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.

Cannon–Thurston maps for $${{\,\mathrm{CAT}\,}}(0)$$ groups with isolated flats

Mahan Mitra (Mj) proved Cannon–Thurston maps exist for normal hyperbolic subgroups of a hyperbolic group (Mitra in Topology, 37(3):527–538, 1998). We prove that Cannon–Thurston maps do not exist for infinite normal hyperbolic subgroups of non-hyperbolic $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups with isolated flats with respect to the visual boundaries. We also show Cannon–Thurston maps do not exist for infinite infinite-index normal $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) subgroups with isolated flats in non-hyperbolic $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups with isolated flats. We obtain a structure theorem for the normal subgroups in these settings and show that outer automorphism groups of hyperbolic groups have no purely atoroidal $$\mathbb {Z}^2$$ Z 2 subgroups.

Conformal dimension of hyperbolic groups that split over elementary subgroups

We study the (Ahlfors regular) conformal dimension of the boundary at infinity of Gromov hyperbolic groups which split over elementary subgroups. If such a group is not virtually free, we show that the conformal dimension is equal to the maximal value of the conformal dimension of the vertex groups, or 1, whichever is greater, and we characterise when the conformal dimension is attained. As a consequence, we are able to characterise which Gromov hyperbolic groups (without 2-torsion) have conformal dimension 1, answering a question of Bonk and Kleiner.

The relative exponential growth rate of subgroups of acylindrically hyperbolic groups

We introduce a new invariant of finitely generated groups, the ambiguity function, and we prove that every finitely generated acylindrically hyperbolic group has a linearly bounded ambiguity function. We use this result to prove that the relative exponential growth rate lim n → ∞ ⁡ | B H X ⁢ ( n ) | n \lim_{n\to\infty}\sqrt[n]{\lvert\vphantom{1_{1}}{B^{X}_{H}(n)}\rvert} of a subgroup 𝐻 of a finitely generated acylindrically hyperbolic group 𝐺 exists with respect to every finite generating set 𝑋 of 𝐺 if 𝐻 contains a loxodromic element of 𝐺. Further, we prove that the relative exponential growth rate of every finitely generated subgroup 𝐻 of a right-angled Artin group A Γ A_{\Gamma} exists with respect to every finite generating set of A Γ A_{\Gamma} .

Strongly aperiodic subshifts of finite type on hyperbolic groups

We prove that a hyperbolic group admits a strongly aperiodic subshift of finite type if and only if it has at most one end.