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.

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.

Full text

Action Dimension of Lattices in Euclidean Buildings

Kevin Schreve