Generalized torsion for hyperbolic $3$--manifold groups with arbitrary large rank

Abstract

Let G be a group and g a non-trivial element in G. If some nonempty finite product of conjugates of g equals to the trivial element, then g is called a generalized torsion element. To the best of our knowledge, we have no hyperbolic 3-manifold groups with generalized torsion elements whose rank is explicitly known to be greater than two. The aim of this short note is to demonstrate that for a given integer n > 1 there are infinitely many closed hyperbolic 3-manifolds Mn which enjoy the property: (i) the Heegaard genus of Mn is n, (ii) the rank of π 1 (Mn) is n, and (ii) π 1 (Mn) has a generalized torsion element. Furthermore, we may choose Mn as homology lens spaces and so that the order of the generalized torsion element is arbitrarily large.