Related papers: Tits Alternative for 2-dimensional CAT(0) complexe…

We prove the Tits Alternative for groups acting on $2$-dimensional "recurrent" complexes with uniformly bounded cell stabilisers. This class of complexes includes, among others: $2$-dimensional Euclidean buildings, $2$-dimensional systolic…

We prove a Tits alternative theorem for groups acting on CAT(0) cubical complexes. Namely, suppose that $G$ is a group for which there is a bound on the orders of its finite subgroups. We prove that if $G$ acts properly on a…

Given a group action on a finite-dimensional CAT(0) cube complex, we give a simple criterion phrased purely in terms of cube stabilisers that ensures that the group satisfies the strong Tits alternative, provided that each vertex stabiliser…

Let $\Gamma$ be a finitely generated group acting properly discontinuously by isometries on a visibility CAT(0) space $X$ that satisfies the bounded packing property. We prove that $\Gamma$ satisfies the Tits alternative: it is either…

We define Tits rigidity for visual boundaries of CAT(0) groups, and prove that the join of two Cantor sets and its suspension are Tits rigid.

We show that the automorphism group of a one-dimensional full shift (the group of reversible cellular automata) does not satisfy the Tits alternative. That is, we construct a finitely-generated subgroup which is not virtually solvable yet…

We prove the Tits alternative for an almost coherent $PD(3)$ group which is not virtually properly locally cyclic. In particular, we show that an almost coherent $PD(3)$ group which cannot be generated by fewer than four elements always…

We prove a version of the Tits alternative for groups acting on complete, finite rank median spaces. This shows that group actions on finite rank median spaces are much more restricted than actions on general median spaces. Along the way,…

We prove that torsion subgroups of groups defined by C(6), C(4)-T(4) or C(3)-T(6) small cancellation presentations are finite cyclic groups. This follows from a more general result on the existence of fixed points for locally elliptic…

We obtain a sufficient condition for lattices in the automorphism group of a finite dimensional CAT(0) cube complex to have infinite girth. As a corollary, we get a version of Girth Alternative for groups acting geometrically: any such…

We investigate the Tits boundary of locally compact CAT(0) 2-complexes. In particular we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As…

We survey some properties of the automorphism groups of compact Kaehler manifolds. In particular, we present recent results by Keum, Oguiso and Zhang on the structure of these groups from the Tits alternative point of view. Several other…

We revisit the topic of probability measures on CAT(0) cube complexes and prove that an amenable group acting on a CAT(0) cube complex, regardless of dimension, necessarily preserves an interval in the Roller compactification. In the finite…

We prove the following rank rigidity result for proper CAT(0) spaces with one-dimensional Tits boundaries: Let $\Gamma$ be a group acting properly discontinuously, cocompactly, and by isometries on such a space $X$. If the Tits diameter of…

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…

A theorem of Cantat and Urech says that an analog of the classical Tits alternative holds for the group of birational automorphisms of a compact complex Kaehler surface. We established in our previous paper the following Tits-type…

Let $G=G_1\ast\dots\ast G_k\ast F$ be a countable group which splits as a free product, where all groups $G_i$ are freely indecomposable and not isomorphic to $\mathbb{Z}$, and $F$ is a finitely generated free group. If for all…

For group actions on hyperbolic CAT(0) square complexes, we show that the acylindricity of the action is equivalent to a weaker form of acylindricity phrased purely in terms of stabilisers of points, which has the advantage of being much…

We prove a Tits alternative for topological full groups of minimal actions of finitely generated groups. On the one hand, we show that topological full groups of minimal actions of virtually cyclic groups are amenable. By doing so, we…

We exhibit 3-generator Artin groups which have finite 2-dimensional Eilenberg-Mac Lane spaces, but which do not act properly discontinuously by semi-simple isometries on a 2-dimensional CAT(0) complex. We prove that infinitely many of these…