Random Dehn function of groups

TL;DR

This paper investigates the behavior of the random Dehn function in finitely presented acylindrically hyperbolic groups, establishing that it is typically smaller than the classical Dehn function, often at most quadratic.

Contribution

It provides the first asymptotic upper bounds for the random Dehn function in these groups and confirms Gromov's intuition in a new probabilistic model.

Findings

Random Dehn function is at most quadratic in certain groups.

In non-hyperbolic cases, the random Dehn function is strictly smaller than the classical Dehn function.

Confirmed Gromov's intuition in a different probabilistic setting.

Abstract

In this note, we study the notion of random Dehn function and compute an asymptotic upper bound for finitely presented acylindrically hyperbolic groups whose Dehn function is at most polynomial. By showing that in these cases, if the group is not hyperbolic, then the random Dehn function is strictly smaller than the usual Dehn function we confirm Gromov's intuition albeit in a different model. In fact, we show that in these cases the random Dehn function is at most quadratic.