Reconstructible graphs, simplicial flag complexes of homology manifolds and associated right-angled Coxeter groups

In this paper, we investigate a relation between finite graphs, simplicial flag complexes and right-angled Coxeter groups, and we provide a class of reconstructible finite graphs. We show that if $\Gamma$ is a finite graph which is the 1-skeleton of some simplicial flag complex $L$ which is a homology manifold of dimension $n \ge 1$, then the graph $\Gamma$ is reconstructible.Comment: 7 page

On splitting theorems for CAT(0) spaces and compact geodesic spaces of non-positive curvature

In this paper, we show some splitting theorems for CAT(0) spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non-positive curvature. A CAT(0) group $\Gamma$ is said to be {\it rigid}, if $\Gamma$ determines the boundary up to homeomorphisms of a CAT(0) space on which $\Gamma$ acts geometrically. C.Croke and B.Kleiner have constructed a non-rigid CAT(0) group. As an application of the splitting theorems for CAT(0) spaces, we obtain that if $\Gamma_1$ and $\Gamma_2$ are rigid CAT(0) groups then so is $\Gamma_1\times \Gamma_2$.Comment: 14 page