Related papers: Fixed points for actions of Aut(Fn) on CAT(0) spac…

We study some dynamical properties of the canonical Aut(F_n)-action on the space R_n(G) of redundant representations of the free group F_n in G, where G is the group of rational points of a simple algebraic group over a local field. We show…

Let $\Gamma$ be a finitely generated group of matrices over $\mathbb{C}$. We construct an isometric action of $\Gamma$ on a complete CAT(0) space $X$ such that the restriction of this action to any subgroup of $\Gamma$ containing no…

We shall generalize the concept of $z=(1-t)x\oplus ty$ to $n$ times which contains to verifying some their properties and inequalities in CAT(0) spaces. In the sequel with introducing of $\alpha$-nonexpansive mappings, we obtain some fixed…

Let $\Gamma$ be an irreducible lattice of $\Q$-rank $\geq 2$ in a semisimple Lie group of noncompact type. We prove that any action of $\Gamma$ on a $\CAT(0)$ cubical complex has a global fixed point.

Let F_n be the free group of rank n and let Aut^+(F_n) be its special automorphism group. For an epimorphism pi : F_n -> G of the free group F_n onto a finite group G we call Gamma^+(G,pi) = {f in Aut^+(F_n) | pi*f = pi} the standard…

We show that if a 1-ended group $G$ acts geometrically on a CAT(0) space $X$ and $\bd X$ is separated by $m$ points then either $G$ is virtually a surface group or $G$ splits over a 2-ended group. In the course of the proof we study nesting…

We show that finitely generated groups which are Liouville and without infinite finite-dimensional linear representations must have a global fixed point whenever they act by isometry on a finite-dimensional complete CAT(0)-space. This…

We study the fixed point set in the ideal boundary of a parabolic isometry of a proper CAT(0)-space. We show that the radius of the fixed point set is at most pi/2, and study its centers. As a consequence, we prove that the set of fixed…

We investigate the size of fixed point sets of automorphisms of bounded domains in $\mathbb{C}^n$. In one complex variable, a nontrivial automorphism has at most two fixed points, but in higher dimensions fixed point sets need not be…

For every positive integer $n$, we construct, using algebraic groups, an infinite family of irreducible algebraic varieties $X$,whose automorphism group ${\rm Aut}(X)$ contains the automorphism group ${\rm Aut}(F_n)$ of a free group $F_n$…

Let T be a d-regular tree (d > 2) and A=Aut(T), its automorphism group. Let G be a group generated by n independent Haar-random elements of A. We show that almost surely, every nontrivial element of G has finitely many fixed points on T.

In positive characteristic, algebraic curves can have many more automorphisms than expected from the classical Hurwitz's bound. There even exist algebraic curves of arbitrary high genus g with more than 16g^4 automorphisms. It has been…

We refine Cohen and Lustig's description of centralisers of Dehn twists of free groups. We show that the centraliser of a Dehn twist of a free group has a subgroup of finite index that has a finite classifying space. We describe an…

We consider an action of the automorphism group $\mathrm{Aut}(F_n)$ of the free group $F_n$ of rank $n$ on the filtered vector space $A_d(n)$ of Jacobi diagrams of degree $d$ on $n$ oriented arcs. This action induces on the associated…

Motivated by the theory of Riemann surfaces, we classify all possibilities for finite simple groups acting faithfully on a compact Riemann surface of genus at least 2 in such a way that all non-trivial elements have at most three fixed…

We study the automorphisms group action on a bounded domain in $\CC^n$ having a boundary point that is exponentially flat. Such a domain typically has a compact automorphism group. Our results enable us to generate many new examples.

We study groups of isometries of packed, geodesically complete, CAT$(0)$-spaces for which the systole at every point is smaller than a universal constant depending only on the packing, deducing strong rigidity results. We show that if a…

Let S be an orientable surface of finite type and let Mod(S) be its mapping class group. We consider actions of Mod(S) by semisimple isometries on complete CAT(0) spaces. If the genus of S is at least 3, then in any such action all Dehn…

We study global fixed points for actions of Coxeter groups on nonpositively curved singular spaces. In particular, we consider property FA_n, an analogue of Serre's property FA for actions on CAT(0) complexes. Property FA_n has implications…

The automorphism group $\operatorname{Aut}(F_n)$ of the free group $F_n$ acts on a space $A_d(n)$ of Jacobi diagrams of degree $d$ on $n$ oriented arcs. We study the $\operatorname{Aut}(F_n)$-module structure of $A_d(n)$ by using two…