all the xi belongs to A. Let A be a subset of G. We say that
A is Γ-connected if any two points in A can be connected by a Γ-path in A. Let ∂Γ A = {(g, h) ∈
Γ ∣ ∣{g, h} ∩ A∣ = 1} be the Γ-boundary of A. For all n ∈ N, let A⩽n = A ⋅ F n be the set of points of
G whose “Γ-distance” from A is at most n, it is easy to see that the projection ∣∂Γ A∣ of ∂Γ A on any
component in G × G is exactly the set ∣∂Γ A∣ = A⩽1 ∩ (G ∖ A)⩽1 .
Recall that U G is the set of all “compactly almost invariant” subsets of G and that B is the family
of bounded subsets of G (see Notations 16).
Proposition 23.
• G is Γ-connected.
• Any bounded subset of G is contained in a compact Γ-connected subset.
• The family UΓ = {A ⊆ G ∣ ∣∂Γ A∣ ∈ B} coincides with U G .
In [2] the third point is stated but it is proved only in the case where G is discrete.
Proof. Let g, h ∈ G. As F generates G, there exists n ∈ N and f1 , . . . , fn ∈ F such that g −1 ⋅h = f1 . . . fn .
This means that 1, f1 , f1 ⋅ f2 , . . . , f1 ⋅ . . . ⋅ fn = g −1 ⋅ h is a Γ-path connecting 1 to g −1 ⋅ h, so that the
“translated” path g = g ⋅ 1, g ⋅ f1 , . . . , g ⋅ (g −1 ⋅ h) = h is a Γ-path connecting g to h. We have the first
point.
Recall that F is a compact symmetric neighborhood of 1 ∈ G. If K is a fixed compact Γ-connected
subset of G (for instance a singleton), we easily check that (K ⩽n )n∈N is an increasing sequence of
compact Γ-connected subsets of G whose interiors cover G, so that any bounded subset of G is
contained in some K ⩽n . We deduce the second point.

Let us show that U G = UΓ . If A ∈ U G , we check that ∣∂Γ A∣ ⊆ (A + A ⋅ F ) ⋅ F ∈ B (recall that
+ stands for the symmetric difference) hence A ∈ UΓ . Conversely, let A ∈ UΓ . We first check that
A + A ⋅ F ⊆ ∣∂Γ A∣ = ⋃f ∈F (A + A ⋅ f ) hence A + A ⋅ F ∈ B and A + A ⋅ f ∈ B for all f ∈ F . Using the fact
that F generates G, we get that A + A ⋅ g ∈ B and A + A ⋅ F ⋅ g ∈ B for all g ∈ G. If W is a non-empty
compact subset of G, there exists a finite subset V ⊆ W such that W ⊆ ⋃v∈V F ⋅ v, and we can show
that for any w ∈ W we have A + A ⋅ W ⊆ ⋃v∈V (A + A ⋅ F ⋅ v) ∪ (A + A ⋅ w) ∈ B. As A ⋅ F (resp. (G ∖ A) ⋅ F )
is a neighborhood of A (resp. of G ∖ A) in G, we get ∂G A ⊆ ∣∂Γ A∣ so that ∂G A ∈ B. We conclude that
A ∈ U G.
14

Until the end of the section, we fix a Specker compactification X of G (Definitions 15) and we
write Λ = X ∖ G.
The following lemma is proven in [2, Lemma 3.1] :
Lemma 24. The map

L̂ ∶ G × X → X
is continuous.
(g, x) ↦ Lˆg (x)

We need the following lemma (generalizing Lemma 18) to prove the convergence property. It is
adapted from [1, 3.5 and 4.4] and [2, 3.4].
Lemma 25. Let λ ∈ Λ.

• The family Bλ = {V ∈ NX (λ) ∣ ∣∂Γ (V ∩ G)∣ ∈ B and G ∖ V is Γ-connected} is a local basis at λ.

• If V ∈ Bλ and if B ⊆ X satisfies ∣∂Γ (B ∩ G)∣ ∈ B, there exists a neighborhood W of λ in X such
that for all g ∈ W ∩ G, either g ⋅ B ⊆ V or g ⋅ B ⊇ X ∖ V .
Proof. In the proof, we will use the following notations: V and V ′ will be subsets of X, U and U ′
will be their traces of G, i.e. U = V ∩ G and U ′ = V ′ ∩ G, and Q and Q′ will be their complement in
G, i.e. Q = G ∖ V = G ∖ U and Q′ = G ∖ V ′ = G ∖ U ′ .
Recall that Proposition 17 gives U (X) ⊆ U G and that Proposition 23 gives UΓ = U G . We deduce
that for any subset B ⊆ X, if ∂X B ⊆ G then ∣∂Γ (B ∩ G)∣ ∈ B.

Let V ′ ∈ NX (λ), by Lemma 18 we can assume that ∂X V ′ ⊆ G so that ∣∂Γ U ′ ∣ ∈ B. There exists a
compact Γ-connected subset K ⊆ G containing ∣∂Γ U ′ ∣ by Proposition 23, we claim that V = V ′ ∖ K
belongs to Bλ .
As K is a compact subset of G we already have V ∈ NX (λ). For all A, B ⊆ G we check that
(A ∩ B) ⋅ F ⊆ A ⋅ F ∪ B ⋅ F so that ∣∂Γ (A ∩ B)∣ ⊆ ∣∂Γ A∣ ∪ ∣∂Γ B∣. Noting then that ∣∂Γ (G ∖ K)∣ = ∣∂Γ K∣ ⊆
K ⩽1 ∈ B and that U = U ′ ∩ (G ∖ K), we get that ∣∂Γ U∣ is bounded.
In order to show that Q = G ∖ U is Γ-connected, we show that every Γ-path connecting points of
Q and “going out Q” can be “diverted” so that it stays in Q. Let then a, b ∈ Q, u, v ∈ U = G ∖ Q, n ∈ N
and g1 , . . . , gn ∈ G be such that (a, u), (u, g1 ), . . . , (gn , v), (v, b) ∈ Γ, we want to show that a, b ∈ K.
Note that Q = Q′ ∪ K and that if a ∈ Q′ , as (a, u) ∈ Γ and as Q′ = G ∖ U ′ ⊆ (G ∖ U ′ ) ⋅ F there exists
f ∈ F such that u = a ⋅ f hence a = u ⋅ f −1 ∈ U ⋅ F ⊆ U ′ ⋅ F and finally a ∈ ∣∂Γ U ′ ∣ ⊆ K. In all cases we
have a ∈ K and the same holds for b. As K is Γ-connected there exists a Γ-path connecting a to b in
K hence in Q.
Let now a, b be any points in Q. By Γ-connectedness of G there exists a Γ-path connecting a to
b and we can “divert all Γ-subpaths going out Q” as earlier (there are only finitely many of them) so
that the resulting path is a Γ-path in Q. We deduce that Q is Γ-connected and the first point.
Let V ∈ Bλ and let B ⊆ X be such that ∣∂Γ A∣ is bounded, where A = B ∩ G. Recall that R̂ is
continuous on G × X and that R̂(G × {λ}) = {λ}. For g ∈ ∣∂Γ A∣ there exists then Og ∈ NG (g) and
Wg ∈ NX (λ) such that R̂(Og × Wg ) ⊆ V , and by compacity there exists a finite family D ⊆ ∣∂Γ A∣ such
that ∣∂Γ A∣ ⊆ ⋃g∈D Og . Then W = ⋂g∈D Wg is a neighborhood of λ in X such that for any g ∈ W ∩ G
we have R̂(∣∂Γ A∣ × {g}) ⊆ V i.e. g ⋅ ∣∂Γ A∣ ⊆ V .
Let us assume that there exists g ∈ W ∩ G and a, b ∈ G ∖ V such that {g −1 (a), g −1 (b)} intersects
both A and G ∖A. As G ∖V is Γ-connected, there exists n ∈ N and a Γ-path a = x0 , x1 , . . . , xn = b in
G ∖ V . The translated path g −1 ⋅ x0 , . . . , g −1 ⋅ xn connects a point in A and a point in G ∖ A, it follows
from the definition of ∣∂Γ A∣ that there exists i such that g −1 ⋅ xi ∈ ∣∂Γ A∣, but then xi = g ⋅ (g −1 ⋅ xi ) ∈
G ∖ V ∩ g ⋅ ∣∂Γ A∣ = ∅, a contradiction.
This means that for all g ∈ W ∖ G there are two cases: either g −1 (G ∖ V ) ⊆ G ∖ A, hence
G ∖ V ⊆ g ⋅ (G ∖ A) and g ⋅ A ⊆ G ∩ V ⊆ V ; or g −1 (G ∖ V ) ⊆ A, hence g ⋅ A ⊇ G ∖ V . In both these
inclusions we can change A to B and G to X because x ↦ g ⋅ x is an homeomorphism of X letting G
and Λ invariant and because P ∩ Λ = P ∩ G ∩ Λ for all P ∈ U +, hence for P ∈ {B, V, X ∖ V } (see the
proof of Proposition 17). We have the second point.
15

We can now prove Theorem D:
Theorem 26. If G is a locally compact compactly generated topological group and if X is a Specker
compactification of G then the action of G on X is convergence and its limit set is the boundary
Λ = X ∖ G.
Proof. Let (gn )n be a wandering net on G. As X is compact, up to passing to a subnet we can
assume that there exists points λ, µ ∈ X such that gn Ð→ λ and gn−1 Ð→ µ in X. The net (gn )n
n→∞
n→∞
is wandering on G which is open in X and g ↦ g −1 is continuous on G, we then have necessarily
λ, µ ∈ Λ. We claim that gn ∣(X∖{µ}) −−↠ λ, we deduce the convergence property.
n→∞
Assume that it is not the case: there exists a compact set K ⊆ X ∖{µ}, a neighborhood V ∈ NX (λ)
and a subnet (gi )i such that gi ⋅ K ⊈ V for all i. We can assume that ∣∂Γ (V ∩ G)∣ is bounded and that
G ∖ V is Γ-connected by Lemma 25. There exists then a net (xi )i in K such that gi ⋅ xi ∉ V for all i,
and up to passing to a subnet we can assume that (xi )i converges to a point x ∈ K.
Let us fix h ∈ G ∖ V . We show that gi−1 ⋅ h = R̂(h, gi−1 ) Ð→ x ∈ X, which is in contradiction with
i→∞

gi−1 Ð→ µ in X and the continuity of R̂ on G × X (note that x ≠ µ).

Let Vx ∈ NX (x), we can assume that ∣∂Γ (Vx ∩ G)∣ ∈ B (Lemma 25). Now, if W ∈ NX (λ) is as
in the second point of the lemma applied to B = Vx , for i large enough gi ∈ W ∩ G and xi ∈ Vx but
gi ⋅ xi ∉ V hence gi ⋅ Vx ⊇ X ∖ V , so that h ∈ gi ⋅ Vx and gi−1 ⋅ h ∈ Vx for i large enough. This means that
gi−1 ⋅ h Ð→ x.
i→∞

i→∞

The statement concerning the limit set is not hard. Any g ∈ G has a compact neighborhood K
in G hence in X and the set {h ∈ G ∣ h ⋅ K ∩ K ≠ ∅} is exactly K ⋅ K −1 which is compact, so that
the ordinary set contains G. Conversely, as G is dense in X for any λ ∈ Λ there exists a net (gn )n
in G such that gn Ð→ λ, up to passing to a subnet we can assume that there exists µ ∈ X such that
n→∞
gn−1 Ð→ µ, and as above we have µ ∈ Λ and gn ∣(X∖{µ}) −−↠ λ. We deduce that the limit set contains
n→∞
n→∞
Λ and the stated result.
Note that B. Bowditch gives in [?] a simple proof that the action of a finitely generated group on
its space of ends is convergence, this proof can be extended without difficulty to our situation. We
easily deduce that the action of G on any Specker compactification is convergence, but is seems hard
to prove directly that such an action is minimal. For instance, there exists locally compact groups
having infinitely many ends, one of them being fixed by the whole group (see [?]).
Let us now prove Corallary E. We first show a lemma establishing the existence of a “minimal
Specker pullback”:
Lemma 27. Let X and Y be two quasi-Specker compactifications of G. Let S(X, Y ) be the closure
in X × Y of the diagonal ∆(G) = {(g, g) ∣ g ∈ G}, we write π1 ∶ X × Y → X (resp. π2 ∶ X × Y → Y ) the
canonical projection. We have the following:
• The space S(X, Y ) is a quasi-Specker compactification of G (for the natural L̂ action).
• For every quasi-Specker compactification Z of G endowed with two G-equivariant continuous
maps πX ∶ Z → X and πY ∶ Z → Y extending the identity on G there exists a unique surjective
G-equivariant continuous map φ ∶ Z → S(X, Y ) extending the identity on G and such that
π1 ○ φ = πX and π2 ○ φ = πY .
• When X and Y are “complete” Specker compactifications of G, so is S(X, Y ).

Proof. Note that ∆(G) is naturally homeomorphic to G, we identify then g ∈ G and (g, g) ∈ ∆(G).
By construction S(X, Y ) is a compactum and contains G as a dense subset, we have the points
1 and 2 in Definitions 15.
16

Let us write R̂X and L̂X (resp. R̂Y and L̂Y ) the extensions to G × X (resp. to G × Y ) of the right
and left translations of G. By definition of the product topology, the maps
R̂ ∶ G × (X × Y ) → X × Y
L̂ ∶ G × (X × Y ) → X × Y
and
(g, (x, y)) ↦ (R̂X (g, x), R̂Y (g, y))
(g, (x, y)) ↦ (L̂X (g, x), L̂Y (g, y))

are continuous. Using the fact that G is dense in S(X, Y ) it is an exercise to show that these two
maps let S(X, Y ) invariant.
We deduce the points 4 and 5 of Definitions 15: S(X, Y ) is a quasi-Specker compactification of
G.
It is easy to see that S(X, Y ) ∖ G ⊆ (X ∖ G) × (Y ∖ G). As the connected components of a
product are the product of the components and as the components of a subspace are contained in
the components of the space, we deduce the last point of the lemma.
Let now Z be a quasi-Specker compactification of G endowed with G-equivariant continuous maps
φ ∶ Z → S(X, Y )
πX ∶ Z → X and πY ∶ Z → Y extending the identity on G. The map
is
t ↦ (πX (t), πY (t))
the only map Z → S(X, Y ) satisfying πX = π1 ○ φ and πY = π2 ○ φ. By construction the map φ is
continuous, G-equivariant and extend the identity on G, one can also show that it is surjective.
Recall that a pullback of two minimal convergence actions of G on compacta M and N is a
triple (P, πM , πN ) where P is a compactum on which G acts with minimally convergence property
and πM ∶ P → M (resp. πN ∶ P → N) is a G-equivariant surjective continuous map. The pullback
(P, πM , πN ) is minimal if for any other pullback (T, pM , pN ) there exists a G-equivariant continuous
map φ ∶ T → P such that πM ○ φ = pM and πN ○ φ = pN .
We can now prove Corollary E:
Theorem 28. Any two minimal convergence actions of a locally compact compactly generated
topological group on totally disconnected compacta containing more than two points admit a minimal
pullback.
Proof. Assume that G is compactly generated and acts minimally with convergence property on two
totally disconnected compacta M and N containing more than two points. We first construct a
pullback and then show that it is minimal.
By Corollary 12 the attractor sums X = G ⊔ M and Y = G ⊔ N are Specker compactifications of G.
We consider the Specker compactification S(X, Y ) of G and the projections π1 ∶ S(X, Y ) → X and
π2 ∶ S(X, Y ) → Y given by Lemma 27. We set P = S(X, Y ) ∖ G and πM = (π1 )∣P (resp. πN = (π2 )∣P ),
it is a G-equivariant continuous map P → M (resp. P → N). By Corollary ?? the action of G on P
is convergence and minimal so that (P, πM , πN ) is a pullback of the actions of G on M and N.
Let now (T, pM , pN ) be an other pullback of these actions. We first consider the totally disconnected quotient space T ′ of T by the relation “belong to the same component”, the natural action of
G on T ′ is convergence and minimal (see Lemma 7, we have here T = Λ).
For every connected component C of T the image pM (C) is a connected subset of M (the map pM
is continuous) hence a singleton (M is totally disconnected). There exists then a map p′M ∶ T ′ → M
satisfying p′M ○ π = pM (where π ∶ T → T ′ is the canonical projection).
The construction of p′M and the G-equivariance of pM and π induce that p′M is also G-equivariant,
it is not hard to see that it is continuous. Applying the same arguments with N we get a G-equivariant
surjective continuous map p′N ∶ T ′ → N such that p′N ○ π = pN .
It follows that (T ′ , p′M , p′N ) is a pullback of the actions of G on M and N. Note that ∣T ′ ∣ ≥ 3 (the
action of G on M is minimal, ∣M∣ ≥ 3 and p′M is G-equivariant) and that p′m is not constant (it is
surjective).
Let us then consider the attractor sum Z = G ⊔ T ′ , by Corollary 12 again this is a Specker
compactification of G. Let p′X ∶ Z → X (resp. p′Y ∶ Z → Y ) be the extension of p′M (resp. of
17

p′N ) on Z by the identity of G, this map it continuous, surjective and G-equivariant (see Lemma
14). By the second point of Lemma 27 there exists then a surjective G-equivariant continuous map
ψ ∶ Z → S(X, Y ) extending the identity on G and satisfying π1 ○ ψ = p′X and π2 ○ ψ = p′Y .
We have then ψ(T ′ ) = P so that the map ψ ○ π ∶ T → P is G-equivariant, surjective and
continuous. We have the stated result.
We end the section with considerations about the hypothesis in Theorem 26. The following two
facts are known:
• there exists two minimal convergence actions of the discrete free group of countable rank on
two totally disconnected compacta admitting no pullback space ([15, Prop 5.5]),
• there exists two minimal convergence actions of the discrete free group of rank 3 on two compacta (one of them not being totally disconnected) admitting no pullback space ([15, Prop
5.2]).
The last result follows directly from an example due to O. Baker and T. Riley in [3]: the authors
exhibit a hyperbolic group G and a free subgroup H < G of rank 3 such that the inclusion do not
induce a Cannon-Thurston map ∂H → ∂G. Note that in [15] the authors study 2-cocompact actions
(i.e. geometrically finite), but we do not need this property in our paper.
This two results show that both hypothesis “the group is compactly generated” and “the Specker
boundary is totally disconnected” cannot be removed from the statement of Theorem 26.

7

The ends of a compactly generated group

The aim of this section is to give an other interpretation of the space of ends of a compactly generated
group. This construction is inspired by a remark of V. Gerasimov (in the context of a finitely
generated discrete group) [14, 4.1].
We first recall some results about uniform spaces. Recall that a filter on a set X is a non-empty
family F of subsets of X stable by finite intersection (∀A, B ∈ F , A ∩ B ∈ F ) and by “growth” (if
A ∈ F and if B ⊆ X satisfies A ⊆ B then B ∈ F ).
Definition 29. A uniformity V on a set X is a filter of X × X such that:
• for all V ∈ V the set V −1 = {(y, x) ∣ (x, y) ∈ V } belongs to V,

• every V ∈ V contains the diagonal ∆ = {(x, x) ∣ x ∈ X},

• for all V ∈ V there exists U ∈ V such that V contains the set
U ○ U = {(x, z) ∈ X × X ∣ ∃y ∈ X, (x, y), (y, z) ∈ U}.
The uniformity V is exact if ⋂ V = ∆, i.e. ∀x ≠ y ∈ X, ∃V ∈ V, (x, y) ∉ V.
Definition-Proposition 30. Let V be a uniformity on X.
• For x ∈ X and V ∈ V let V [x] = {y ∈ X ∣ (x, y) ∈ V }. There exists a unique topology on X
(the uniform topology) such that the family (V [x])V ∈V is a local basis at every x ∈ X. This
topology is Hausdorff if and only if V is exact.
• For V ∈ V we say that a subset A ⊆ X is V -small if A × A ⊆ V and we write Small(V ) the set
of all V -small subsets of X.
18

• A Cauchy filter on X is a filter on X containing V -small sets for every V ∈ V. A Cauchy filter
is minimal if it is minimal for the inclusion, we write X̃ the set of all minimal Cauchy filters
on X. The filters NX (x) for x ∈ X are minimal Cauchy filters, they are called convergent.
• For V ∈ V we set Ṽ = {(F , G) ∈ X̃ × X̃ ∣ F ∩ G ∩ Small(V ) ≠ ∅}, the family {Ṽ ∣ V ∈ V} generates
a uniformity on X̃. This uniform space is called the completion of X by the uniformity
V. When V is exact the map x ↦ NX (x) is a homeomorphism of X on a dense open subset
of X̃ (both spaces endowed with the uniform topology) and we can identify any x ∈ X with
NX (x) ∈ X̃.
In the whole section we fix a locally compact compactly generated topological group G and we
use the notations and results of the previous section. The Cayley graph Γ of G is seen as a subspace
of G × G. Recall that B denotes the set of all bounded subsets of G and let BΓ be the set of all
bounded subsets of Γ for this topology.
Let us fix a subset M ⊆ Γ. A M-path is a sequence g0 , . . . , gn (where n ∈ N) such that (gi , gi+1 ) ∈ M
for all i. A subset A ⊆ G is M-connected if any two points in A can be connected with a M-path.
The M-components of G are the maximal connected subsets of G, we write π0 (M) the set of all
the M-components.
We can now define a “visibility” on G.
Definition-Proposition 31.
For M ∈ BΓ let vM = {(g, h) ∈ G×G ∣ there exists a (Γ∖M)-path connecting g and h}. The family
V = {v ⊆ G × G ∣ ∃M ∈ BΓ , vM ⊆ V } is an exact uniformity on G.
−1
= vM ○ vM , moreover for
Proof. For all M ∈ BΓ we have vM = ⋃C∈π0 (Γ∖M ) C × C so that ∆ ⊆ vM = vM
all M, N ∈ BΓ we have vM ∩ vN ⊇ vM ∪N . We deduce that V is a uniformity on G.
This also shows that a subset A ⊆ G is vM -small if and only if there exists some C ∈ π0 (Γ ∖ M)
such that A ⊆ C.

For all g ∈ G the set F (g) = {(h1 , h2 ) ∈ Γ ∣ h1 = g or h2 = g} is bounded and the component of
Γ ∖ F (g) to which g belongs is exactly {g}. We deduce that V is exact.
Note that the topology induced by V on G is the discrete topology.
For M ∈ BΓ let uM be the set of all (g, h) ∈ G × G such that every shortest Γ-path connecting g
and h avoids M. As G is Γ-connected we have uM ⊆ vM for all M ∈ BΓ . This is the definition of a
visibility on Γ, see [14].
The main result of this section is the following theorem (see Propositions 17 and 19 for the
definitions):
Theorem 32. For every non-convergent minimal Cauchy filter F on G the family F ∩ U G is an
unbounded ultrafilter on U G , i.e. a point of the set of ends E(G) of G.
f ∶ G̃ ∖ G → E(G)
The map
is a homeomorphism.
F
↦ f (F ) = F ∩ U G
Proof. Let F be a non-convergent Cauchy filter on G, we first show that f (F ) ∈ E(G). The points
(F0), (F1) and (F2) of the proof of Proposition 17 are immediate.
Assume that F contains a bounded subset B ∈ B. The set F (B) = {(h1 , h2 ) ∈ Γ ∣ h1 ∈ B or h2 ∈ B}
is bounded and F is Cauchy so that there exists A ∈ Small(vF (B) ) ∩ F . We have then A ∩ B ∈ F ,
but F (B) contains all the edges with one end in B so that necessarily A ∩ B is a singleton and F is
convergent. We deduce the point (NB).
In order to show the point (U) we need to prove the following fact: every A ⊆ G such that
∂Γ A ∈ BΓ is a union of (Γ ∖ ∂Γ A)-components (and so is G ∖ A). Indeed, let a ∈ A and b ∈ G be in
19

the same (Γ ∖ ∂Γ A)-component. There exists then a (Γ ∖ ∂Γ A)-path connecting a and b, and if some
point on this path is not in A then this path has an edge in ∂Γ A, a contradiction.
For all A ⊆ G such that ∂Γ A ∈ BΓ , as every P ∈ Small(v∂Γ A ) is contained in a (Γ ∖∂Γ A)-component
every such P is a subset of A or of G ∖ A.
Let then A ∈ U G , recall that ∂Γ A ∈ BΓ (see Proposition 23). As F is Cauchy it contains a (v∂Γ A )small set P , we have then P ⊆ A or P ⊆ G ∖ A so that A ∈ F or G ∖ A ∈ F . We deduce the point
(U).
The following lemma, inspired by [4, I,p.117,ex.19], enables us to give the inverse map of f .
Lemma 33. Let us fix M ∈ BΓ ∖ {∅, G}, we write p(M) = {g ∈ G ∣ ∃h ∈ G, (g, h) ∈ M or (h, g) ∈ M}
(we have ∅ ≠ p(M) ∈ B). Let K be a compact subset of G such that p(M)⩽1 ⊆ K̊ (for instance
K = {1}⩽n for some n ∈ N big enough, see Proposition 23) and L = K ⩽1 ∖ K̊, this is a compact subset.
1. For all C ∈ π0 (Γ ∖ M) we have C ∩ p(M)⩽1 ≠ ∅.
2. For all C ∈ π0 (Γ ∖ M) such that C ∩ L = ∅ we have C ⊆ K̊.
3. The set {C ∈ π0 (Γ ∖ M) ∣ C ∩ L ≠ ∅} is finite.
4. For all F ∈ E(G), there exists a unique C(M, F ) ∈ π0 (Γ ∖ M) ∩ F , moreover C(M, F ) ∉ B.
Proof of the lemma. Let C ∈ π0 (Γ∖M) (then C ≠ ∅) and assume C ⊈ p(M)⩽1 . Pick any c ∈ C∖p(M)⩽1
and any g ∈ p(M), and let c = g0 , . . . , gn = g be a Γ-path. The set {i ∈ J1, nK ∣ gi ∈ p(M)} is not empty
and has a smallest element i > 0, and as g0 , . . . , gi−1 ∉ p(M) we have (g0 , g1 ), . . . , (gi−2 , gi−1 ) ∉ M so
that gi−1 ∈ C (the path from c = g0 to gi−1 stays in the same (Γ ∖ M)-component, namely C). As
gi ∈ p(M) we have gi−1 ∈ C ∩ p(M)⩽1 , we deduce the first point.
Assume that C ∈ π0 (Γ ∖ M) satisfies C ∩ L = ∅. Fix c0 ∈ C ∩ p(M)⩽1 (then c0 ∈ C ∩ K̊) and let c
be any point in C. By definition of a (Γ ∖ M)-component there exists a Γ-path c0 , . . . , cn = c in C.
Assume that this path intersects G ∖ K̊ and let i > 0 be the smallest index such that ci ∉ K̊, we have
then ci ∈ C ∩ (K̊ ⩽1 ∖ K̊) ⊆ C ∩ L = ∅, a contradiction. As a consequence we have c ∈ K̊, we deduce the
second point.
Let C ∈ π0 (Γ ∖ M) be such that C ∩ L ≠ ∅, we first show that C ∩ L is open in L. Let c ∈ C ∩ L.
If g ∈ L is such that (c, g) ∈ Γ, as c ∉ p(M) we have (c, g) ∉ M so that g ∈ C. The set c ⋅ F is an open
neighborhood of c in G so that (c ⋅ F ) ∩ L is a neighborhood of c in L, contained in C ∩ L.
As L is compact, the open covering L = ⋃C∈π0 (Γ∖M ) (C ∩L) has a finite subcovering, and as different
components are disjoints, we have the third point.
Let now F ∈ E(G). Every C ∈ π0 (Γ ∖ M) such that C ∩ L = ∅ is contained in K by the third
point, so that X is covered by the finite union K ∪ ⋃{C ∈ π0 (Γ ∖ M) ∣ C ∩ L ≠ ∅}. Note that every
C ∈ π0 (Γ ∖ M) satisfy ∂Γ C ⊆ M so that C ∈ U G , as X ∈ F the properties (U) and (NB) give the fourth
point.
The last point of the lemma is in fact also true for any non-convergent Cauchy filter F on G,
indeed F must contain a vM -small set, hence a (unbounded) (Γ ∖ M)-component, for every M ∈ BΓ .
Note then that C(M, f (F )) = C(M, F ) for all M ∈ BΓ .
Let F ∈ E(G), we define g(F ) = {A ⊆ G ∣ ∃M ∈ BΓ , C(M, F ) ⊆ A}. For all M, N ∈ BΓ we have
C(M, F ) ∩ C(N, F ) ⊆ C(M ∪ N, F ), we easily deduce that g(F ) is a filter on G. As the C(M, F ) are
vM -small and unbounded this filter is Cauchy and non-convergent.
Let F0 be the unique minimal Cauchy filter contained in g(F ) (see [4, II]), we have F0 ⊆ g(F ).
As different component are disjoint we necessarily have C(M, F0 ) = C(M, F ) so that g(F ) ⊆ F0 and
the equality. We deduce that g(F ) ∈ G̃.
This shows in fact that for every Cauchy filter A on G, the family (C(M, A))M ∈BΓ is a basis for
the unique minimal Cauchy filter contained in A.
20

Note that for every F ∈ E(G) and every A ∈ U G , we have A ∈ F if and only if C(∂Γ A, F ) ⊆ A
(recall that either ∂Γ A ⊆ A or ∂Γ A ⊆ G ∖ A and that either A ∈ F or G ∖ A ∈ F ). The previous
construction “preserves” the family (C(M, ⋅))M ∈BΓ , we deduce that for all A ∈ U G we have A ∈ F if
and only if A ∈ f (g(F )). This shows that F = f (g(F )).
Similarly, let F be a minimal Cauchy filter on G, then F and g(f (F )) are minimal Cauchy filters
on G with the “same family (C(M, ⋅))M ∈BΓ ”, they are then equal.
We have shown that the map f is a bijection and that g is its inverse.
Let us show that f is continuous: let F ∈ G̃ ∖ G and let O be a neighborhood of f (F ) in E(G),
we show that f −1 (O) is a neighborhood of F in G̃ ∖ G. We can assume that O = {G ∈ E(G) ∣ A ∈ G}
for some open subset A ∈ U G , note then that A ∈ f (F ) = F ∩ U G ⊆ F . Let us write M = ∂Γ A and
V = v˜M [F ] ∖ G = {G ∈ G̃ ∖ G ∣ (F , G) ∈ v˜M }, it is a neighborhood of F . For all G ∈ G̃ ∖ G we have
G ∈ V ⇔ F ∩ G ∩ Small(vM ) ≠ ∅ ⇔ C(M, F ) = C(M, G), and as A ∈ F we deduce that G ∈ V imply
A ∈ G and f (G) ∈ O, we have then F ∈ V ⊆ f −1 (O).
Conversely, let F ∈ G̃ ∖ G and let U be a neighborhood of F in G̃ ∖ G, we can assume that there
exists M ∈ BΓ such that U = v˜M [F ] ∖ G. Let us write C = C(M, F ), we can show that ∣∂Γ C̊∣ ⊆ ∣∂Γ C∣⩽1
so that C̊ ∈ U G , the set O = {G ∈ E(G) ∣ C̊ ∈ G} is then a neighborhood of f (F ) in E(G). As
previously we show that for all G ∈ G̃ ∖ G such that f (G) ∈ O we have G ∈ U, we conclude that
f (F ) ∈ O ⊆ f (U) and that f is open.
Finally, f is a homeomorphism as stated.
For any M ⊆ N ∈ BΓ let us consider the natural map ιM,N ∶ π0 (Γ ∖ N) → π0 (Γ ∖ M) which
sends a (Γ ∖ N)-component to the (Γ ∖ M)-component containing it, we have then an inverse system
({π0 (Γ ∖ M) ∣ M ∈ BΓ }, {ιM,N ∣ M ⊆ N ∈ BΓ }). It follows from the previous proof that the map
F ↦ (C(M, F ))M ∈BΓ is a bijection from the set G̃ ∖ G of the minimal non-convergent Cauchy filters
to the inverse limit of this inverse system.
Theorem 32 can then be rephrased in the following “unformal” way: the set of ends of a compactly
generated group coincides with the sets of the components of the “ideal boundary” of its Cayley-graph.

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