- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:04:59
Groups, Geometry, and Dynamics
Groups Geom. Dyn.
GGD
1661-7207
1661-7215
Group theory and generalizations
10.4171/GGD
http://www.ems-ph.org/doi/10.4171/GGD
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
6
2012
1
Pattern rigidity in hyperbolic spaces: duality and PD subgroups
Kingshook
Biswas
RKM Vivekananda University, Dist. HOWRAH, West Bengal, INDIA
Mahan
Mj
Tata Institute of Fundamental Research, MUMBAI, INDIA
Quasi-isometric rigidity, pattern rigidity, homology manifold, quasiconformal map
body { counter-reset: list; } ol li { list-style: none; } ol li:before { content: "("counter(list, decimal) ") "; counter-increment: list; } For $i= 1,2$, let $G_i$ be cocompact groups of isometries of hyperbolic space $\mathbf{H}^n$ of real dimension $n$, $n \geq 3$. Let $H_i \subset G_i$ be infinite index quasiconvex subgroups satisfying one of the following conditions: The limit set of $H_i$ is a codimension one topological sphere. The limit set of $H_i$ is an even dimensional topological sphere. $H_i$ is a codimension one duality group. This generalizes (1). In particular, if $n = 3$, $H_i$ could be any freely indecomposable subgroup of $G_i$. $H_i$ is an odd-dimensional Poincaré duality group PD$(2k+1)$. This generalizes (2). We prove pattern rigidity for such pairs extending work of Schwartz who proved pattern rigidity when $H_i$ is cyclic. All this generalizes to quasiconvex subgroups of uniform lattices in rank one symmetric spaces satisfying one of the conditions (1)–(4), as well as certain special subgroups with disconnected limit sets. In particular, pattern rigidity holds for all quasiconvex subgroups of hyperbolic 3-manifolds that are not virtually free. Combining this with results of Mosher, Sageev, and Whyte, we obtain quasi-isometric rigidity results for graphs of groups where the vertex groups are uniform lattices in rank one symmetric spaces and the edge groups are of any of the above types.
Group theory and generalizations
Topological groups, Lie groups
Manifolds and cell complexes
General
97
123
10.4171/GGD/152
http://www.ems-ph.org/doi/10.4171/GGD/152