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Register a new userA `transversal for minimal invariant sets in the boundary of a CAT(0) group
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We introduce new techniques for studying boundary dynamics of CAT(0) groups. For a group $G$ acting geometrically on a CAT(0) space $X$ we show there is a flat $Fsubset X$ of maximal dimension whose boundary sphere intersects every minimal $G$-invariant subset of $partial_infty X$. As a result we derive a necessary and sufficient dynamical condition for $G$ to be virtually-Abelian, as well as a new approach to Ballmanns rank rigidity conjecture.
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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.
We prove that the automorphism group of the braid group on four strands acts faithfully and geometrically on a CAT(0) 2-complex. This implies that the automorphism group of the free group of rank two acts faithfully and geometrically on a CAT(0) 2-complex, in contrast to the situation for rank three and above.
We study abstract group actions of locally compact Hausdorff groups on CAT(0) spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris-Nickolas for actions on trees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion free CAT(0) group is continuous.
Let $V$ be a finite vector space over a finite field of order $q$ and of characteristic $p$. Let $Gleq GL(V)$ be a $p$-solvable completely reducible linear group. Then there exists a base for $G$ on $V$ of size at most $2$ unless $q leq 4$ in which case there exists a base of size at most $3$. The first statement extends a recent result of Halasi and Podoski and the second statement generalizes a theorem of Seress. An extension of a theorem of Palfy and Wolf is also given.
We study the general theory of asymptotically CAT(0) groups, explaining why such a group has finitely many conjugacy classes of finite subgroups, is $F_infty$ and has solvable word problem. We provide techniques to combine asymptotically CAT(0) groups via direct products, amalgams and HNN extensions. The universal cover of the Lie group $PSL(2,mathbb{R})$ is shown to be an asymptotically CAT(0) metric space. Therefore, co-compact lattices in $widetilde{PSL(2,mathbb{R})}$ provide the first examples of asymptotically CAT(0) groups which are neither CAT(0) nor hyperbolic. Another source of examples is shown to be the class of relatively hyperbolic groups.
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