Concise formulae in groups of non-positive curvature

TL;DR

This paper demonstrates the conciseness of first-order formulae in various classes of groups, including acylindrically hyperbolic and Burnside groups, and explores properties of definable sets within them.

Contribution

It extends the understanding of formula conciseness to a broad range of groups and analyzes definable set properties based on formula types.

Findings

First-order formulae are concise in acylindrically hyperbolic groups.

Conciseness is established for groups like Burnside, icc, and torus knot groups.

Properties of definable sets, such as finiteness, depend on the formula type.

Abstract

We show that first-order formulae are concise in acylindrically hyperbolic groups and certain extensions thereof. We study further classes of groups, including Burnside groups, icc groups, groups with the `Big Powers' condition, torus knot groups and more, and prove conciseness for wide classes of formulae. We also explore properties of definable sets in these groups, such as their finiteness, depending on the type of formula considered.