Hyperbolic Groups and Their Quotients of Bounded Exponents

In 1987, Gromov conjectured that for every non-elementary hyperbolic group G there is an n = n(G) such that the quotient group G/Gn is infinite. The article confirms this conjecture. In addition, a description of finite subgroups of G/Gn is given, it is proven that the word and conjugacy problem are solvable in G/Gn and that ⋂∞ k=1G k = {1}. The proofs heavily depend upon prior authors’ results on the Gromov conjecture for torsion free hyperbolic groups and on the Burnside problem for periodic groups of even exponents. 0. Introduction Let A be a finite set of generators for a group G. By |g| = |W | denote the length of a shortest word W in the alphabet A that represents an element g ∈ G. One of definitions of a hyperbolic group G is given by means of the Gromov product (g · h) = 12 (|g|+ |h| − |g −1h|) as follows: A group G is called hyperbolic [Gr] if there exists a constant δ ≥ 0 such that for every triple g, h, f ∈ G (1) (g · h) ≥ min((g · f), (h · f))− δ. In turns out [Gr], [GH] that the property of being hyperbolic does not depend on a particular generating set A (but the constant δ does depend on A). The concept of hyperbolicity has its origins in geometry [Gr]. However, hyperbolic groups deserve the special attention of group theory as well: The class of hyperbolic groups is generic so that almost every (in a definite statistical meaning) finitely presented group is hyperbolic [Gr], [Ol4]. Basic properties of hyperbolic groups are the subject of books [GH], [CDP]. For the free group over the alphabet A inequality (1) is satisfied with δ = 0, because in this case (g·h) is the length of the maximal common beginning of reduced words g, h in A. Similar to free groups, an arbitrary non-elementary hyperbolic group has many homomorphic images (recall a group Γ is termed in [Gr] elementary if Γ has a cyclic subgroup of finite index). Discussing an approach to construction of an infinite periodic quotient group Ḡ of a non-elementary hyperbolic group, Gromov Received by the editors April 5, 1995. 1991 Mathematics Subject Classification. Primary 20F05, 20F06, 20F32, 20F50. The second author was supported in part by Russian Fund for Fundamental Research, Grant 010-15-41, and by International Scientific Foundation, Grant MID 000. c ©1996 American Mathematical Society