Divergence and quasi-isometry classes of random Gromov's monsters

TL;DR

This paper investigates the geometric properties of random Gromov's monsters, showing they have linear divergence, lack Morse quasigeodesics, and form uncountably many quasi-isometry classes, highlighting their diverse large-scale geometry.

Contribution

It introduces the concept of random Gromov's monsters from i.i.d. labellings, analyzes their divergence, and establishes the existence of uncountably many quasi-isometry classes, expanding understanding of their geometric diversity.

Findings

Random Gromov's monsters have linear divergence along a subsequence.

They do not contain Morse quasigeodesics.

There are uncountably many quasi-isometry classes of these monsters.

Abstract

We show that Gromov's monsters arising from i.i.d. random labellings of expanders (that we call random Gromov's monsters) have linear divergence along a subsequence, so that in particular they do not contain Morse quasigeodesics, and they are not quasi-isometric to Gromov's monsters arising from graphical small cancellation labellings of expanders. Moreover, by further studying the divergence function, we show that there are uncountably many quasi-isometry classes of random Gromov's monsters.