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Register a new userQuasiflats in CAT(0) complexes
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We show that if X is a piecewise Euclidean 2-complex with a cocompact isometry group, then every 2-quasiflat in X is at finite Hausdorff distance from a subset which is locally flat outside a compact set, and asymptotically conical.
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It is known that a cocompact special group $G$ does not contain $mathbb{Z} times mathbb{Z}$ if and only if it is hyperbolic; and it does not contain $mathbb{F}_2 times mathbb{Z}$ if and only if it is toric relatively hyperbolic. Pursuing in this direction, we show that $G$ does not contain $mathbb{F}_2 times mathbb{F}_2$ if and only if it is weakly hyperbolic relative to cyclic subgroups, or cyclically hyperbolic for short. This observation motivates the study of cyclically hyperbolic groups, which we initiate in the class of groups acting geometrically on CAT(0) cube complexes. Given such a group $G$, we first prove a structure theorem: $G$ virtually splits as the direct sum of a free abelian group and an acylindrically hyperbolic cubulable group. Next, we prove a strong Tits alternative: every subgroup $H leq G$ either is virtually abelian or it admits a series $H=H_0 rhd H_1 rhd cdots rhd H_k$ where $H_k$ is acylindrically hyperbolic and where $H_i/H_{i+1}$ is finite or free abelian. As a consequence, $G$ is SQ-universal and it cannot contain subgroups such that products of free groups and virtually simple groups.
We provide a necessary and sufficient condition on a finite flag simplicial complex, L, for which there exists a unique CAT(0) cube complex whose vertex links are all isomorphic to L. We then find new examples of such CAT(0) cube complexes and prove that their automorphism groups are virtually simple. The latter uses a result, which we prove in the appendix, about the simplicity of certain subgroups of the automorphism group of a rank-one CAT(0) cube complex. This result generalizes previous results by Tits and by Haglund and Paulin.
We prove that any group acting essentially without a fixed point at infinity on an irreducible finite-dimensional CAT(0) cube complex contains a rank one isometry. This implies that the Rank Rigidity Conjecture holds for CAT(0) cube complexes. We derive a number of other consequences for CAT(0) cube complexes, including a purely geometric proof of the Tits Alternative, an existence result for regular elements in (possibly non-uniform) lattices acting on cube complexes, and a characterization of products of trees in terms of bounded cohomology.
Let $G$ be a group acting properly and essentially on an irreducible, non-Euclidean finite dimensional CAT(0) cube complex $X$ without fixed points at infinity. We show that for any finite collection of simultaneously inessential subgroups ${H_1, ldots, H_k}$ in $G$, there exists an element $g$ of infinite order such that $forall i$, $langle H_i, grangle cong H_i * langle grangle$. We apply this to show that any group, acting faithfully and geometrically on a non-Euclidean possibly reducible CAT(0) cube complex, has property $P_{naive}$ i.e. given any finite list ${g_1, ldots, g_k}$ of elements from $G$, there exists $g$ of infinite order such that $forall i$, $langle g_i, grangle cong langle g_i rangle *langle grangle$. This applies in particular to the Burger-Moses simple groups that arise as lattices in products of trees. The arguments utilize the action of the group on its Poisson boundary and moreover, allow us to summarise equivalent conditions for the reduced $C^*$-algebra of the group to be simple.
In this paper we start the inquiry into proving uniform exponential growth in the context of groups acting on CAT(0) cube complexes. We address free group actions on CAT(0) square complexes and prove a more general statement. This says that if $F$ is a finite collection of hyperbolic automorphisms of a CAT(0) square complex $X$, then either there exists a pair of words of length at most 10 in $F$ which freely generate a free semigroup, or all elements of $F$ stabilize a flat (of dimension 1 or 2 in $X$). As a corollary, we obtain a lower bound for the growth constant, $sqrt[10]{2}$, which is uniform not just for a given group acting freely on a given CAT(0) cube complex, but for all groups which are not virtually abelian and have a free action on a CAT(0) square complex.
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