The Commensurability Relation for Finitely Generated Groups

J. Group Theory 12 (2009), 901–909 Journal of Group Theory DOI 10.1515/JGT.2009.022 ( de Gruyter 2009

The commensurability relation for ﬁnitely generated groups

Simon Thomas (Communicated by A. V. Borovik)

Abstract. There does not exist a Borel way of selecting an isomorphism class within each commensurability class of ﬁnitely generated groups.

1 Introduction

Two groups G1, G2 are (abstractly) commensurable, written G1 QC G2, if and only if there exist subgroups Hi c Gi of finite index such that H1 G H2. In this paper, we shall consider the commensurability relation on the space G of finitely generated groups. Here G denotes the Polish space of finitely generated groups introduced by Grigorchuk [4]; i.e., the elements of G are the isomorphism types of marked groups 3G; c4, where G is a finitely generated group and c is a finite sequence of generators. (For a clear account of the basic properties of the space G, see either Champetier [1] or Grigorchuk [6].) Our starting point is the following result, which intuitively says that the isomorphism and commensurability relations on G have exactly the same complexity.

Theorem 1.1. The isomorphism relation G and commensurability relation QC on the space G of ﬁnitely generated groups are Borel bireducible.

In particular, there exists a Borel reduction from QC to G; i.e., a Borel map f : G ! G such that if G, H a G, then

G QC H if and only if f ðGÞ G f ðHÞ:

However, while the proof of Theorem 1.1 shows that there exists a Borel reduction from QC to G, it does not provide an explicit example of such a reduction. (On the other hand, as we shall see in Section 2, it is straightforward to ﬁnd an explicit example of a Borel reduction from G to QC.)

This research was partially supported by NSF Grants DMS 0600940. 902 S. Thomas

Open Problem 1.2. Find an explicit ‘group-theoretic’ Borel reduction f : G ! G from the commensurability relation QC to the isomorphism relation G.

Of course, one approach to this problem would be to seek an explicit ‘group- theoretic’ Borel map which selects an isomorphism class within each commensurability class of ﬁnitely generated groups. However, the main result of this paper shows that no such map exists.

Theorem 1.3. There does not exist a Borel reduction f : G ! G from QC to G such that f ðGÞQC G for all G a G.

This paper is organized as follows. In Section 2, after recalling some of the basic theory of countable Borel equivalence relations, we shall present the proof of Theo- rem 1.1. In Section 3, after a short discussion of the branch groups constructed in Segal [11], we shall present the proof of Theorem 1.3. Finally, in Section 4, we shall consider the analogues of Theorem 1.3 for the virtual isomorphism and quasi- isometry relations on the space G of ﬁnitely generated groups.

2 Countable Borel equivalence relations In this section, we shall present the proof of Theorem 1.1. But ﬁrst we need to recall some of the basic theory of countable Borel equivalence relations. Suppose that ðX; TÞ is a Polish space; i.e., a separable completely metrizable topo- logical space. Then the corresponding standard Borel space is ðX; BÞ, where B is the s-algebra of Borel subsets of X. (Recall that a subset Z M X n is Borel if and only if Z is an element of the s-algebra generated by the collection of open subsets of X n.) If X, Y are standard Borel spaces, then a map f : X ! Y is Borel if and only if f 1ðZÞ is a Borel subset of X for each Borel subset Z M Y. Equivalently, f : X ! Y is Borel if and only if graphðf Þ is a Borel subset of X Y. Let X be a standard Borel space. Then a Borel equivalence relation on X is an equivalence relation E M X 2 which is a Borel subset of X 2.IfE, F are Borel equivalence relations on the standard Borel spaces X, Y respectively, then we say that E is Borel reducible to F and write E cB F if there exists a Borel map f : X ! Y such that xEyif and only if f ðxÞ FfðyÞ. Such a map f is called a Borel reduction from E to F. We say that E and F are Borel bireducible and write E PB F if both E cB F and F cB E. Finally we write E

xE0 y if and only if xðnÞ¼yðnÞ for all but ﬁnitely many n:

More precisely, from [8], if E is any (not necessarily countable) Borel equivalence relation, then E is non-smooth if and only if E0 cB E. It turns out that there is also a most complex countable Borel equivalence relation El, which is universal in the sense that F cB El for every countable Borel equivalence relation F. (Clearly this universality property uniquely determines El up to Borel bireducibility.) El has a number of natural realizations in many areas of mathematics, including algebra, topology and recursion theory. In particular, by Thomas and Velickovic [14], the isomorphism relation G on the space G of ﬁnitely generated groups is a universal countable Borel equivalence.

Proof of Theorem 1.1. It is well known that if G is a finitely generated group, then there exist only countably many groups H up to isomorphism such that G QC H. (For example, see Erschler [2].) It follows that the commensurability relation QC is a countable Borel equivalence relation on the space G of finitely generated groups. Since the isomorphism relation is a universal countable Borel equivalence, it follows that ðQCÞ cB ðGÞ. In order to see that ðGÞ cB ðQCÞ, let S be a fixed infinite finitely generated simple group and consider the Borel map h : G ! G defined by

h G N ðAltð5Þ wr GÞ wr S:

(Here wr denotes the restricted wreath product.) Combining [12, Lemma 2.2 (b)] and [12, Theorem 2.5], we obtain that if G, H are arbitrary groups, then

G G H if and only if ðAltð5Þ wr GÞ wr S QC ðAltð5Þ wr HÞ wr S:

In particular, h is a Borel reduction from G to QC. r

Remark 2.1. Strictly speaking, in the proof of Theorem 1.1, we should first fix some finite sequences a, b of generators of Altð5Þ, S respectively. Clearly if G is a finitely generated group and c is a finite sequence of generators of G, then the concatenation a^b^c is a finite sequence of generators of ðAltð5Þ wr GÞ wr S. Hence we can define a Borel map h : G ! G on the space of marked finitely generated groups by

h 3G; c4 N 3ðAltð5Þ wr GÞ wr S; a^b^c4:

Having done this, it is easily checked that h is actually a continuous reduction from G to QC. 904 S. Thomas

On the other hand, the above argument for the existence of a Borel reduction f : G ! G from QC to G ultimately relies on the Feldman–Moore theorem [3] that every countable Borel equivalence relation can be realized as the orbit equivalence relation of a Borel action of some countable group; and this argument provides no information concerning the Borel complexity of such a reduction.

Question 2.2. Does there exist a continuous reduction f : G ! G from the commensurability relation QC to the isomorphism relation G?

3 Some ﬁnitely generated branch groups In this section, we shall present the proof of Theorem 1.3. But ﬁrst we need to recall some of the basic notions from the theory of branch groups. (A detailed account of the theory can be found in Grigorchuk [5].)

Let ðlnÞn d 0 be a sequence of integers such that ln d 2 for all n. Then the rooted tree of type ðlnÞn d 0 is a tree T with a distinguished vertex v0 (called the root) of valency l0 such that every vertex at distance n d 1 from v0 has valency ln þ 1. The distance from v0 to a vertex v is called the level of v and the set of vertices of level n is called the nth layer of T. If we picture the tree T with v0 at the top and with ln edges descending from each vertex of level n, then each vertex v of level m is the root of a rooted sub- tree Tv of T of type ðlnÞn d m. Let G be a subgroup of AutðTÞ. Then for each vertex v, the corresponding rigid stabilizer ristGðvÞ is the subgroup of G consisting of those automorphisms which ﬁx every vertex of TnTv; and for each n, the corresponding rigid level stabilizer ristGðnÞ is the group generated by the subgroups ristGðvÞ with v of level n. In other words, if fv1; ...; vtg is the nth layer of T, then

ristGðnÞ¼ristGðv1ÞristGðvtÞ:

We say that G is a branch group on T if and only if for each n d 0, (i) G acts transitively on the nth layer of T; and (ii) ristGðnÞ has ﬁnite index in G.

From now on, let ðpnÞn d 0 be the increasing enumeration of the primes p d 5 and let ln ¼ pn þ 1. Let T be the rooted tree of type ðlnÞn d 0; and for each m d 0, let Tm be the rooted tree of type ðlnÞn d m. For each n d 0, let

0 1 Gn ¼ PSLð2; pnÞ and Gn ¼ AltðlnÞ:

Using the natural permutation representation of PSLð2; pnÞ on the points of the pro- jective line over Fpn , we can regard

PSLð2; pnÞ < AltðlnÞ < SymðlnÞ: The commensurability relation for ﬁnitely generated groups 905

For later use, note that the groups

fPSLð2; pnÞjn d 0g U fAltðlnÞjn d 0g are pairwise non-isomorphic. Let D ¼ SLð2; Z½1=6Þ SLð2; Z½1=6Þ. Then D is a per- fect 4-generator group; and, by the proof of Segal [11, Lemma 8], for each n d 0 and e ¼ 0; 1, there exists a surjective homomorphism

e e yn : D ! Gn:

Let d1; ...; d4 be a ﬁxed set of generators of D and for each n d 0 and e ¼ 0; 1, deﬁne

e e e aðiÞn ¼ ynðdiÞ a Gn:

Deﬁnition 3.1. Fix some j a 2N and for each n d 0, let

j jðnÞ jðnÞ jðnÞ Sn ¼ Gn ¼ 3að1Þn ; ...; að4Þn 4:

Then Aj is the corresponding branch group on T, as in Segal [11, Lemma 1].

Remark 3.2. This paper has been written so as to be intelligible to readers who are unfamiliar with the details of Segal’s construction. In particular, Lemma 3.4 sum- marizes the algebraic properties of the groups Aj that are needed in the proof of Theorem 1.3.

Deﬁnition 3.3. Let s : 2N ! 2N be the shift map deﬁned by

sðe0; e1; e2; ...Þ¼ðe1; e2; e3; ...Þ:

N If j a 2 and m d 0, then Asmj denotes the subgroup of AutðTmÞ deﬁned in the corresponding way from the sequence smj.

In the statement of the next result, an upper composition factor of a group G means a composition factor of some ﬁnite quotient G=N of G.

Lemma 3.4. The branch group Aj satisﬁes the following properties.

(a) If G c Aj is a subgroup of ﬁnite index, then there exists m d 0 such that

ristAj ðmÞ t G.

(b) For each m d 1, the rigid level stabilizer ristAj ðmÞ is a product of l0 ...lm1 copies of the subgroup Asmj. (c) For each m d 0, the upper composition factors of the subgroup Asmj are exactly j fSn j n d mg.

Proof. Each of these properties is proved in Segal [11, Section 2]. r 906 S. Thomas

At this point, it is easy to compute the Borel complexity of the isomorphism and N commensurability problems for the space of groups fAj j j a 2 g.

N Lemma 3.5. If j, c a 2 , then Aj G Ac if and only if j ¼ c.

Proof. This is an immediate consequence of Lemma 3.4 (c). r

N Lemma 3.6. If j, c a 2 , then Aj QC Ac if and only if j E0 c.

m m Proof. First suppose that j E0 c. Then there exists m d 1 such that s j ¼ s c. Ap- G plying Lemma 3.4 (b), we see that ristAj ðmÞ ristAc ðmÞ and hence Aj QC Ac. Next suppose that Aj QC Ac; say, G G H, where ½Aj : G, ½Ac : H < l.By

Lemma 3.4 (a), we can suppose that G ¼ ristAj ðmÞ for some m d 0. Hence the upper j composition factors of G are exactly fSn j n d mg. Similarly, there exists t d 0 such that ristAc ðtÞ t H and so there exists a ﬁnite set F of ﬁnite simple groups such that U c the upper composition factors of H are F fSn j n d tg. Clearly if j is not E0- j U c equivalent to c, then there exists an n such that Sn c F fSn j n d tg, which is impossible. Hence j E0 c. r

From now on, let X M G be the Borel subset consisting of those groups G a G such N N that G QC Aj for some j a 2 . Note that both X and fAj j j a 2 g are standard Borel spaces.

N Lemma 3.7. There exists a Borel map b : X !fAj j j a 2 g such that for all G, H a X,

(i) G QC bðGÞ, and (ii) if G G H, then bðGÞ¼bðHÞ.

Proof. Let G a X. Then there is a minimal d d 1 such that there exists a subgroup H c G such that

(a) ½G : H¼d, and N G (b) there exists j a 2 and m d 0 such that H ristAj ðmÞ.

j Note that the upper composition factors of H are precisely fSn j n d mg. Thus H uniquely determines both m and smj. In particular, for each such H, there are only finitely many possibilities for j. Since G contains only finitely many subgroups of index d, there are also only finitely many possibilities for H and hence we can let bðGÞ¼Aj, where j is the lexicographically least of the finitely many elements of 2N that arise in this fashion. Clearly bðGÞ depends only on the isomorphism class of G. r

We are now ready to present the proof of Theorem 1.3. So suppose that f : G ! G is a Borel reduction from QC to G such that f ðGÞQC G for all G a G. Then N f ðAjÞQC Aj for all j a 2 . Hence, after adjusting f by the map b given by Lemma The commensurability relation for ﬁnitely generated groups 907

3.7, we can suppose that there exists a Borel map h : 2N ! 2N such that N N f ðAjÞ¼AhðjÞ for all j a 2 . Now notice that if j, c a 2 , then j E0 c if and only if Aj QC Ac (by Lemma 3.6)

if and only if AhðjÞ G AhðcÞ (by the assumptions on f ) if and only if hðjÞ¼hðcÞ (by Lemma 3.5).

But this means that E0 is smooth, which is a contradiction.

4 Concluding remarks In this final section, we shall consider the analogues of Theorem 1.3 for the virtual isomorphism and quasi-isometry relations on the space G of finitely generated groups. Recall that two groups G1, G2 are said to be virtually isomorphic or commensurable up to finite kernels if and only if there exist subgroups Ni c Hi c Gi for i ¼ 1; 2 sat- isfying the following conditions:

(a) ½G1 : H1, ½G2 : H2 < l; (b) N1, N2 are ﬁnite normal subgroups of H1, H2 respectively; and (c) H1=N1 G H2=N2.

Let QVI be the virtual isomorphism relation on the space G of ﬁnitely generated groups. Then it is easily checked that QVI is a Borel equivalence relation on G.In Thomas [12], it was shown that ðGÞ

Conjecture 4.1. There does not exist a Borel way of selecting an isomorphism class within each quasi-isometry class.

Conjecture 4.2. There does not exist a Borel way of selecting a virtual isomorphism class within each quasi-isometry class.

It seems likely that Conjecture 4.1 can be proved using a suitable modiﬁcation of N the arguments of Section 3. For example, for each t a 3 , let Gt be the corresponding Grigorchuk group, as deﬁned in [4]. Then a routine variation of the proof of Grigorchuk [4, Theorem 7.2] yields the following analogue of Lemma 3.6. 908 S. Thomas

Lemma 4.3. There exists a Borel map f : 2N ! 3N such that the following conditions are equivalent for all x, y a 2N:

(a) xE0 y; (b) Gf ðxÞ, Gf ðyÞ are commensurable; (c) Gf ðxÞ, Gf ðyÞ have the same growth rate. N r In particular, if x, y a 2 , then x E0 y if and only if Gf ðxÞ, Gf ðyÞ are quasi-isometric.

Furthermore, by Nekrashevych [10, Theorem 2.10.13], the isomorphism relation N on the space of groups fGf ðxÞ j x a 2 g is smooth and so the analogue of Lemma 3.5 also holds. Unfortunately, I have been unable to prove the (presumably much more di‰cult) analogue of Lemma 3.7 in this setting. Finally it should be mentioned that very little is known concerning the Borel complexity of the quasi-isometry relation on the space G of ﬁnitely generated groups. In fact, the following result sums up the current state of knowledge regarding this problem.

Theorem 4.4 (Thomas [13]). The quasi-isometry relation on the space G of ﬁnitely generated groups is not smooth.

Proof. This is an immediate consequence of Lemma 4.3. r

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Received 29 September, 2008 Simon Thomas, Mathematics Department, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, U.S.A. E-mail: [email protected]