Cross ratios and cubulations of hyperbolic groups

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.

Folding-like techniques for CAT(0) cube complexes

In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings’s methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani–Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fundamental groups of non-positively curved cube complexes.

Rank-one isometries of CAT(0) cube complexes and their centralisers

If G is a group acting geometrically on a CAT(0) cube complex X and if g ∈ G has infinite order, we show that exactly one of the following situations occurs: (i) g defines a rank-one isometry of X; (ii) the stable centraliser SCG (g) = {h ∈ G ∣ ∃ n ≥ 1, [h, gn ] = 1} of g is not virtually cyclic; (iii) Fix Y (gn ) is finite for every n ≥ 1 and the sequence (Fix Y (gn )) takes infinitely many values, where Y is a cubical component of the Roller boundary of X which contains an endpoint of an axis of g. We also show that (iii) cannot occur in several cases, providing a purely algebraic characterisation of rank-one isometries.

Drinfeld–Manin solutions of the Yang–Baxter equation coming from cube complexes

The most common geometric interpretation of the Yang–Baxter equation is by braids, knots and relevant Reidemeister moves. So far, cubes were used for connections with the third Reidemeister move only. We will show that there are higher-dimensional cube complexes solving the [Formula: see text]-state Yang–Baxter equation for arbitrarily large [Formula: see text]. More precisely, we introduce explicit constructions of cube complexes covered by products of [Formula: see text] trees and show that these cube complexes lead to new solutions of the Yang–Baxter equations.