Systolic geometry and simplicial complexity for groups

TL;DR

This paper introduces simplicial complexity, a new combinatorial invariant for finitely presentable groups, providing insights into Gromov's question on the boundedness of groups' systolic area and revealing new topological behaviors.

Contribution

The paper defines simplicial complexity for groups and applies it to answer Gromov's question, linking it to systolic area and topological properties.

Findings

Simplicial complexity effectively bounds the set of groups with bounded systolic area.

New relations between systolic area and topological features of groups.

Enhanced understanding of the topological behavior of groups with bounded systolic area.

Abstract

Twenty years ago Gromov asked about how large is the set of isomorphism classes of groups whose systolic area is bounded from above. This article introduces a new combinatorial invariant for finitely presentable groups called {\it simplicial complexity} that allows to obtain a quite satisfactory answer to his question. Using this new complexity, we also derive new results on systolic area for groups that specify its topological behaviour.