$z$-Classes of Isometries of The Hyperbolic Space

TL;DR

This paper investigates the structure of isometries of hyperbolic space by classifying them into $z$-classes, showing that there are finitely many such classes, which correspond to different dynamical types, and provides a new parametrization method.

Contribution

It establishes the finiteness of $z$-classes in hyperbolic isometry groups and introduces a novel parametrization of conjugacy classes using the linear model.

Findings

Number of $z$-classes in $I( ext{H}^n)$ is finite.

Explicit computation of the number of $z$-classes.

New parametrization of conjugacy classes using the linear model.

Abstract

Let $G$ be a group. Two elements $x, y$ are said to be {\it $z$-equivalent} if their centralizers are conjugate in $G$. The class equation of $G$ is the partition of $G$ into conjugacy classes. Further decomposition of conjugacy classes into $z$-classes provides an important information about the internal structure of the group. Let $I(\H^n)$ denote the group of isometries of the hyperbolic $n$-space. We show that the number of $z$-classes in $I(\H^n)$ is finite. We actually compute their number, cf. theorem 1.3. We interpret the finiteness of $z$-classes as accounting for the finiteness of "dynamical types" in $I(\H^n)$. Along the way we also parametrize conjugacy classes. We mainly use the linear model for the hyperbolic space for this purpose. This description of parametrizing conjugacy classes appears to be new.