Dehn functions and the space of marked groups

TL;DR

This paper investigates how Dehn functions behave under limits in the space of marked groups, establishing inequalities and implications for hyperbolicity preservation in converging sequences.

Contribution

It provides a new inequality relating Dehn functions of converging groups and proves hyperbolicity is preserved under certain limits.

Findings

Dehn functions of converging groups satisfy specific inequalities.

Hyperbolicity is retained in limits of hyperbolic marked groups.

The results connect geometric properties with algebraic limits in group theory.

Abstract

In the space of marked group, we suppose that a sequence $(G_i, X_i)$ converges to $(G,X)$, where $G$ is finitely presented. We obtain an inequality which connects Dehn functions of $G_i$s and $G$. As a result, we show that if a sequence $(K, X_i)$ converges to a hyperbolic marked group, then $K$ is already hyperbolic.