# Action Dimension of Lattices in Euclidean Buildings

## TL;DR

This paper investigates the minimal dimension of contractible manifolds on which groups acting on Euclidean buildings can act properly discontinuously, providing bounds and specific computations for certain arithmetic groups.

## Contribution

It establishes a lower bound for the action dimension of groups acting on Euclidean buildings and computes the action dimension for S-arithmetic groups over number fields.

## Key findings

- Action dimension is at least twice the building's dimension.
- Computed action dimension for S-arithmetic groups over number fields.
- Partially answers a question by Bestvina, Kapovich, and Kleiner.

## Abstract

The action dimension of a group G is the minimal dimension of a contractible manifold that G acts on properly discontinuously. We show that if G acts properly and cocompactly on a thick Euclidean building, then the action dimension is bounded below by twice the dimension of the building. We also compute the action dimension of S-arithmetic groups over number fields, partially answering a question of Bestvina, Kapovich, and Kleiner.