There are (essentially) two types of groups containing expanders, also known as Gromov's mosters. The "random Gromov's monsters" are constructed, following Gromov and Arzhantseva-Delzant, using random labellings on suitable expanders. On the other hand, Osajda (again using probabilistic arguments) showed that there are labellings on suitable expanders satisfying a graphical small-cancellation condition, yielding the "combinatorial Gromov's monsters".
These two types of Gromov's monsters look superficially similar, but in fact they are quite different. More specifically, I will discuss the fact that combinatorial Gromov's monsters are acylindrically hyperbolic, while random Gromov's monsters cannot act non-elementarily on hyperbolic spaces. In particular, groups of the latter type cannot be isomorphic to groups of the former type. What is more, random Gromov's monsters cannot even be quasi-isometric to combinatorial Gromovs's monsters, since, as I will explain, the latter have linear divergence along a subsequence. This same phenomenon can also be used to show that there are uncountably many quasi-isometry types of random Gromov's monsters.
The proofs rely on comparing random walks on expander graphs with random walks on the corresponding groups, following Naor-Silberman.
Based on joint works with Dominik Gruber and Romain Tessera.