Orbihedra of nonpositive curvature

Abstract

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Lectures on Spaces of Nonpositive Curvature

“…In the case of 2-dimensional polyhedra with piecewise smooth metrics, the answer to Question 1.6 is affirmative, see [BB3]. In this reference, one also finds a natural generalization of the Liouville measure for geodesic flows on polyhedra with piecewise smooth metrics.…”

“…Of course one might wonder how serious such a restriction is and how many interesting examples are excluded by it. A rich source of examples are locally finite polyhedra with piecewise smooth metrics of nonpositive Alexandrov curvature and, in dimension 2, such a polyhedron always contains a homotopy equivalent subpolyhedron which is geodesically complete and of nonpositive Alexandrov curvature with respect to the induced length metric, see [BB3]. Whether this or something similar holds in higher dimensions is open (as far as I know).…”

On the SQ-universality of groups with special presentations

“…By Lemma 1 there is a l-polygonal presentation K over the star graph G of P whose corresponding polyhedron has one vertex such that the link is G, that is, the incidence graph of a finite projective plane (or generalized 3-gon). Now the universal cover of such a polyhedron is anà A 2 -building on which the group G, being the fundamental group of the polyhedron, acts cocompactly; see, for example, [2] or [3]. It follows that if G is defined by a triangle presentation as defined in [6] then we can appeal to the Margulis normal subgroup theorem [19] to deduce that G is just infinite.…”

Boundaries and JSJ Decompositions of CAT(0)-Groups

“…We give a brief introduction to CAT(0) spaces (see [9], [3] for details). Let Y be a metric space, and I be an interval of R. A path γ :…”

“…In particular: if G doesn't fix a point of ∂X and doesn't have rank 1, and I is a minimal closed invariant set for the action of G on ∂X, then for any x ∈ ∂X, d T (x, I) ≤ π 2 . then G contains a free subgroup of rank 2 ( [3], theorem A). We remark that the Tits alternative is not known for CAT(0) groups.…”