JournalsggdVol. 6, No. 1pp. 97–123

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
Pattern rigidity in hyperbolic spaces:  duality and PD subgroups cover

Abstract

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{ content: "("counter(list, decimal) ") "; counter-increment: list; } For i=1,2i= 1,2, let GiG_i be cocompact groups of isometries of hyperbolic space Hn\mathbf{H}^n of real dimension nn, n3n \geq 3. Let HiGiH_i \subset G_i be infinite index quasiconvex subgroups satisfying one of the following conditions:

  1. The limit set of HiH_i is a codimension one topological sphere.
  2. The limit set of HiH_i is an even dimensional topological sphere.
  3. HiH_i is a codimension one duality group. This generalizes (1). In particular, if n=3n = 3, HiH_i could be any freely indecomposable subgroup of GiG_i.
  4. HiH_i is an odd-dimensional Poincaré duality group PD(2k+1)(2k+1). This generalizes (2).

We prove pattern rigidity for such pairs extending work of Schwartz who proved pattern rigidity when HiH_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.

Cite this article

Kingshook Biswas, Mahan Mj, Pattern rigidity in hyperbolic spaces: duality and PD subgroups. Groups Geom. Dyn. 6 (2012), no. 1, pp. 97–123

DOI 10.4171/GGD/152