Topic
Cocompact group action
About: Cocompact group action is a research topic. Over the lifetime, 8 publications have been published within this topic receiving 53 citations.
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TL;DR: In this article, it was shown that a cocompact CAT(0) space is almost geodesically complete under mild conditions, and they proved the same result for the case of mild conditions.
41 citations
TL;DR: In this article, the large time limit of the fibrewise heat operator for Bismut-Lott type superconnections in the L 2 setting has been shown to imply refined L 2-index formulas.
Abstract: We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut-Lott type superconnections in the L^2-setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of the heat operator at large time implies refined L^2-index formulas. As applications, we prove a local L^2-index theorem for families of signature operators and an L^2-Bismut-Lott theorem, expressing the Becker-Gottlieb transfer of flat bundles in terms of Kamber-Tandeur classes. With slightly stronger regularity we obtain the respective refined versions: we construct L^2-eta forms and L^2-torsion forms as transgression forms.
5 citations
TL;DR: In this article, it was shown that the Izeki-Nayatani invariance of a graph model on a Bruhat-Tits building associated to a semi-simple algebraic group has a global fixed point.
Abstract: We prove that if a geodesically complete CAT(0) space X admits a proper cocompact isometric action of a group, then the Izeki-Nayatani invariant of X is less than 1. Let G be a finite connected graph, μ1(G) be the linear spectral gap of G, and λ1(G,X) be the nonlinear spectral gap of G with respect to such a CAT(0) space X. Then, the result implies that the ratio λ1(G,X)/μ1(G) is bounded from below by a positive constant which is independent of the graph G. It follows that any isometric action of a random group of the graph model on such X has a global fixed point. In particular, any isometric action of a random group of the graph model on a Bruhat-Tits building associated to a semi-simple algebraic group has a global fixed point.
4 citations
TL;DR: In this paper, the authors studied affine maps between CAT(0) spaces with cocompact group actions and showed that they essentially split as products of dilations and linear maps on the Euclidean factor.
Abstract: We study affine maps between CAT(0) spaces with cocompact group actions, and show that they essentially split as products of dilations and linear maps (on the Euclidean factor). This extends known results from the Riemannian case. Furthermore, we prove a splitting lemma for the Tits boundary of a CAT(0) space with cocompact group action, a variant of a splitting lemma for geodesically complete CAT(1) spaces by Lytchak.
2 citations
TL;DR: In this article, the large time limit of the fibrewise heat operator for Bismut-Lott type superconnections in the L 2 setting was shown to imply refined L 2-index formulas.
Abstract: We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut-Lott type superconnections in the L-2-setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of the heat operator at large time implies refined L-2-index formulas.
As applications, we prove a local L-2-index theorem for families of signature operators and an L-2-Bismut-Lott theorem, expressing the Becker-Gottlieb transfer of flat bundles in terms of Kamber-Tondeur classes. With slightly stronger regularity we obtain the respective refined versions: we construct L-2-eta forms and L-2-torsion forms as transgression forms.
2 citations