REDES SOCIALES

(B) There exists a nontrivial totally geodesic subset S $ bordX such that Λ_⊆S.

(C) There exists a nontrivial totally geodesic subset S $ bordX such that

CΛ⊆S.

(D) There exists a nontrivial totally geodesic subset S $ bordX such that

Co⊆S for someo∈X.

Proof of(A)_⇒(B). Let S $ bordX be a nontrivial totally geodesic G- invariant subset. Fixo∈S∩X. Then Λ⊆G(o)⊆S. Proof of(B) _⇒(C). IfS is any totally geodesic set which contains Λ, then

S is a closed convex set containing Λ, so by Proposition 7.5.2,CΛ⊆S. Proof of (C)⇒(D). Since G is nonelementary, CΛ 6=. Fix o∈ CΛ; then

Co⊆ CΛ.

Proof of (D)⇒(A). Let S be the smallest totally geodesic subset of X

which contains_Co, i.e.

S:=\_{W :W _{⊇ C}o totally geodesic}.

Then our hypothesis implies that S $bordX. Sinceo ∈S, S is nontrivial. It is obvious from the definition thatS isG-invariant. This completes the proof. Remark7.6.4. IfGIsom(X) is nonelementary, then Proposition 7.6.3 gives us a way to find a nontrivial totally geodesic set on whichGacts reducibly; namely, the smallest totally geodesic set containing Λ, or equivalently _CG, will have this property (cf. Lemma 2.4.5). On the other hand, there exists a parabolic group

G _≤ Isom(H∞_{) such that G does not act irreducibly on any nontrivial totally}

geodesic subsetS ⊆bordH∞ _{(Remark 11.2.19).}

7.7. Semigroups of compact type

Definition 7.7.1. We say that a semigroup GIsom(X) is ofcompact type if its limit set Λ is compact.

Proposition7.7.2. For GIsom(X), the following are equivalent: (A) Gis of compact type.

(B) Every sequence (xn)∞1 in G(o) with kxnk → ∞ has a convergent subse- quence.

Furthermore, ifX is regularly geodesic, then (A)-(B) are equivalent to: (C) The set_Co is a proper metric space.

(D) The setCΛ is a proper metric space.

Proof of (A)_⇒(B). Fix a sequence (gn)∞1 in Gwithkgnk → ∞. The exis- tence of such a sequence implies thatGis not elliptic. If Gis parabolic or inward focal, then the proof of Proposition 7.3.1(ii) shows thatgn(o)→ξ, where Λ ={ξ}. So we may assume thatGis lineal, outward focal, or of general type, in which case Proposition 7.3.1 gives #(Λ)≥2.

Fix distinct ξ1, ξ2 ∈ Λ, and let (nk)∞1 be a sequence such that (gnk(ξi))

∞

1

converges fori= 1,2, and such that

hg−nk1(o)|ξ1io≤ hg

−1

nk(o)|ξ2io

for allk. (If this is not possible, switchξ1 andξ2.) We have

0_≍+,ξ1,ξ2 hξ1|ξ2io&+min hg −1 nk(o)|ξ1io,hg −1 nk(o)|ξ2io =_hg−1 nk(o)|ξ1io and thus hgnk(o)|gnk(ξ1)io≍+,ξ1,ξ2 kgnkk −→ n ∞. On the other hand, there existsη_∈Λ such thatgnk(ξ1)−→

k η, and thus

hgnk(ξ1)|ηio−→

n ∞. Applying Gromov’s inequality yields

hgnk(o)|ηio −→

n ∞ and thusgnk(o)−→

k η. This completes the proof.

Proof of (B)⇒(A). Fix a sequence (ξn)∞1 in Λ. For each n ∈ N, choose gn∈Gwith

hgn(o)|ξnio≥n. In particularkgnk ≥n −→

n ∞. Thus by our hypothesis, there exists a convergent subsequencegnk(o)−→ k η ∈Λ. Now D(ξnk, η)≤D(gnk(o), ξnk) +D(gnk(o), η).×b −nk_+D(g nk(o), η)−→ k 0, i.e. ξnk−→ k η.

Proof of (A)⇒(C). Let (xn)∞1 be a bounded sequence in Co. For each

n _∈ N, there exist yn(1), y(2)n ∈ G(o) such that xn ∈ N1/n([y(1)n , yn(2)]). Choose a sequence (nk)∞1 on whichy (1) nk −→ k αandy (2) nk −→

k β. SinceX is regularly geodesic we have

[yn(1)k, y (2)

nk]−→_k [y

(1)_{, y}(2)_].

For eachk, choosezk ∈[yn(1)k, y (2)

nk] withd(xnk, zk)≤1/nk. Since the sequence (zk)

∞

1

7.7. SEMIGROUPS OF COMPACT TYPE 125

it follows that the corresponding subsequence of (xnk)

∞

1 is also convergent. Thus

every bounded sequence in_Cohas a convergent subsequence, soCo is proper.

Proof of(C) _⇒(B). Obvious sinceG(o)_{⊆ C}o.

Proof of(A) ⇒(D). Note first of all that we cannot get (A)⇒(D) imme- diately from Proposition 7.5.3, since theτ-thickening of a compact set is no longer compact.

By [35, Proposition 1.5], there exists a metricρon bordX compatible with the topology such that the mapF _7→Hull1(F) is a semicontraction with respect to the

Hausdorff metric of (bordX, ρ). (Finite-dimensionality is not used in any crucial way in the proof of [35, Proposition 1.5],2_{and in any case for algebraic hyperbolic}

spaces it can be proven by looking at finite-dimensional subsets, as we did in the proof of Proposition 7.5.3.) We remark that if F = R, then such a metric ρ can be prescribed explicitly: if X =Bis the ball model, then the Euclidean metric on bordB_{⊆ H}has this property, due to the fact that geodesics in the ball model are line segments in_H(cf. (2.2.3)).3

Now let us demonstrate (D). It suffices to show that bordCΛ = Hull∞(Λ) is

compact. Since Hull∞(Λ) is by definition closed, it suffices to show that Hull∞(Λ)

is totally bounded with respect to the ρ metric. Indeed, fix ε > 0. Since Λ is compact, there is a finite setFε⊆Λ such that

Λ_⊆Nε/2(Fε).

(In this proof, all neighborhoods are taken with respect to theρmetric.) LetXε⊆

X be a finite-dimensional totally geodesic set containingFε. Then Λ⊆Nε/2(Xε). On the other hand, sinceXεis compact, there exists a finite setFε′⊆Xεsuch that

Xε⊆Nε/2(Fε′).

Now, our hypothesis onρimplies that

Hull1(Nε/2(Xε))⊆Nε/2(Hull1(Xε)) =Nε/2(Xε),

and thus thatNε/2(Xε) is convex. But Λ⊆Nε/2(Xε), so Hull∞(Λ)⊆Nε/2(Xε). Thus

Hull∞(Λ)⊆Nε/2(Xε)⊆Nε(Fε′).

Sinceεwas arbitrary, this shows that Hull∞(Λ) is totally bounded, completing the

proof.

Proof of (D)⇒(B). Since property (B) is clearly basepoint-independent, we may without loss of generality supposeo_{∈ C}Λ. Then (D)⇒(C)⇒(B). 2_{One should keep in mind that the Cartan–Hadamard theorem [119, IX, Theorem 3.8] can be}

used as a substitute for the Hopf–Rinow theorem in most circumstances.

As an example of an application we prove the following corollary.

Corollary 7.7.3. Suppose thatX is regularly geodesic. Then any moderately discrete subgroup of Isom(X)of compact type is strongly discrete.

Proof. IfGis a moderately discrete group, thenG↿_Cois moderately discrete by Observation 5.2.14, and therefore strongly discrete by Propositions 5.2.5 and 7.7.2. Thus by Observation 5.2.14,Gis strongly discrete. A well-known characterization of the complement of the limit set in the Stan- dard Case is that it is the set of points where the action ofGis discrete. We extend this characterization to hyperbolic metric spaces for groups of compact type:

Proposition7.7.4. Let G_≤Isom(X)be a strongly discrete group of compact type. Then the action ofGonbordX\Λis strongly discrete in the following sense: For any set S_⊆bordX_\Λ satisfying

(7.7.1) D(S,Λ)>0,

we have

#{g∈G:g(S)∩S6=}_<∞_.

Proof. By contradiction, suppose that there exists a sequence of distinct (gn)∞1 such that gn(S)∩S 6= for all n ∈ N. Since G is strongly discrete, we have_kgnk → ∞, and sinceGis of compact type there exist an increasing sequence (nk)∞1 andξ+, ξ− ∈Λ such thatgnk(o)→ξ+ andg

−1

nk(o)→ξ−. In the remainder

of the proof we restrict to this subsequence, so thatgn(o)→ξ+ andgn−1(o)→ξ−.

For eachn, fix xn∈g−n1(gn(S)∩S), so thatxn, gn(xn)∈S. Then

D(xn, ξ−), D(gn(xn), ξ+)≥D(S,Λ)≍×1,

and so

hxn|ξ−io,hgn(xn)|ξ+io≍+ 0.

On the other hand,hg−1

n (o)|ξ−io,hgn(o)|ξ+io→ ∞. Applying Gromov’s inequality gives

hxn|gn−1(o)io,hgn(xn)|gn(o)io≍+0

for allnsufficiently large. But then

kgnk=hgn(xn)|oign(o)+hgn(xn)|gn(o)io≍+0,

Part 2