Parabolic isometries of visible CAT(0) spaces and metrics on moduli space

TL;DR

This paper investigates the properties of isometries in visible CAT(0) spaces, showing that parabolic isometries have zero translation length and classifying all isometries, while also proving certain moduli spaces lack complete visible CAT(0) metrics.

Contribution

It establishes that parabolic isometries in semi-uniformly visible CAT(0) spaces have zero translation length and classifies isometries, additionally proving the non-existence of certain metrics on moduli spaces.

Findings

Parabolic isometries have zero translation length in semi-uniformly visible CAT(0) spaces.

Classification of isometries based on translation lengths.

Moduli space $ ext{M}(S_{g,n})$ admits no complete visible CAT(0) Riemannian metric for $3g+n extgreater{}5$.

Abstract

We show that the translation length of any parabolic isometry on a complete semi-uniformly visible CAT(0) space is always zero. As a consequence, we will classify the isometries on visible CAT(0) spaces in terms of translation lengths. We will also show that the moduli space $\mathbb{M}(S_{g,n})$ of surface $S_{g,n}$ of $g$ genus with $n$ punctures admits no complete visible CAT(0) Riemannian metric if $3g+n\geq 5$, which answers the Brock-Farb-McMullen question in the visible case.