Fractional Isoperimetric Inequalities and subgroup distortion

TL;DR

This paper constructs finitely presented groups with specific isoperimetric and isodiametric functions, demonstrating a wide range of subgroup distortions and Dehn functions, including non-integer exponents.

Contribution

It introduces new constructions of finitely presented groups with prescribed polynomial Dehn functions and subgroup distortions, expanding understanding of geometric group theory.

Findings

Existence of groups with Dehn functions approximately n^r for infinitely many non-integers r>2.

Construction of pairs of groups with subgroup distortion approximately n^s for any positive rational s.

Groups with isodiametric functions approximately n^s for any s ≥ 1.

Abstract

It is shown that there exist infinitely many non-integers $r>2$ such that the Dehn function of some finitely presented group is $\simeq n^r$. For each positive rational number $s$ we construct pairs of finitely presented groups $H\subset G$ such that the distortion of $H$ in $G$ is $\simeq n^s$. And for each $s\ge 1$ we also construct finitely presented groups whose isodiametric function is $\simeq n^s$.