Compactly presented groups

TL;DR

This survey introduces compactly presented groups as a broad class of locally compact groups, emphasizing their geometric properties, invariance under quasi-isometry, and providing examples related to Lie groups and Cayley graphs.

Contribution

It offers an elementary introduction to compactly presented groups, connecting algebraic and geometric perspectives, and includes an example of a Lie group not quasi-isometric to any homogeneous graph.

Findings

Compact presentation relates to coarse simple connectedness.

Compactly presented groups are quasi-isometry invariants.

An example of a Lie group not quasi-isometric to homogeneous graphs is provided.

Abstract

This survey purports to be an elementary introduction to compactly presented groups, which are the analogue of finitely presented groups in the broader realm of locally compact groups. In particular, compact presentation is interpreted as a coarse simple connectedness condition on the Cayley graph, and in particular is a quasi-isometry invariant. In the appendix, an example of a Lie group, not quasi-isometric to any homogeneous graph, is given; the short argument relies on results of Trofimov and Pansu, anterior to~1990.