Divergence and quasi-isometry classes of random Gromov’s 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.

Quasi‐isometric diversity of marked groups

“…Then the lattice R in the group Isom(DL(n, n)) corresponding to G N will have an infinite center Z. In the notation introduced in the previous paragraph, condition (13) obviously implies that Z L ∩ L , which contradicts (14). Thus G N ∼ q.i.…”

“…Example We leave it to the reader to verify that the proof of quasi‐isometric diversity of groups constructed by Kropholler–Leary–Soroko [20] and Gruber–Sisto [14] can also be simplified to Corollary 1.2. More precisely, our result allows one to avoid computing Bowditch's invariant in [20] and divergence functions in [14] for the purpose of distinguishing non‐quasi‐isometric groups. Note, however, that we do need to use some technical lemmas from these papers to establish that the corresponding subspaces of are perfect.…”

“…A continuous family of pairwise non‐quasi‐isometric groups with property was constructed by Kropholler, Leary, and Soroko [20]. Gruber and Sisto showed that Gromov's construction of groups associated to a random labeling of certain expanders yields quasi‐isometry classes [14].…”

“…The proof of Theorem 1.1, given in Section 3, is fairly short and combines an observation, originally made by Thomas [36], that ‐quasi‐isometry defines a closed relation on , with a purely topological argument. Nevertheless, Theorem 1.1 allows one to avoid non‐trivial computations of particular invariants used to distinguish the quasi‐isometry classes in [3, 8, 14, 20] via the following corollary. For a more detailed discussion, we refer to Examples 3.2–3.4.…”

Controlled Analytic Properties and the Quantitative Baum-Connes Conjecture

“…Currently, it is possible to use either a geometric small cancellation labelling (as in Gromov [Gro03]) or a graphical small cancellation labelling, and these have been shown to have a very different character, for instance the groups that come from these labellings cannot be quasi-isometric (this is due to Gruber-Sisto [GS18]). The method itself is very flexible and compatible with other small cancellation constructions possible in the literature -see for instance the remarks of Arzhantseva-Delzant [AD08].…”