Infinite Groups with Large Balls of Torsion Elements and Small Entropy

We exhibit infinite, solvable, virtually abelian groups with a fixed number of generators, having arbitrarily large balls consisting of torsion elements. We also provide a sequence of 3-generator non-virtually nilpotent polycyclic groups of algebraic entropy tending to zero. All these examples are obtained by taking appropriate quotients of finitely presented groups mapping onto the first Grigorchuk group. The Burnside Problem asks whether a finitely generated group all of whose elements have finite order must be finite. We are interested in the following related question: fix n sufficiently large; given a group Γ, with a finite symmetric generating subset S such that every element in the n-ball is torsion, is Γ finite? Since the Burnside problem has a negative answer, a fortiori the answer to our question is negative in general. However, it is natural to ask for it in some classes of finitely generated groups for which the Burnside Problem has a positive answer, such as linear groups or solvable groups. This motivates the following proposition, which in particular answers a question of Breuillard to the authors. Proposition 1. For every n, there exists a group G, generated by a 3-element subset S consisting of elements of order 2, in which the n-ball consists of torsion elements, and which satisfies one of the additional assumptions: (i) G is solvable, virtually abelian, and infinite (more precisely, it has a free abelian normal subgroup of finite 2-power index); in particular it is linear. (ii) G is polycyclic but not virtually nilpotent. (iii) G is solvable but not polycyclic. Remark 2. (1) The groups in Proposition 1 can actually be chosen to be 2-generated: indeed, if G is generated by three involutions a, b, c, then the subgroup generated by ab and bc has index at most two. (2) Natural stronger hypotheses are the following: being linear in fixed dimension; being solvable of given solvability length. We have no answer in these cases. It is also natural to ask what happens it we fix a torsion exponent. (3) By [Se, Corollaire 2, p. 90], if G is a group and S is any finite generating subset for which the 2-ball of G consists of torsion elements, then G has Property (FA): every action of G on a tree has a fixed point. In particular, if G is infinite, then by Stallings’ Theorem [St] it cannot be virtually free. (4) For every sufficiently large prime p, and for all n, there exists a non-elementary, 2-generated word hyperbolic group in which the n-ball consists of elements of p-torsion [Ol]. Date: October 7, 2005. 2000 Mathematics Subject Classification. Primary 20F50; Secondary 20F16, 20F05.