Related papers: Tits Alternative for 2-dimensional CAT(0) complexe…
We explore the geometry of nonpositively curved spaces with isolated flats, and its consequences for groups that act properly discontinuously, cocompactly, and isometrically on such spaces. We prove that the geometric boundary of the space…
We establish obstructions for groups to act by homeomorphisms on dendrites. For instance, lattices in higher rank simple Lie groups will always fix a point or a pair. The same holds for irreducible lattices in products of connected groups.…
We prove a quantitative version of the classical Tits' alternative for discrete groups acting on packed Gromov-hyperbolic spaces supporting a convex geodesic bicombing. Some geometric consequences, as uniform estimates on systole, diastole,…
A tubular group is a group that acts on a tree with $\mathbb{Z}^2$ vertex stabilizers and $\mathbb{Z}$ edge stabilizers. This paper develops further a criterion of Wise and determines when a tubular group acts freely on a finite dimensional…
We prove that every limit group acts geometrically on a CAT(0) space with the isolated flats property.
A generalized triangle group is a group that can be presented in the form $G = < x,y | x^p=y^q=w(x,y)^r=1>$, where $p,q,r\geq 2$ and $w(x,y)$ is a cyclically reduced word of length at least 2 in the free product $\Z_p*\Z_q=< x,y |…
A finite-dimensional CAT(0) cube complex $X$ is equipped with several well-studied boundaries. These include the Tits boundary (which depends on the CAT(0) metric), the Roller boundary (which depends only on the combinatorial structure),…
We study low-dimensional representations of matrix groups over general rings, by considering group actions on CAT(0) spaces, spheres and acyclic manifolds.
We study groups acting on CAT(0) square complexes. In particular we show if Y is a nonpositively curved (in the sense of A. D. Alexandrov) finite square complex and the vertex links of Y contain no simple loop consisting of five edges, then…
Whenever the mapping class group of a closed orientable surface of genus g acts by semisimple isometries on a complete CAT(0) space of dimension less than g it fixes a point.
We study the group Tame(SL$_2$) of tame automorphisms of a smooth affine 3-dimensional quadric, which we can view as the underlying variety of SL(2,$\mathbb{C}$). We construct a square complex on which the group admits a natural cocompact…
We show that groups satisfying Kazhdan's property (T) have no unbounded actions on finite dimensional CAT(0) cube complexes, and deduce that there is a locally CAT(-1) Riemannian manifold which is not homotopy equivalent to any finite…
A recent article of J.P. Bell, K. Huang, W. Peng and T.J. Tucker establishes an analog of the Tits alternative for semigroups of endomorphisms of the projective line. The proof involves a ping-pong argument on arithmetic height functions.…
Let G be a one-ended group acting discretely and co-compactly on a CAT(0) space X. We show that the boundary of X has no cut points and that one can detect splittings of $G$ over two-ended groups and recover its JSJ decomposition from the…
In this short note, we show that a group acting geometrically on a CAT(0) cube complex with virtually abelian hyperplane-stabilisers must decompose virtually as a free product of free abelian groups and surface groups.
We prove a uniform version of the Tits alternative. As a consequence, we obtain uniform lower bounds for the Cheeger constant of Cayley grahs of finitely generated non virtually solvable linear groups in arbitrary characteristic. Also we…
For each $n$ we construct examples of finitely presented $C'(1/6)$ small cancellation groups that do not act properly on any $n$-dimensional CAT(0) cube complex.
In this note, we discuss and motivate the use of the terminology ``median graphs'' in place of ``CAT(0) cube complexes'' in geometric group theory.
The classical Tits construction provides models of the exceptional simple Lie algebras in terms of a unital composition algebra and a degree three simple Jordan algebra. A couple of actions of the symmetric group of degree 4 on this…
We investigate the Tits buildings of the paramodular groups with or without canonical level structure, respectively. These give important combinatorical information about the boundary of the toroidal compactification of the moduli spaces of…