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Pattern rigidity in hyperbolic spaces: duality and PD subgroups
| Content Provider | Semantic Scholar |
|---|---|
| Author | Biswas, Kingshook Mj, Mahan |
| Copyright Year | 2008 |
| Abstract | For i = 1, 2, let G i be cocompact groups of isometries of hyperbolic space Hn of real dimension n, n ≥ 3. Let H i ⊂ G i be infinite index quasiconvex subgroups satisfying one of the following conditions: 1) limit set of H i is a codimension one topological sphere. 2) limit set of H i is an even dimensional topological sphere. 3) 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 . 4) H i is an odd-dimensional Poincare Duality group PD(2k + 1). This gener alizes (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 the main result of Mosher-Sageev-Whyte [MSW04], we get quasi-isometric rigidity results for graphs of groups where the vertex groups are uniform lattices in rank one symmetric spaces and edge groups are of any of the above types. |
| Starting Page | 97 |
| Ending Page | 123 |
| Page Count | 27 |
| File Format | PDF HTM / HTML |
| DOI | 10.4171/GGD/152 |
| Volume Number | 6 |
| Alternate Webpage(s) | http://arxiv.org/pdf/0809.4449v2.pdf |
| Alternate Webpage(s) | https://arxiv.org/pdf/0809.4449v3.pdf |
| Alternate Webpage(s) | http://repository.ias.ac.in/89536/1/12-a.pdf |
| Alternate Webpage(s) | http://www.ems-ph.org/journals/show_pdf.php?iss=1&issn=1661-7207&rank=4&vol=6 |
| Alternate Webpage(s) | https://doi.org/10.4171/GGD%2F152 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
