Commensurability and Qi Classification of Free Products of Finitely Generated Abelian Groups

We give the commensurability classifications of free products of finitely many finitely generated abelian groups. We show this coincides with the quasi-isometry classification and prove that this class of groups is quasiisometrically rigid. The following gives the complete commensurability and quasi-isometry classification of free products of finitely generated abelian groups. The quasi-isometry classification is a special case of Papasoglu and Whyte [4]. Theorem 1. Let Gi be a nontrivial free product of a finite set Si of finitely generated abelian groups for i = 1, 2, excluding Z/2 ∗ Z/2. Then the following are equivalent: (1) The sets of ranks ≥ 2 of groups in S1 and S2 are equal (the rank of a finitely generated group is the rank of its free abelian part). (2) G1 and G2 are commensurable. (3) G1 and G2 are quasi-isometric. Proof. The main step is to show (1) implies (2). By going to finite index subgroups of G1 and G2 we can assume the groups in S1 and S2 are free abelian (take the kernel of the map ofGi to a product of finite quotients of the groups in Si by torsionfree normal subgroups, or see the lemma below for a more general statement). Let n1 = 1 and let n2, . . . , nk be the ranks ≥ 2 of the groups in S1 and in S2. Let ri and si be the number of rank ni groups in S1 and S2 respectively. We identify G1 with the fundamental group of the topological space W1 consisting of the wedge of ri ni–dimensional tori for each i; similarly we let G2 = π1(W2), where W2 is defined similarly using the si’s. A finite cover of such a wedge of tori is homotopy equivalent to a wedge of tori. We proceed in two steps. First, using finite covers we replace (r1, r2, . . . , rk) and (s1, s2, . . . , sk) by the sequences (R1, Y, Y, . . . , Y ) and (S1, Y, Y, . . . , Y ), respectively, where R1, S1, and Y are positive integers. Then we show that, again taking finite covers, we can leave the Y ’s unchanged and replace both R1 and S1 by a positive integer X, making the two sequences equal and completing the argument. Received by the editors December 6, 2007, and, in revised form, February 13, 2008. 2000 Mathematics Subject Classification. Primary 20E06, 20F65, 20F36. This research was supported under NSF grants no. DMS-0604524, DMS-0706259, and DMS-