# Commensurating actions of birational groups and groups of pseudo-automorphisms

## TL;DR

This paper shows that groups of birational transformations with certain fixed point properties can be conjugated to groups acting as pseudo-automorphisms, aiding classification of surface birational groups.

## Contribution

It introduces a geometric group theory approach to classify birational groups with fixed point properties as pseudo-automorphisms.

## Key findings

- Groups with fixed point properties are birationally conjugate to pseudo-automorphism groups.
- Application to classify surface birational transformation groups with fixed point properties.
- Uses CAT(0) cubical complexes and Kazhdan Property (T) in the analysis.

## Abstract

Pseudo-automorphisms are birational transformations acting as regular automorphisms in codimension 1. We import ideas from geometric group theory to prove that a group of birational transformations that satisfies a fixed point property on CAT(0) cubical complexes, for example a discrete countable group with Kazhdan Property (T), is birationally conjugate to a group acting by pseudo-automorphisms on some non-empty Zariski-open subset. We apply this argument to classify groups of birational transformations of surfaces with this fixed point property up to birational conjugacy.