Recognition of Subgroups of Direct Products of Hyperbolic Groups

We give examples of direct products of three hyperbolic groups in which there cannot exist an algorithm to decide which finitely presented subgroups are isomorphic. In [1] Baumslag, Short and the present authors constructed an example of a biautomatic group G in which there is no algorithm that decides isomorphism among the finitely presented subgroups of G. The group G was a direct product of a certain (word) hyperbolic group H with an HNN extension B of H ; the group B was CAT(0) and biautomatic but not hyperbolic. At the time of writing [1] we were unable to construct, more simply, a direct product of hyperbolic groups in which the isomorphism problem for finitely presented subgroups was unsolvable. In the course of our project on finiteness properties of subdirect products [7], we uncovered a new criterion for the finite presentability of certain semidirect products (see Section 2). That, combined with an improved understanding of subgroups of direct products, has enabled us to prove the following result: Theorem 1. Let 1 → K → Γ → L → 1 be an exact sequence of groups. Suppose that (1) Γ is torsion-free and (word) hyperbolic, (2) K is infinite and finitely generated, and (3) L is a non-abelian free group. If F is a non-abelian free group, then there is no algorithm to decide which pairs of finitely presented subgroups of Γ× Γ× F are isomorphic. In more detail, there is a recursive sequence ∆i, where i = 0, 1, . . ., of finite subsets of Γ×Γ×F together with finite presentations 〈∆i|Θi〉 of the subgroups they generate such that there is no algorithm to determine whether or not 〈∆i|Θi〉 ∼= 〈∆0|Θ0〉. An example of such a group Γ is given, for instance, by applying the construction of Rips [14] to a non-abelian free group. Received by the editors January 31, 2002 and, in revised form, September 9, 2002. 2000 Mathematics Subject Classification. Primary 20F10, 20F67.