SUFFICIENT CONDITIONS FOR POINCAR ´ E REGULARITY

Proposition 9.3.1. Let G ≤Isom(X) be nonelementary, and assume either that

(1) X is regularly geodesic andGis moderately discrete,

(2) X is an algebraic hyperbolic space and Gis weakly discrete, or that (3) X is an algebraic hyperbolic space and G is COT-discrete and acts irre-

ducibly.

Then Gis Poincar´e regular.

Remark 9.3.2. Example 13.4.2 shows that Proposition 9.3.1 cannot be im- proved by replacing “COT” with “UOT”, Example 13.4.9 shows that Proposition 9.3.1 cannot be improved by removing the assumption thatGacts irreducibly, Ex- ample 13.4.1 shows that Proposition 9.3.1 cannot be improved by removing the hypothesis thatX is an algebraic hyperbolic space from (ii), and Example 13.4.4 shows that Proposition 9.3.1 cannot be improved by removing the assumption that

X is regularly geodesic.

We begin with the following observation:

Observation9.3.3. If (3) implies thatGis Poincar´e regular, then (2) does as well.

Proof. Suppose (2) holds, and letS be the smallest totally geodesic subset of bordX which contains Λ (cf. Lemma 2.4.5). SinceGis nonelementary,V :=S_∩X

is nonempty; it is clear thatV isG-invariant. By Observation 5.2.14, the actionG↿ V is weakly discrete. By Proposition 5.2.7,G↿V is COT-discrete. Furthermore,G

acts irreducibly onV because of the wayV was defined (cf. Proposition 7.6.3). Thus (3) holds for the actionG↿V, which by our hypothesis implies ∆G =∆eG(since the Poincar´e set and modified Poincar´e set are clearly stable under restrictions). We now proceed to prove that (1) and (3) each imply thatGis Poincar´e regular. By contradiction, let us suppose thatGis Poincar´e irregular. By Proposition 8.2.4(ii), we have that Gis not strongly discrete and thus

e

δG< δG=∞.

This gives us two contrasting behaviors: On one hand, by Proposition 8.2.4(iii), there exist ρ > 0 and a maximal ρ-separated set Sρ ⊆G(o) so that Sρ does not contain an bounded infinite set. On the other hand, sinceGis not strongly discrete, there existsσ >0 such that #(Gσ) =∞, where

Gσ :={g∈G:g(o)∈B(o, σ)}.

Proof. Suppose not. Then the set Gσ(ξ) is complete (with respect to the metric D) but not compact. It follows that Gσ(ξ), and thus also Gσ(ξ), is not totally bounded. So there existsε >0 and an infiniteε-separated subset (gn(ξ))∞1 .

Fix L large to be determined. Since ξ _∈ Λ, we can find x_∈ G(o) such that

hx_|ξ_io≥L.

Subclaim 9.3.5. By choosingL large enough we can ensure

d(gm(x), gn(x))≥2ρ ∀m, n∈N.

Proof. By (d) of Proposition 3.3.3,

hgn(x)|gn(ξ)io≍+,σhgn(x)|gn(ξ)ign(o)=hx|ξio≥L,

and thus

D(gn(x), gn(ξ)).×,σb−L. IfLis large enough, then this implies

D(gn(x), gn(ξ))≤ε/3.

Since by construction the sequence (gn(ξ))∞1 isε-separated, we also have D(gm(ξ), gn(ξ))≥ε

and then the triangle inequality gives

D(gm(x), gn(x))≥ε/3, or, taking logarithms,

hgm(x)|gn(x)io.+ −logb(ε/3). Now we also have

kgn(x)k ≍+,σkxk ≥ hx|ξio≥L and therefore

d(gm(x), gn(x)) = kgm(x)k+kgn(x)k −2hgm(x)|gn(x)io

&+,σ2L−2(−logb(ε/3)).

Thus by choosingL sufficiently large, we ensure thatd(gm(x), gn(x))≥2ρ.

⊳ Recall thatSρ is a maximalρ-separated set. Thus for eachn∈N, we can find

yn ∈Sρwithd(gn(x), yn)< ρ. Then the subclaim implies yn6=ymforn6=m. But on the other hand

kynk ≤ kxk+σ+ρ ∀n∈N,

9.3. SUFFICIENT CONDITIONS FOR POINCAR ´E REGULARITY 151

We now proceed to disprove the hypotheses (1) and (3) of Proposition 9.3.1. Thus if either of these hypotheses are assumed, we have a contradiction which finishes the proof.

Proof that (1) cannot hold. SinceGis assumed to be nonelementary, we can find distinct pointsξ1, ξ2∈Λ. By Claim 9.3.4, there exist a sequence (gn)∞1 in Gσ and pointsη1, η2∈Λ such that

gn(ξi)−→ n ηi.

Next, choose a point x∈[ξ1, ξ2]. For eachn∈N, we have gn(x)∈[gn(ξ1), gn(ξ2)].

Thus since X is regularly geodesic there exist a sequence (nk)∞1 and a point z ∈

[η1, η2] such that

gnk(x)−→

k z.

Since gn ∈ Gσ ∀n, the sequence (gn(x))∞1 is bounded and thus z ∈ X. By

contradiction, suppose that G is moderately discrete, and fix ε > 0 satisfying (5.2.2). For all m, n∈ Nlarge enough so that gm(x), gn(x)∈ B(z, ε/2), we have

d(x, g−1

mgn(x)) =d(gm(x), gn(x))≤ε. Thus for someN∈N, we have #_{g−1

m gn:m, n≥N}<∞.

This is clearly a contradiction.

Proof that (3) cannot hold. Now we assume thatX is an algebraic hy- perbolic space, say X = H = Hα_F, and that G acts irreducibly on X. Using the identification

Isom(H)_≡O∗(_L;_Q)/_∼,

(Theorem 2.3.3), for eachg_∈GσletTg∈O∗(L;Q) be a representative ofg. Recall (Lemma 2.4.11) that

kTgk=kTg−1k=ekgk,

so sinceg ∈Gσ we have kTgk =kTg−1k ≤bσ. In particular, the family (Tg)g∈Gσ

acts equicontinuously onL.

For simplicity of exposition, in the following proof we will assume that X

is separable. (In the non-separable case, the reader should use nets instead of sequences.) It follows that Λ_⊆∂X is also separable; let (ξk = [xk])∞1 be a dense

Claim 9.3.6. There exists a sequence of distinct elements (gn)∞1 in Gσ such that the following hold:

Tgn[xk] −→ n y (+) k ∈ L \ {0} Tg−n1[xk] −→_n y (−) k ∈ L \ {0} σ(Tgn)−→ n σ∈Aut(F).

Proof. For eachk∈Nlet

Kk={y∈ L \ {0}: [y]∈Gσ(ξk) andb−σ≤ kyk ≤bσ}, and let K:= Y k∈N Kk !2 ×Aut(F).

Then by Claim 9.3.4 (and general topology),Kis a compact metrizable space, and is in particular sequentially compact. Now for eachg_∈Gσ,

b−σ_{≤ k}Tg[xk]k ≤bσ and b−σ≤ kTg−1[xk]k ≤bσ and thus

φg:= (Tg(xk))∞1 ,(Tg−1(xk))∞1 , σ(Tg)∈ K,

and so since #(Gσ) =∞, there exists a sequence of distinct elements (gn)∞1 inGσ so that the sequence (φgn)

∞ 1 converges to a point (y(+)k )∞1 ,(y (−) k )∞1 , σ ∈ K.

Writing out what this means yields the claim. ⊳

Let Tn = Tgn and σn = σ(Tn) → σ. We claim that the sequence (Tn)

∞

1 is

convergent in the strong operator topology. Let K={a∈F:σ(a) =a}

V ={x∈ L: the sequence (Tn[x])∞1 converges}.

ThenKis anR-subalgebra ofF, andV is aK-module. Givenx,y_∈V, by Obser- vation 2.3.6 we have

σn(BQ(x,y)) =BQ(Tnx, Tny)−→

n BQ(x,y),

so B(x,y) _∈ K. Thus V satisfies (2.4.1). On the other hand, since the family (Tn)∞1 acts equicontinuously onL, the setV is closed. Thus [V]∩bordX is totally

geodesic. But by construction,ξk ∈[V] for allk, and so Λ⊆[V]

9.3. SUFFICIENT CONDITIONS FOR POINCAR ´E REGULARITY 153

Therefore, since by hypothesis G acts irreducibly, it follows that [V] = X, i.e.

V =_L (Proposition 7.6.3). So for everyx_{∈ L}, the sequence (Tn(x))∞1 converges.

Thus

Tn−→ n T

(+) ∈L(_L)

in the strong operator topology. (The boundedness of the operator T(+) _follows

from the uniform boundedness of the operators (Tn)∞1 .) We do not yet know that T(+)_{is invertible. But a similar argument yields that}

Tn−1−→ n T

(−)

∈L(_L),

and since the sequences (Tn)∞1 and (Tn−1)∞1 are equicontinuous, we have T(+)_T(−)_{= lim}

n→∞TnT −1

n =I,

and similarlyT(−)_T(+)_{=I. ThusT}(+)_andT(−)_{are inverses of each other and in}

particular

T(+)_∈O∗(_L;_Q).

Leth= [T(+)_{]∈Isom(X). By Proposition 5.1.2, we haveg}

n →hin the compact- open topology. Thus, Lemma 5.2.8 completes the proof.

Part 3

isometries which can be written as the “Schottky product” of two subsemigroups. Next, we analyze in detail the class of parabolic groups of isometries in Chapter 11. In Chapter 12, we define a subclass of the class of groups of isometries which we call geometrically finite, generalizing known results from the Standard Case. In Chapter 13, we provide a list of examples whose main importance is that they are counterexamples to certain implications; however, these examples are often geometrically interesting in their own right. Finally, in Chapter 14, we consider methods of constructingR-trees which admit natural group actions, including what we call the “stapling method”.

CHAPTER 10