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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…

The automorphism group of an elliptic curve over an algebraically closed field is well known. However, for various applications in coding theory and cryptography, we usually need to apply automorphisms defined over a finite field. Although…

Let $\Phi:F\rightarrow F$ be an automorphism of the finite-rank free group $F$. Suppose that $G=F\rtimes_\Phi\mathbb Z$ is word-hyperbolic. Then $G$ acts freely and cocompactly on a CAT(0) cube complex.

We develop the foundations of the theory of relatively geometric actions of relatively hyperbolic groups on CAT(0) cube complexes, a notion introduced in our previous work [5]. In the relatively geometric setting we prove: full relatively…

Shephard groups are common generalizations of Coxeter groups, Artin groups, and graph products of cyclic groups. Their definition is similar to that of a Coxeter group, but generators may have arbitrary order rather than strictly order 2.…

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…

We show that if G is an infinitely generated locally (polycyclic-by-finite) group with cohomology almost everywhere finitary, then every finite subgroup of G acts freely and orthogonally on some sphere.

We show that if a group G acts isometrically on a locally finite leafless R-tree inducing a two-transitive action on its ends, then this tree is determined by the action of G on the boundary. As a corollary we obtain that locally finite…

We study algebraic properties of groups of PL or smooth homeomorphisms of unit cubes in any dimension, fixed pointwise on the boundary, and more generally PL or smooth groups acting on manifolds and fixing pointwise a submanifold of…

We show that if a (locally compact) group $G$ acts properly on a locally compact $\sigma$-compact space $X$ then there is a family of $G$-invariant proper continuous finite-valued pseudometrics which induces the topology of $X$. If $X$ is…

We show that every real analytic action of a connected supersoluble Lie group on a compact surface with nonzero Euler characteristic has a fixed point. This implies that E. Lima's fixed point free $C^{\infty}$ action on $S^2$ of the affine…

We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space $X$ has a non-empty intersection in the visual bordification $ \bar{X} = X \cup \partial X$. Using this fact, several results known for proper…

Let G be a cyclic group of order 3, 5 or 7, and X=E(n) be the relatively minimal elliptic surface with rational base. In this paper, we prove that under certain conditions on n, there exists a locally linear G-action on X which is…

If $\mathcal C$ is a class of complexes closed under taking full subcomplexes and covers and $\mathcal G$ is the class of groups admitting proper and cocompact actions on one-connected complexes in $\mathcal C$, then $\mathcal G$ is closed…

Let G be a finitely generated group having the property that any action of any finite-index subgroup of G by homeomorphisms of the circle must have a finite orbit. (By a theorem of E.Ghys, lattices in simple Lie groups of real rank at least…

Given an action of a Compact Quantum Group (CQG) on a finite dimensional Hilbert space, we can construct an action on the associated Cuntz algebra. We study the fixed point algebra of this action, using Kirchberg classification results.…

We generalize ideas of Jahncke from trees to square complexes. We introduce the notion of progression in $\mathrm{CAT}(0)$ square complexes. Using progression, we are able to build on the proof strategy of Dahmani-Guirardel-Przytycki to…

Consider a finite l-group acting on the affine space of dimension n over a field k, whose characteristic differs from l. We prove the existence of a fixed point, rational over k, in the following cases: --- The field k is p-special for some…

Suppose that $\{T_{a}:a\in G\}$ is a group of uniformly $L$-Lipschitzian mappings with bounded orbits $\left\{T_{a}x:a\in G\right\}$ acting on a hyperconvex metric space $M$. We show that if $L<\sqrt{2}$, then the set of common fixed points…

Consider the following property of a topological group G: every continuous affine G-action on a Hilbert space with a bounded orbit has a fixed point. We prove that this property characterizes amenability for locally compact sigma-compact…