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

Let $G$ be a group and $g$ a non-trivial element in $G$. If some non-empty 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 $M_n$ which enjoy the property: (i) the Heegaard genus of $M_n$ is $n$, (ii) the rank of the fundamental group of $M_n$ is $n$, and (ii) the fundamental group of $M_n$ has a generalized torsion element. Furthermore, we may choose $M_n$ as homology lens spaces and so that the order of the generalized torsion element is arbitrarily large.