Cross ratios and cubulations of hyperbolic groups

Abstract

Many geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichmüller space, Hitchin representations and geodesic currents. We add to this picture by studying cocompact cubulations of arbitrary Gromov hyperbolic groups G. Under weak assumptions, we show that the space of cubulations of G naturally injects into the space of G-invariant cross ratios on the Gromov boundary $$\partial _{\infty }G$$ ∂ ∞ G . A consequence of our results is that essential, hyperplane-essential, cocompact cubulations of hyperbolic groups are length-spectrum rigid, i.e. they are fully determined by their length function. This is the optimal length-spectrum rigidity result for cubulations of hyperbolic groups, as we demonstrate with some examples. In the hyperbolic setting, this constitutes a strong improvement on our previous work [4]. Along the way, we describe the relationship between the Roller boundary of a $$\mathrm{CAT(0)}$$ CAT ( 0 ) cube complex, its Gromov boundary and—in the non-hyperbolic case—the contracting boundary of Charney and Sultan. All our results hold for cube complexes with variable edge lengths.

Comparing topologies on the Morse boundary and quasi-isometry invariance

“…Combining Theorem 1.1 with results from [CM18] and [BF18a] (see section 5 for details), we obtain Corollary 1.3. Let Y be a uniformly locally finite CAT(0) cube complex.…”

“…The Roller boundary induces a subspace topology on ∂ R,M Y . The following result is part of Theorem 3.10 in [BF18a].…”

“…Theorem 5.2 ( [BF18a]). Let Y be a uniformly locally finite CAT(0) cube complex and equip ∂ M Y with the visual topology.…”

Sublinearly Morse boundaries from the viewpoint of combinatorics

“…We remark that the above theorem provides an extension of Theorem D of Beyrer, Fioravanti in [BF18] and Theorem 1.1 of Zalloum in [Zal18]. More precisely, Theorem D in [BF18] shows Theorem 1.1 for the special case where κ = 1 (the Morse boundary of a CAT(0) cube complex).…”

“…We remark that the above theorem provides an extension of Theorem D of Beyrer, Fioravanti in [BF18] and Theorem 1.1 of Zalloum in [Zal18]. More precisely, Theorem D in [BF18] shows Theorem 1.1 for the special case where κ = 1 (the Morse boundary of a CAT(0) cube complex). On the other hand, Theorem 1.1 in [Zal18] shows that for any proper geodesic metric space X, an appropriate subspace of the horofunctions boundary of X continuously surjects on the Morse boundary of X.…”