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On Residual Properties of Word Hyperbolic Groups
MINASYAN, Ashot
Abstract
For a fixed word hyperbolic group we compare different residual properties related to quasiconvex subgroups.
MINASYAN, Ashot. On Residual Properties of Word Hyperbolic Groups. Journal of Group Theory , 2006, vol. 9, no. 5, p. 695-714
arxiv : math/0506606v2
Available at:
http://archive-ouverte.unige.ch/unige:12139
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1 / 1
arXiv:math/0506606v2 [math.GR] 13 Nov 2006
GROUPS
ASHOT MINASYAN
Abstract. For a fixed word hyperbolic group we compare different residual properties related to quasiconvex subgroups.
1. Introduction
Any groupGcan be equipped with aprofinite topologyPT(G), whose basic open sets are cosets to normal finite index subgroups. It is easy to see that the group operations are continuous in PT(G). The group is residually finite if and only if the profinite topology is Hausdorff.
A subgroupH ≤Gis closed in PT(G) if and only if it is equal to an intersec- tion of finite index subgroups; equivalently, for any elementg /∈ H there exists a homomorphismϕfrom Gto a finite groupLsuch that ϕ(g)∈/ ϕ(H). In this case the subgroupH is calledG-separable.
The profinite closure of a subgroup H ≤ G, i.e., the smallest closed subset containingH, is equal to the intersection of all finite index subgroupsKofGsuch thatH ≤K.
A groupG is said to be LERF if every finitely generated subgroup is closed in PT(G). The class of all LERF groups includes free groups [6], surface groups [20]
and fundamental groups of certain 3-manifolds [20], [3].
LetGbe a (word) hyperbolic group with a finite symmetrized generating setA and let Γ(G,A) be the corresponding Cayley graph ofG. A subsetQ⊆Gis said to bequasiconvexif there exists a constantη≥0 such that for any pair of elements u, v ∈ Q and any geodesic segment p connecting u and v, pbelongs to a closed η-neighborhood of the subsetQ in Γ(G,A). Quasiconvex subgroups are precisely those finitely generated subgroups which are embedded inGwithout distortion [10, Lemma 1.6].
As it was noted in [13], in the context of word hyperbolic groups instead of studying LERF-groups it makes sense to study GFERF-groups. A hyperbolic group is called GFERF if each quasiconvex subgroup is closed in PT(G). Thus, within the class of hyperbolic groups, the notion of GFERF is more general than LERF:
every LERF hyperbolic group is GFERF but not vice versa.
Unfortunately, it is absolutely unclear how to decide if a random word hyperbolic group is GFERF (or LERF). With this purpose, D. Long [8] and, later, G. Niblo and B. Williams [16] suggested to utilize the engulfingproperty. They say that a
2000Mathematics Subject Classification. Primary 20F67, Secondary 20E26.
Key words and phrases. Word Hyperbolic Groups, Profinite Topology, Engulfing, GFERF.
This work was partially supported by the NSF grant DMS #0245600 of A. Ol’shanskii and M.
Sapir.
1
subgroupH ≤Gisengulfed if it is contained in a proper finite index subgroup of G.
The author was mainly interested in the following two theorems established by Niblo and Williams in 2002:
Theorem A.([16, Thm. 4.1])Let Gbe a word hyperbolic group and suppose that Gengulfs every finitely generated free subgroup with limit set a proper subset of the boundary of G. Then the intersection of all finite index subgroups ofGis finite. If Gis torsion-free then it is residually finite.
Theorem B. ([16, Thm. 5.2]) Let G be a word hyperbolic group which engulfs every finitely generated subgroupK such that the limit set Λ(K)is a proper subset of the boundary of G. Then every quasiconvex subgroup of Ghas a finite index in its profinite closure in G.
The main goal of this paper is to generalize Theorems A and B by weakening their assumptions and, in certain situations, strengthening their conclusions.
In a hyperbolic groupGthe structure of a distorted subgroup can be very com- plicated. Thus, the basic idea is to use assumptions which concern only quasiconvex subgroups. We prove the following results in Section 5:
Theorem 1. LetGbe a hyperbolic group with a generating set of cardinalitys∈N.
Suppose that each proper free quasiconvex subgroup of ranksis engulfed inG. Then the intersection of all finite index subgroups ofGis finite. IfGis torsion-free then it is residually finite.
Theorem 2. SupposeGis a hyperbolic group which engulfs each proper quasiconvex subgroup. Let H be an arbitrary quasiconvex subgroup ofG. ThenH has a finite index in its profinite closureK. Moreover,K=HQwhereQis the intersection of all finite index subgroups of G.
The assumptions of Theorems 1 and 2 are, indeed, less restrictive than the assumptions of Theorems A and B, because if H is a quasiconvex subgroup of a hyperbolic group Gwith|G:H|=∞then the limit set Λ(H) is a proper subset of∂G([22, Thm. 4], [14, Lemma 8.2]).
In the residually finite case Theorem 2 can be reformulated as follows:
Theorem 3. Let Gbe a residually finite hyperbolic group where every proper qua- siconvex subgroup is engulfed. Then Gis GFERF.
Combining together the claims of Theorems 1 and 3 one obtains
Corollary 1. Let Gbe a torsion-free hyperbolic group where each proper quasicon- vex subgroup is engulfed. ThenGis GFERF.
N. Romanovskiˇı [19] and, independently, R. Burns [2] showed that a free product of two LERF groups is again a LERF group. We suggest yet another way for constructing GFERF groups by proving a similar result for them (see Section 6):
Theorem 4. Suppose G1 and G2 are GFERF hyperbolic groups. Then the free product G=G1∗G2 is also a GFERF hyperbolic group.
Acknowledgements
I would like to thank Dr. Alexander Ol’shanskii for useful discussions and Dr.
Graham Niblo for referring me to his paper [16].
2. Preliminaries
Let Gbe a group with a finite symmetrized generating set A. Naturally, this generating set gives rise to a word length function|g|G for every elementg ∈ G.
The (left-invariant)word metric d: G×G → N∪ {0} is defined by the formula d(x, y) =|x−1y|G for anyx, y∈G. This metric can be canonically extended to the Cayley graph Γ(G,A) by making each edge isometric to the interval [0,1]⊂R.
For any three points x, y, w ∈ Γ(G,A), the Gromov product of x and y with respect tow is defined as
(x|y)w def= 1
2
d(x, w) +d(y, w)−d(x, y) .
Since the metric is left-invariant, for arbitraryx, y, w∈Gwe have (x|y)w= (w−1x|w−1y)1G.
The group G is called (word) hyperbolic according to M. Gromov [5] if there existsδ≥0 such that for anyx, y, z, w∈Γ(G,A) their Gromov products satisfy
(x|y)w≥min{(x|z)w,(y|z)w} −δ.
Equivalently,Gis hyperbolicif there exists δ≥0 such that each geodesic triangle
∆ in Γ(G,A) isδ-slim, i.e., every side of ∆ is contained an aδ-neighborhood of the two other sides (see [1]).
From now on we will assume thatGis a hyperbolic group and the constantδis large enough so that it satisfies both of the definitions above.
For any two points x, y ∈ Γ(G,A) we fix a geodesic path between them and denote it by [x, y]. Let pbe a path in the Cayley graph of G. Then p−, p+ will denote the startpoint and the endpoint ofp,kpk– its length; lab(p), as usual, will mean the word in the alphabetAwritten onp. elem(p)∈Gwill denote the element of the groupGrepresented by the wordlab(p). If W is a word in theA,elem(W) will denote the corresponding element of the groupG. For a subsetA⊂Γ(G,A) its closedε-neighborhood will be denoted byOε(A).
Theδ-slimness of geodesic triangles implies 2δ-slimness of all geodesic quadran- glesabcdin Γ(G,A):
[a, b]⊂ O2δ [b, c]∪[c, d]∪[a, d]
.
A pathqis called (λ, c)-quasigeodesic if there exist 0< λ≤1,c≥0, such that for any subpathpofqthe inequalityλkpk −c≤d(p−, p+) holds. A wordW is said to be (λ, c)-quasigeodesic if some (equivalently, every) pathq in Γ(G,A) labelled byW is (λ, c)-quasigeodesic.
Lemma 2.1. ([4, 5.6,5.11],[1, 3.3])There is a constant ν=ν(δ, λ, c)such that for any(λ, c)-quasigeodesic path pinΓ(G,A)and a geodesicqwithp−=q−,p+=q+, one has p⊂ Oν(q)andq⊂ Oν(p).
Lemma 2.2. ([14, Lemma 4.1]) Consider a geodesic quadrangle x1x2x3x4 in the Cayley graph Γ(G,A)with d(x2, x3)> d(x1, x2) +d(x3, x4). Then there are points u, v∈[x2, x3] such that d(x2, u)≤d(x1, x2),d(v, x3)≤d(x3, x4)and the geodesic subsegment[u, v] of[x2, x3] lies 2δ-close to the side [x1, x4].
Ifx, g∈G, we definexg =gxg−1. For a subsetAof the groupG,Ag=gAg−1, and the notationAGwill be used to denote the subset{gag−1|a∈A, g∈G} ⊂G.
Remark 2.1. [10, Remark 2.2] LetQ⊆Gbe quasiconvex,g∈G. Then the subsets gQ,QgandgQg−1are quasiconvex as well.
Thus, a conjugate of a quasiconvex subgroup in a hyperbolic group is quasiconvex itself. Another important property of hyperbolic groups states that any cyclic subgroup is quasiconvex (see [1], for instance).
In this paper we will also use the concept ofGromov boundary of a hyperbolic groupG(for a detailed theory the reader is referred to the corresponding chapters in [4] or [1]).) A sequence (xi)i∈Nof elements of the groupGis said to beconverging to infinity if
i,j→∞lim (xi|xj)1G =∞.
Two sequences (xi)i∈N,(yj)i∈Nconverging to infinity are said to be equivalent if
i→∞lim(xi|yi)1G=∞.
The points of the boundary ∂G are identified with the equivalence classes of se- quences converging to infinity. It is easy to see that this definition does not depend on the choice of a basepoint: instead of 1G one could use any fixed point p of Γ(G,A) (see [1]). Ifαis the equivalence class of (xi)i∈N, we will write lim
i→∞xi=α.
The space ∂G can be topologized so that it becomes compact, Hausdorff and metrizable (see [1],[4]).
Left multiplication by an element of the group induces a homeomorphic action ofGon its boundary: for anyg∈Gand [(xi)i∈N]∈∂Gset
g◦[(xi)i∈N]def= [(gxi)i∈N]∈∂G.
Ifg∈Gis an element of infinite order then the sequences (gi)i∈N and (g−i)i∈N
converge to infinity and we will use the notation
i→∞lim gi=g∞∈∂G, lim
i→∞g−i=g−∞∈∂G.
For a subsetA⊆Gthelimit setΛ(A) ofAis the collection of the pointsα∈∂G that are limits of sequences (converging to infinity) fromA.
Let us recall an auxiliary binary relation defined between subsets of an arbitrary groupGin [14]: supposeA, B ⊆G, we will writeAB if and only if there exist elementsx1, . . . , xn∈Gsuch that
A⊆Bx1∪Bx2∪ · · · ∪Bxn.
It is not difficult to see that the relation ”” is transitive and for any g ∈ G, AB impliesgAgB.
Lemma 2.3. ([14, Lemma 2.1])Let A, B be subgroups ofG. Then AB if and only if the index |A: (A∩B)|is finite.
The basic properties of limit sets are described in the following statement:
Lemma 2.4. ([7],[22],[14, Lemma 6.2]) SupposeA, B are arbitrary subsets of G, g∈G. Then
(a) Λ(A) =∅ if and only if Ais finite;
(b) Λ(A)is a closed subset of the boundary∂G;
(c) Λ(A∪B) = Λ(A)∪Λ(B);
(d) Λ(Ag) = Λ(A),g◦Λ(A) = Λ(gA);
(e)if AB thenΛ(A)⊆Λ(B).
The following property of limit sets of quasiconvex subgroups was first proved by E. Swenson:
Lemma 2.5. ([22, Thm. 8], [11, Lemma 9.1]) Let A,B be quasiconvex subgroups of a hyperbolic group G. ThenΛ(A)∩Λ(B) = Λ(A∩B)in∂G.
3. Auxiliary Facts
Lemma 3.1. AssumeH is an η-quasiconvex subgroup of aδ-hyperbolic group G, X is a word overA representing an element of infinite order inG,0< λ≤1 and c≥0. Letν=ν(δ, λ, c)be the constant given by Lemma 2.1. There exists a number N =N(δ, η, ν, G)∈Nsuch that for anym∈Nthe following property holds.
If a wordW ≡U1XnU2 is (λ, c)-quasigeodesic and satisfies kU1k,kU2k>(m+ ν+c)/λ,n≥Nandelem(V1W V2)∈H for some wordsV1, V2withkV1k,kV2k ≤m, then there exist k∈Nanda∈Gsuch that elem(Xk)∈Ha and|a|G ≤2δ+ν+η.
Proof. Consider a path q in Γ(G,A) starting at 1G and labelled by V1W V2. By our assumptions, q+ ∈ H. Let p and r be its (λ, c)-quasigeodesic subpaths with lab(p)≡W andlab(r)≡Xn respectively. Choose an arbitraryphasevertex u∈r such that the subpath ofrfrom r− touis labelled by some power ofX.
According to Lemma 2.1 there is v∈[p−, p+] satisfyingd(u, v)≤ν. Using the assumptions and the triangle inequality one can achieve
d(p−, v)≥d(p−, u)−d(u, v)≥λkU1k −c−ν > m, and d(p+, v)≥d(p+, u)−d(u, v)≥λkU2k −c−ν > m.
Hence, by Lemma 2.2,v∈ O2δ([1G, q+]).
Thus,
u∈ O2δ+ν([1G, q+])⊂ O2δ+ν+η(H),
i.e., there is an elementa=a(u)∈Gsuch that|a|G≤2δ+ν+η andu∈Ha.
Now, since the alphabetA is finite, there are only finitely many elements inG having length at most (2δ+ν+η). Hence, ifnis large enough, there should be two different phase verticesu1, u2∈rwitha(u1) =a(u2) =a. By the construction,
u−11 u2=elem(Xk)∈a−1Ha
for some positive integerk(Xkis a label of the segment ofrfromu1tou2, provided these points are chosen in a correct order). Q.e.d.
Lemma 3.2. AssumeGis a hyperbolic group andH ≤Gis a quasiconvex subgroup.
If g∈GandgHH, thenHgH.
Proof. IfH is finite, the statement is trivial. Our assumptions immediately imply gk−1H gk−2H · · · gH H
for allk∈N. Hence,
(1) gk−1H H.
If g has order k ∈ N in the group G, then to achieve the desired result it is enough to multiply both sides of the above formula byg.
Thus, we can further assume that H is infinite and the element g has infinite order. Therefore,H has at least one limit pointα∈Λ(H). Observe that (1) implies
gn◦Λ(H) = Λ(gnH)⊆Λ(H),
thusgn◦α∈Λ(H) for alln∈N.
Ifα6=g−∞ in∂G, it is well known (see, for instance, [4, 8.16]) that the sequence (gn◦α)n∈N converges to g∞. Since Λ(H) is a closed subset of the boundary∂G, in either case we achieve
Λ(hgi∞) ={g∞, g−∞} ∩Λ(H)6=∅.
By Lemma 2.5, the latter impliesgk∈H for somek∈N. Combining this fact with (1) we getH =gkHgH, which concludes the proof.
The previous lemma has an interesting corollary:
Lemma 3.3. SupposeH, Kare subgroups of a hyperbolic groupGsuch thatH ≤K,
|K:H|=∞andH is quasiconvex. Then |K: (K∩Hg)|=∞for any g∈G.
Proof. If |K : (K∩Hg)| < ∞ for some g ∈ G then H K gHg−1 gH according to Lemma 2.3. Consequentlyg−1H H. Hence H g−1H by Lemma 3.2, implyinggHg−1gHH. But the latter leads toKH which contradicts
the condition|K:H|=∞(see Lemma 2.3).
If G is a hyperbolic group, each element g ∈ G of infinite order belongs to a uniquemaximal elementary subgroupE(g). By [17, Lemmas 1.16, 1.17] it has the following description:
(2) E(g) =
x∈G| xgkx−1=gl for somek, l∈Z\{0} =
x∈G|xgnx−1=g±n for somen∈N . Note that the subgroup E+(g)def=
x∈G| xgnx−1 =gn for somen∈ N has index at most 2 inE(g).
LetW1, W2, . . . , Wlbe words in Arepresenting elementsg1, g2, . . . , glof infinite order, whereE(gi)6=E(gj) fori6=j. The following lemma will be useful:
Lemma 3.4. ([17, Lemma 2.3])There exist constantsλ=λ(W1, W2, . . . , Wl)>0, c = c(W1, W2, . . . , Wl) ≥0 and N =N(W1, W2, . . . , Wl) >0 such that any path pin the Cayley graph Γ(G,A)with label Wim11Wim22. . . Wimss is(λ, c)-quasigeodesic if ik 6=ik+1 for k = 1,2, . . . , s−1, and |mk|> N for k= 2,3, . . . , s−1 (each ik
belongs to{1, . . . , l}).
For a subgroupH ofGdenote byH0 the set of elements of infinite order inH; ifA⊆G, the subgroup CH(A) = {h∈H |gh=g, ∀g ∈A} is thecentralizer of Ain H.
SetE(H) =T
x∈H0E(x). IfH is a non-elementary subgroup ofG, thenE(H) is the unique maximal finite subgroup ofGnormalized byH ([17, Prop. 1]).
Ifg ∈G0, T(g) will be used to denote the set of elements of finite order in the subgroupE(g).
LetGbe a hyperbolic group andH be its non-elementary subgroup. Recalling the definition from [17] (and using the terminology from [12]), an elementg∈H0 will be calledH-suitableifE(H) =T(g) and
E(g) =E+(g) =CG(g) =T(g)× hgi.
In particular, if the elementgisH-suitable theng∈CH E(H) .
Two elementsg, h∈Gof infinite order are calledcommensurableifgk=ahla−1 for some non-zero integersk, land somea∈G. The following important statement was proved by A. Ol’shanskii in 1993:
Lemma 3.5. ([17, Lemma 3.8]) Every non-elementary subgroup H of a hyper- bolic group G contains an infinite set of pairwise non-commensurable H-suitable elements.
Suitable elements can be modified in a natural way:
Lemma 3.6. ([12, Lemma 4.3])LetHbe a non-elementary subgroup of a hyperbolic groupG, andg be anH-suitable element. Ify∈CH E(H)
\E(g)then there exists N ∈ N such that the element ygn has infinite order in H and is H-suitable for every n≥N.
In [14] the author studied properties of quasiconvex subgroups of infinite index and showed
Lemma 3.7. ([14, Prop. 1])SupposeH is a quasiconvex subgroup of a hyperbolic group G and K is any subgroup of G that satisfies |K : (K∩Hg)| = ∞ for all g ∈ G. Then there exists an element x ∈ K having infinite order, such that hxi∞∩HG={1G}.
Later we will utilize a stronger fact:
Lemma 3.8. Assume H, K are subgroups of a δ-hyperbolic group G, H is η- quasiconvex, K is non-elementary and |K : (K ∩Hg)| = ∞ for every g ∈ G.
Then there exists aK-suitable elementy∈K such that hyi∞∩HG={1G}.
Proof. SetK′=CK E(K)
. SinceE(K) is a finite subgroup normalized byK, we have|K:K′|<∞. Therefore|K′: (K′∩Hg)|=∞for allg∈G. Applying Lemma 3.7, we can find an element of infinite orderx∈K′ such thathxi∞∩HG={1G}.
By Lemma 3.5 there is a K-suitable element z ∈K which is non-commensurable withx, hencehxi∞∩E(z) ={1G}.
Choose some wordsXandZin the alphabetArepresentingxandzrespectively.
Then one is able to find the numbersλ=λ(X, Z),c=c(X, Z) andN1=N1(X, Z) from the claim of Lemma 3.4.
Define ν=ν(δ, λ, c) andN2=N2(δ, η, ν, G) as in Lemmas 2.1 and 3.1. Denote N = max{N1, N2} and apply Lemma 3.6 to obtainn≥N1 such that the element y=xNzn ∈KisK-suitable.
It remains to check thathyi∞∩HG ={1G}. Assume the contrary, i.e., there existt∈Nandg∈Gsuch thatyt∈Hg. Then for eachl∈N, the elementytl∈Hg will be represented by the (λ, c)-quasigeodesic word W ≡ (XNZn)tl. And if the number l is chosen sufficiently large (compared tom =|g−1|G =|g|G), it should be possible to find a subword of the formXN in the ”middle” ofW which satisfies all the assumptions of Lemma 3.1. Hence, xk =elem(Xk)∈HG for somek∈N.
The latter contradicts to the construction ofx. Thus the lemma is proved.
As usual, letGbe aδ-hyperbolic group andH– itsη-quasiconvex subgroup.
Lemma 3.9. Suppose the elementsx1, x2∈Ghave infinite order,E(x1)6=E(x2), and for each i = 1,2, satisfy hxii∞∩HG = {1G}. Then there exists a number N ∈ N such that for any m, n ≥ N the elements xm1 , xn2 freely generate a free subgroupF of rank 2in G. Moreover,F is quasiconvex and F∩HG={1G}.
Proof. Choose some wordsX1andX2in the alphabetArepresenting the elements x1 and x2. Apply Lemma 3.4 to find the corresponding λ = λ(X1, X2), c = c(X1, X2) and N1 = N1(X1, X2). Then one can find the constant ν = ν(δ, λ, c)
from the claim of Lemma 2.1 and define N2 = N2(δ, η, ν, G) in accordance with Lemma 3.1.
SetN = max{N1, N2,⌊c/λ⌋+ 1}. Consider arbitrary integersm, n≥N and the subgroupF =hxm1, xn2i ≤G. By Lemma 3.4 any non-empty (freely) reduced word W in the generators{X1m, X2n}is (λ, c)-quasigeodesic. Hence
|elem(W)|G≥λkWk −c≥λN−c >0.
Consequently,elem(W)6= 1GinG, implying thatF is free with the free generating set{xm1 , xn2}. By the construction ofν, F will beε-quasiconvex where
ε=ν+1
2max{mkX1k, nkX2k}.
Consider a non-empty cyclically reduced wordW in the generators{X1m, X2n}.
For establishing the last claim, it is sufficient to demonstrate that elem(W) ∈/ HG. Arguing as in Lemma 3.8, suppose elem(W)∈ Hg for some g ∈ G. Then (elem(W))l ∈ Hg for every l ∈ N. By choosingl sufficiently large and applying Lemma 3.1 one will obtain a contradiction with the assumptionhxii∞∩HG ={1G}, i= 1,2, as in Lemma 3.8. ThereforeF∩HG ={1G}. Q.e.d.
Corollary 2. With the assumptions of Lemma 3.8, K has a free subgroup F of rank 2 which is quasiconvex inG,E(F) =E(K)andF∩HG={1G}.
Proof. Choose aK-suitable element x1 ∈K according to Lemma 3.8. Since K is non-elementary, there exists y ∈ K\E(x1). Therefore, x2
def= yx1y−1 ∈ K0 and E(x2)6=E(x1) (as it can be seen from (2)). By the construction,hxii∞∩HG = {1G}, i = 1,2. Hence the subgroup F can be found by applying Lemma 3.9.
EvidentlyE(K)≤E(F), andE(F)⊆T(x1) =E(K). ThusE(F) =E(K).
4. Free products of quasiconvex subgroups
Below we will assume thatGis aδ-hyperbolic group generated by a finite setA.
First let us recall some properties of the hyperbolic boundary.
Lemma 4.1. ([12, Lemma 2.14])SupposeAandB are arbitrary subsets ofGand Λ(A)∩Λ(B) =∅. Then sup
a∈A, b∈B
{(a|b)1G}<∞.
Remark 4.1. Supposeg, x∈Gandghas infinite order. If (x◦ {g±∞})∩{g±∞} 6=∅ in the boundary∂G, thenx∈E(g). IfE(g) =E+(g) theng∞∈/ G◦ {g−∞}.
Note that x◦ {g±∞} = {(xgx−1)±∞} = Λ(hxgx−1i) ⊂ ∂G. Since any cyclic subgroup in a hyperbolic group is quasiconvex, we can apply Lemma 2.5 to show thathgi ∩ hxgx−1i 6={1G}. Hencex∈E(g).
LetE(g) =E+(g) and suppose thatg∞=x◦g−∞ for somex∈G. Then, as we showed above,x∈E(g) =E+(g). Hence
x◦g−∞ = lim
n→−∞(xgx−1)n= lim
n→−∞gn=g−∞. Thus we achieve a contradiction with the inequalityg∞6=g−∞.
Lemma 4.2. Let g, x ∈G where g has infinite order and E(g) = E+(g). Then there isN1∈Nsuch that for everyn≥N1 the elementxgn∈Ghas infinite order.
Proof. Ifx /∈E(g) then the claim follows by [11, Lemma 9.14].
So, assume x∈E(g). SinceE(g) =E+(g), the center ofE(g) has finite index in it, thus E(g) is an FC-group. By B.H. Neumann’s Theorem [15] the elements of finite order form a subgroup T(g) ≤ E(g). Therefore the cardinality of the intersection {xgk | k ∈ Z} ∩T(g) can be at most 1. Thus xgn ∈/ T(g) for each
sufficiently largen.
The main result of this paper is based on the following statement concerning broken lines in aδ-hyperbolic metric space:
Lemma 4.3. ([18, Lemma 21], [14, Lemma 3.5]) Letp= [y0, y1, . . . , yn]be a broken line in Γ(G,A) such that ||[yi−1, yi]||> C1 ∀ i= 1, . . . , n, and (yi−1|yi+1)yi < C0
∀ i= 1, . . . , n−1, where C0 ≥14δ,C1>12(C0+δ). Then the geodesic segment [y0, yn] is contained in the closed14δ-neighborhood ofpandk[y0, yn]k ≥ kpk/2.
Supposea, b, c anddare arbitrary points in Γ(G,A). Considering the geodesic triangle with the vertices 1G, aandab, one can observe that
(a|ab)1G=|a|G−(1G|ab)a=|a|G−(a−1|b)1G. Now, since Γ(G,A) isδ-hyperbolic,
(3) (a|c)1G ≥min
(a|ab)1G,(ab|c)1G −δ= min
|a|G−(a−1|b)1G,(ab|c)1G −δ.
Replacingc withcdin the above formula, we get (4) (a|cd)1G ≥min
|a|G−(a−1|b)1G,(ab|cd)1G −δ.
The Gromov product is symmetric, therefore we are able to combine formulas (3) and (4) to achieve
(5) (a|c)1G ≥min
|c|G−(c−1|d)1G,(a|cd)1G −δ≥ min
|a|G−(a−1|b)1G,|c|G−(c−1|d)1G,(ab|cd)1G −2δ.
Theorem 5. Consider some elementsg1, x1, g2, x2, . . . , gs, xs∈Gand anη-quasi- convex subgroup H≤G. Suppose the following three conditions are satisfied:
• g1, . . . , gs have infinite order and are pairwise non-commensurable;
• E(gi) =E+(gi)for each i= 1,2, . . . , s;
• E(gi)∩H =E(gi)∩x−1i Hxi={1G}for each i= 1,2, . . . , s.
Then there exists a numberN ∈Nsuch that for everyn≥N the elementsxigni ∈G, i= 1,2. . . , s, have infinite order, and the subgroup
M def= hH, x1g1n, . . . , xsgsni ≤G
is quasiconvex in G and isomorphic (in the canonical way) to the free product H∗ hx1g1ni ∗ · · · ∗ hxsgsni.
Proof. Choose arbitrary elements w1, w2 ∈M and definew=w−11 w2∈M. Then this element has a presentation
(6) w=h0(xi1gin1)ǫ1h1(xi2gin2)ǫ2· · ·hl−1(xilgnil)ǫlhl, wherehj∈H,ij ∈ {1, . . . , s},ǫj∈ {1,−1},j= 1,2, . . . , l,l∈N∪ {0}.
Moreover, we can assume that the presentation (6) is reduced in the following sense: if 1≤j≤l−1,ij=ij+1 andǫj+1=−ǫj thenhj6= 1G.
Consider a geodesic broken line [y0, y1, . . . , yl+1] in Γ(G,A) withy0 = w1 and elem([yk, yk+1]) = hk(xik+1gink+1)ǫk+1, k = 0,1, . . . , l −1, elem([yl, yl+1]) = hl. Thereforeelem([y0, yl+1]) =wandyl+1=w2.
Now we are going to find upper bounds for the Gromov products (yk−1|yk+1)yk = (y−1k yk−1|y−1k yk+1)1G, k= 1, . . . , l.
By the assumptions of the theorem, Lemma 2.5 implies that
xi◦g∞i = (xigix−1i )∞∈/Λ(H) and gi−∞∈/ Λ(H), i= 1, . . . , s.
Since
Λ {xigim, g−mi x−1i |m∈N,1≤i≤s}
={xi◦g∞i , g−∞i |1≤i≤s} ⊂∂G, we are able to apply Lemma 4.1 to define
αdef= max
(h|xigmi )1G,(h|gi−mx−1i )1G
h∈H,1≤i≤s, m∈N <∞.
Similarly, sincegi andgj are non-commensurable if i6=j andE(gi) =E+(gi), we have (according to Lemma 2.5 and Remark 4.1)
G◦ {g±∞i } ∩G◦ {gj±∞}=∅, G◦ {gi∞} ∩G◦ {gi−∞}=∅, i, j∈ {1, . . . , s}, i6=j.
Hence, the following numbers are also finite:
β1
def= maxn
(xigim)−1|hxjgjt
1G, xigmi |h(xjgjt)−1
1G
h∈H, |h|G≤2α+ 2δ, 1≤i, j≤s, m, t∈No ,
β2
def= maxn
xigmi |hxjgjt
1G, (xigmi )−1|h(xjgjt)−1
1G
h∈H, |h|G≤2α+ 2δ, 1≤i, j≤s, i6=j, m, t∈No .
Note that if h∈H\{1G} then, according to our assumptions, x−1i hxi ∈/ E(gi).
Therefore, {g±∞i } ∩(x−1i hxi)◦ {gi±∞} = ∅ (Remark 4.1), implying xi ◦gi∞ 6=
(hxi)◦g∞i ,i= 1, . . . , s. We can also use Remark 4.1 to show thatg−∞i 6=h◦g−∞i for eachi. Consequently, by Lemma 4.1,
β3
def= maxn
xigmi |hxigit
1G, (xigmi )−1|h(xigti)−1
1G
h∈H, |h|G≤2α+ 2δ, h6= 1G, 1≤i≤s, m, t∈No
<∞.
Finally, defineβ= max{β1, β2, β3}<∞,
(7) C0= max{α+ 2δ, β+δ,14δ}+ 1 and C1= 12(C0+δ) + 1.
Since eachgi, i= 1, . . . , s, has infinite order in Gthere existsN ∈Nsuch that for anyi∈ {1, . . . , s}one has
(8) |gni|G>max{α, β,2C1}+α+|xi|G+ 2δ, ∀ n≥N.
By Lemma 4.2, without loss of generality, we can assume that the order ofxigin, i = 1,2, . . . , s, is infinite for every n≥N. Fix an integer n≥N and choose any k∈ {1, . . . , l−1}. Then
(yk−1|yk+1)yk= (xikgink)−ǫkh−1k−1
hk(xik+1gink+1)ǫk+1
1G.
To simplify the notation, let us denote a= (xikgink)−ǫk, b =h−1k−1, c =hk and d= (xik+1gnik+1)ǫk+1. By construction
(9) (a|c)1G,(a−1|b)1G,(c−1|d)1G ≤α.
Now we need to consider two separate cases.
Case 1: |hk|G =|c|G ≤2α+ 2δ. Then, due to the definition of the numberβ, the inequality (a|cd)1G ≤β holds. Therefore, applying formulas (4) and (9) one obtains
β≥(a|cd)1G ≥min
|a|G−(a−1|b)1G,(ab|cd)1G −δ≥ min
|gink|G− |xik|G−α,(ab|cd)1G −δ.
By (8),|gink|G− |xik|G−α > β+δ, hence the above inequality implies (yk−1|yk+1)yk = (ab|cd)1G≤β+δ < C0.
Case 2: |hk|G =|c|G>2α+ 2δ. Apply formulas (5) and (9) to achieve α≥(a|c)1G≥min{|a|G−α,|c|G−α,(ab|cd)1G} −2δ.
Observing|a|G−α≥ |gink|G− |xik|G−α > α+ 2δand|c|G−α > α+ 2δ, we can conclude that
(yk−1|yk+1)yk = (ab|cd)1G ≤α+ 2δ < C0.
At last, let us estimate the product (yl−1|yl+1)yl = (xilgnil)−ǫkh−1l−1 hl
1G. De- notea= (xilgnil)−ǫk,b=h−1l−1 andc=hl.
Using formula (3) and the definition ofαone can obtain α≥(a|c)1G≥min
|xilginl|G−α,(ab|c)1G −δ.
As before, the latter implies that
(yl−1|yl+1)yl = (ab|c)1G≤α+δ < C0. Thus, we have shown that
(10) (yk−1|yk+1)yk< C0for eachk= 1,2, . . . , l.
Now we need to find a lower bound for the lengths of the sides in the broken line [y0, y1, . . . , yl+1].
Let 0≤k≤l−1. Note that|ab|G≥ |b|G−(a−1|b)1G for anya, b∈G. Hence k[yk, yk+1]k=|hk(xik+1gink+1)ǫk+1|G ≥
|(xik+1gnik+1)ǫk+1|G− h−1k
(xik+1gnik+1)ǫk+1
1G ≥ |gink+1|G− |xik+1|G−α.
Applying inequality (8) we obtain
(11) k[yk, yk+1]k>2C1 if 0≤k≤l−1.
Depending on the length of the last side, k[yl, yl+1]k =|hl|G, there can occur two different situations.
Case 1: k[yl, yl+1]k = |hl|G ≤ C1. Then we can use inequalities (10) and (11) to apply Lemma 4.3 to the geodesic broken line p′ = [y0, . . . , yl]. Hence [y0, yl]⊂ O14δ(p′) andd(y0, yl)≥ kp′k/2.
Since geodesic triangles in Γ(G,A) areδ-slim, we have [y0, yl+1]⊂ Oδ [y0, yl]∪[yl, yl+1]
⊂ Oδ+C1 [y0, yl]
⊂ O15δ+C1(p′).
Now, ifl≥1 in the presentation (6), one can use (11) to obtain
d(y0, yl+1)≥d(y0, yl)−d(yl, yl+1)≥ kp′k/2−C1≥ k[y0, y1]k/2−C1>0.
Case 2: k[yl, yl+1]k=|hl|G> C1. Then we can apply Lemma 4.3 to the broken linep= [y0, . . . , yl, yl+1], thus achieving
[y0, yl+1]⊂ O14δ(p).
As before, ifl≥1 one has
d(y0, yl+1)≥ kpk/2>0.
Thus, in either case we have established the following properties:
(12) [y0, yl+1]⊂ O15δ+C1(p) and
(13) d(y0, yl+1)>0.
The inequality (13) implies thatw6= 1G in the groupGfor any elementw∈M having a ”reduced” presentation (6) withl≥1. Therefore
M ∼=H∗ hx1gn1i ∗ · · · ∗ hxsgnsi.
Asn≥N is fixed, one is able to define the constants ζ= max
1≤i≤s{|xigin|G}<∞ and ε= 16δ+C1+η+ζ.
We will finish the proof by showing that [y0, yl+1]⊂ Oε(M) which implies thatM isε-quasiconvex.
Consider an arbitraryk∈ {0,1, . . . , l}. Using the construction ofyk ∈M andζ, andδ-hyperbolicity of the Cayley graph we achieve
(14) [yk, yk+1]⊂ Oδ+ζ([yk, ykhk]).
H isη-quasiconvex, therefore [1G, hk]⊂ Oη(H). The metric in Γ(G,A) is invariant under the action ofGby left translations, consequently
[yk, ykhk]⊂ Oη(ykH)⊂ Oη(M).
Combining the latter formula with (14) leads us to [yk, yk+1]⊂ Oδ+η+ζ(M) for eachk.
Finally, an application of (12) yields
[y0, yl+1]⊂ O15δ+C1+δ+η+ζ(M) =Oε(M),
as desired.
5. Hyperbolic groups with engulfing
Proof of Theorem 1. Since every elementary group is residually finite, it is sufficient to consider the case whenGis non-elementary. Letx1, . . . , xsbe the generators of G.
DefineK= \
L≤G,|G:L|<∞
L; thenK is normal inG. SupposeK is infinite. If the subgroup K were elementary, then it would be quasiconvex (this follows directly from Lemmas 3.4 and 2.1). Hence, according to a result proved by Mihalik and Towle [9] (see also [14, Cor. 2]), it would have a finite index inG, thus the group Gwould also have to be elementary. ThereforeK can not be elementary.
Now we can apply Lemma 3.5 to find pairwise non-commensurableK-suitable elements g1, . . . , gs+1 ∈ K. Since the trivial subgroup H = {1G} ≤ G is quasi- convex, we can use Theorem 5 to show that the subgroups M =hx1gn1, . . . , xsgnsi andM′ =hx1gn1, . . . , xsgsn, gns+1iare free (of rankssand (s+ 1) respectively) and quasiconvex inGfor some sufficiently largen∈N.
Note thatM is a proper (infinite index) subgroup ofGbecause |M′ :M|=∞.
According to our assumptions, there exists a proper finite index subgroupLin G withM ≤L. By the construction,K≤L, thusxigin, gi ∈Lfor eachi= 1, . . . , s.
Consequentlyxi∈L for eachi= 1, . . . , s, contradicting with properness ofL.
Therefore K is finite. If G is torsion-free then K is trivial, and, thus, G is
residually finite.
As E(G) is the maximal finite normal subgroup in G, we obtain the following statement right away:
Corollary 3. With the assumptions of Theorem 1, suppose, in addition, that E(G) ={1G}. ThenGis residually finite.
Below it will be convenient to use the following equivalence relation between subsets of a groupGdefined in [14]: for anyA, B⊆Gsuch thatABandBA we will writeA≈B.
Remark 5.1. ([14, Remark 3]) IfA, B⊆G,A≈B andAis quasiconvex, thenB is also quasiconvex.
In particular, if A ≤ B are subgroups of G and A has finite index in B then A≈B. HenceA is quasiconvex if and only ifB is quasiconvex.
Lemma 5.1. LetGbe a residually finite hyperbolic group and letH≤Gbe a qua- siconvex subgroup. Suppose that every proper quasiconvex subgroup ofGis engulfed.
Then H has a finite index in its profinite closureK inG.
Proof. Arguing by the contrary, assume that |K :H|=∞. Since the group Gis residually finite, any finite subset is closed in the profinite topology. Thus H is infinite; henceK is non-elementary.
Choose some generating set x1, . . . , xs of the group G. Since G is residually finite, it has a finite index subgroupG1 satisfying
(15) G1∩ E(K)∪
s
[
i=1
xiE(K)x−1i
!
={1G}.
Define H1 = H ∩G1; then |H : H1| < ∞ and, according to Remark 5.1, H1 is quasiconvex. The profinite closureK1ofH1 inGhas a finite index inK, therefore K1is non-elementary and|K1:H1|=∞. The definition of a finite index subgroup implies that there is l∈Nsuch that gl ∈K1 for eachg ∈K. SinceE(g) =E(gl) for anyg∈K0we have
E(K)⊆E(K1) = \
g∈(K1)0
E(g)⊆ \
g∈K0
E(gl) = \
g∈K0
E(g) =E(K).
ThusE(K1) =E(K).
By Lemma 3.3,|K1: (K1∩H1g)|=∞for everyg∈Gand we can use Corollary 2 to obtain a free subgroup F ≤ K1 of rank 2 satisfying F ∩H1G = {1G} and E(F) =E(K1) =E(K).
According to Lemma 3.5 there exist elementsg1, . . . , gs+1∈F which are pairwise non-commensurable andF-suitable. ConsequentlyE(gi) =hgii∞×E(K) and, since E(K) is finite, formula (15) implies that
E(gi)∩H1=E(K)∩H1={1G}, i= 1,2, . . . , s+ 1, and E(gi)∩x−1i H1xi=E(K)∩x−1i H1xi={1G}, i= 1,2, . . . , s.
Now we apply Theorem 5 to findn∈Nsuch that the subgroups M =hH1, x1gn1, . . . , xsgnsi and M′=hH1, x1g1n, . . . , xsgsn, gns+1i
are quasiconvex inGandM′ =M∗ hgs+1n i∞≤G. Thus,|M′ :M|=∞andM is a proper subgroup ofG.
The subgroupM is engulfed by our assumptions, thereforeGhas a proper finite index subgroupLcontainingM. Observe thatK1≤LbecauseH1≤M ≤L, and, sincexigni, gi∈M∪K1⊂L, we getxi∈Lfor eachi= 1, . . . , s. The latter implies
G=L– a contradiction.
We will now prove Theorem 3 which strengthens the the statement of the previ- ous lemma.
Proof of Theorem 3. We can assume thatGis non-elementary because any elemen- tary group is LERF. Since G is residually finite, there is a finite index subgroup G1≤GwithG1∩E(G) ={1G}.
Take an arbitrary quasiconvex subgroup H ≤G and set H1 =H ∩G1. As it follows from Remark 5.1,H1 is quasiconvex inG. Therefore, according to Lemma 5.1,H1has a finite index in its profinite closureK1in G.
IfH1=K1, i.e.,H1is closed in the profinite topology onG, then so isH. Thus, we can suppose thatH16=K1. Consequently|G:H1|=∞, and thus|G:K1|=∞.
The subgroupK1 is quasiconvex according to Remark 5.1, hence we can apply Lemma 3.8 to find aG-suitable elementg∈Gsuch thathgi∞∩K1={1G}. Since E(g) =hgi ×E(G),E(G) is finite andK1≤G1, we have
E(g)∩K1=E(G)∩K1={1G}.
Now we can apply Theorem 5 to find a number n∈Nsuch that the subgroups M =hH1, gniandM′ =hK1, gniare quasiconvex inGandM′∼=K1∗ hgni∞.
Using properties of free products, we observe thatM ≤M′ and|M′ :M|=∞ becauseH1K1. On the other hand,M′is contained inside of the profinite closure ofM in G. Thus we achieve a contradiction with the claim of Lemma 5.1.
Before proceeding with the next statement, we need to recall some facts concern- ing quasiisometries of metric spaces. Let X andY be geodesic metric spaces with metricsd(·,·) ande(·,·) respectively. A mapf :X → Y is called aquasiisometryif there are constantsD1>0 andD2≥0 such that
D1−1d(a, b)−D2≤e(f(a), f(b))≤D1d(a, b) +D2 ∀a, b∈ X.
The spaces X andY are said to be quasiisometric if there exists a quasiisometry f : X → Y whose image is quasidensein Y, i.e., there exists ε≥0 such that for eachy∈ Y there isx∈ X withe(y, f(x))≤ε.
M. Gromov [5] showed that ifX is hyperbolic and quasiisometric toY (through some map f :X → Y) then the space Y is hyperbolic too. He also noted that in this case the image f(Q) of any quasiconvex subsetQ⊆ X will be quasiconvex in Y.
Proof of Theorem 2. Note thatQis a finite normal subgroup ofGby Theorem 1.
Consider the quotient G1 = G/Q together with the natural homomorphism ψ : G → G1. Since Q is finite, ψ is a quasiisometry between G and G1 (G1 is equipped with the word metric induced by the image of the finite generating set of G). Therefore,G1 is also hyperbolic and any preimage map ¯ψ−1:G1 →G(which maps an element of G1 to some element ofG belonging to the corresponding left coset moduloQ) is a quasiisometry as well.
Choose an arbitrary proper quasiconvex subgroupH1≤G1. Then ¯ψ−1(H1) is a quasiconvex subset ofGand
ψ¯−1(H1)⊆ψ−1(H1)⊆ψ¯−1(H1)·Q, whereψ−1(H1) is the full preimage ofH1in G.
As Q is finite, the above formula implies ¯ψ−1(H1) ≈ ψ−1(H1). Therefore ψ−1(H1) is quasiconvex inGby Remark 5.1. According to our assumptions, there is a proper finite index subgroupL≤Gcontainingψ−1(H1). By definition,Q≤L, henceψ(L) is a proper finite index subgroup ofG1 withH1≤ψ(L).
Thus, we have shown that G1 also engulfs each proper quasiconvex subgroup.
By the construction,G1is residually finite and, therefore, GFERF (Theorem 3).
Consider any quasiconvex subgroupH ≤G. Then ψ(H) is quasiconvex in G1
and, thus, it is closed in the profinite topology on G1. The homomorphism ψis a continuous map ifGandG1 are equipped with their profinite topologies, thus the full preimageψ−1(ψ(H)) =H·Qis closed inG. ObviouslyH, Q≤K (whereK is the profinite closure ofH in G), hence K=HQ. Q.e.d.
6. Free products of GFERF groups
In the previous section we considered hyperbolic groups which engulf every proper quasiconvex subgroup. Let us name themQE-groups, for brevity.
As it can be seen from Theorem 2, any QE-group G is very close to being GFERF. In fact,Gis quasiisometric to the quotientG/E(G) which is GFERF by Corollary 3 and Theorem 3. Nevertheless, the answer to the question whether each QE-group is GFERF is still unclear. Theorem 2 would yield a positive answer if a free product of any two QE-groups were a QE-group itself. Unfortunately, the author is unable to prove this; actually, he doubts if this is true in general.
However, the following statement, proved by R. Burns, can be used to show that a free product of GFERF-groups is, again, GFERF:
Lemma 6.1. ([2, Thm. 1.1]) Suppose G is a free product of its subgroups Gi
indexed by some setI, and letH be a finitely generated subgroup. If for eachi∈I, g∈G, the subgroup(Hg∩Gi) isGi-separable, then H isG-separable.
LetH be a subgroup of a groupG. SupposeAandBare finite generating sets for Gand H respectively and | · |G, | · |H are the corresponding length functions.
Set ˆc= max{|b|G : b∈ B}. Evidently,|h|G ≤ˆc|h|H for allh∈H.
H is called undistorted in G if there exists a constant c ≥0 such that |h|H ≤ c|h|G for every h∈H. In a hyperbolic groupG, a finitely generated subgroup is undistorted if and only if it is quasiconvex ([10, Lemma 1.6]).
Proof of Theorem 4. It is well known that a free product of hyperbolic groups is a hyperbolic group (see, for instance, [4, 1.34]). Thus,Gis hyperbolic. Clearly, the subgroupsG1andG2 are undistorted inG; consequently, they are quasiconvex.
Choose an arbitrary quasiconvex subgroup H ≤ G, an element g ∈ G and i∈ {1,2}. The subgroupHgis quasiconvex by Remark 2.1. Since the intersection of two quasiconvex subgroups is quasiconvex ([21, Prop. 3]), (Hg∩Gi) is quasiconvex inG. Consequently, (Hg∩Gi) is undistorted inG, and, hence, it is undistorted in Gi. Thus (Hg∩Gi) isGi-separable becauseGi is GFERF.
According to Lemma 6.1,H isG-separable. Q.e.d.
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Universit´e de Gen`eve, Section de Math´ematiques, 2-4 rue du Li`evre, Case postale 64, 1211 Gen`eve 4, Switzerland
E-mail address: [email protected]
