Generalized torsion for hyperbolic 3‐manifold groups with arbitrary large rank

Let G$G$ be a group and g$g$ a non‐trivial element in G$G$. If some non‐empty finite product of conjugates of g$g$ equals to the trivial element, then g$g$ 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$n > 1$ there are infinitely many closed hyperbolic 3‐manifolds Mn$M_n$ which enjoy the property: (i) the Heegaard genus of Mn$M_n$ is n$n$, (ii) the rank of π1(Mn)$\pi _1(M_n)$ is n$n$, and (ii) π1(Mn)$\pi _1(M_n)$ has a generalized torsion element. Furthermore, we may choose Mn$M_n$ as homology lens spaces and so that the order of the generalized torsion element is arbitrarily large.