Infinite groups with fixed point properties
We construct finitely generated groups with strong fixed point properties. Let $\mathcal{X}_{ac}$ be the class of Hausdorff spaces of finite covering dimension which are mod-$p$ acyclic for at least one prime $p$. We produce the first examples of infinite finitely generated groups $Q$ with the prope...
| Main Authors: | , , , , , |
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| Format: | Journal article |
| Language: | English |
| Published: |
2007
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| _version_ | 1797069664571883520 |
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| author | Arzhantseva, G Bridson, M Januszkiewicz, T Leary, I Minasyan, A Swiatkowski, J |
| author_facet | Arzhantseva, G Bridson, M Januszkiewicz, T Leary, I Minasyan, A Swiatkowski, J |
| author_sort | Arzhantseva, G |
| collection | OXFORD |
| description | We construct finitely generated groups with strong fixed point properties. Let $\mathcal{X}_{ac}$ be the class of Hausdorff spaces of finite covering dimension which are mod-$p$ acyclic for at least one prime $p$. We produce the first examples of infinite finitely generated groups $Q$ with the property that for any action of $Q$ on any $X\in \mathcal{X}_{ac}$, there is a global fixed point. Moreover, $Q$ may be chosen to be simple and to have Kazhdan's property (T). We construct a finitely presented infinite group $P$ that admits no non-trivial action by diffeomorphisms on any smooth manifold in $\mathcal{X}_{ac}$. In building $Q$, we exhibit new families of hyperbolic groups: for each $n\geq 1$ and each prime $p$, we construct a non-elementary hyperbolic group $G_{n,p}$ which has a generating set of size $n+2$, any proper subset of which generates a finite $p$-group. |
| first_indexed | 2024-03-06T22:27:48Z |
| format | Journal article |
| id | oxford-uuid:5740eba4-51fa-4b0e-8fa4-f84f517f613b |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T22:27:48Z |
| publishDate | 2007 |
| record_format | dspace |
| spelling | oxford-uuid:5740eba4-51fa-4b0e-8fa4-f84f517f613b2022-03-26T16:55:35ZInfinite groups with fixed point propertiesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:5740eba4-51fa-4b0e-8fa4-f84f517f613bEnglishSymplectic Elements at Oxford2007Arzhantseva, GBridson, MJanuszkiewicz, TLeary, IMinasyan, ASwiatkowski, JWe construct finitely generated groups with strong fixed point properties. Let $\mathcal{X}_{ac}$ be the class of Hausdorff spaces of finite covering dimension which are mod-$p$ acyclic for at least one prime $p$. We produce the first examples of infinite finitely generated groups $Q$ with the property that for any action of $Q$ on any $X\in \mathcal{X}_{ac}$, there is a global fixed point. Moreover, $Q$ may be chosen to be simple and to have Kazhdan's property (T). We construct a finitely presented infinite group $P$ that admits no non-trivial action by diffeomorphisms on any smooth manifold in $\mathcal{X}_{ac}$. In building $Q$, we exhibit new families of hyperbolic groups: for each $n\geq 1$ and each prime $p$, we construct a non-elementary hyperbolic group $G_{n,p}$ which has a generating set of size $n+2$, any proper subset of which generates a finite $p$-group. |
| spellingShingle | Arzhantseva, G Bridson, M Januszkiewicz, T Leary, I Minasyan, A Swiatkowski, J Infinite groups with fixed point properties |
| title | Infinite groups with fixed point properties |
| title_full | Infinite groups with fixed point properties |
| title_fullStr | Infinite groups with fixed point properties |
| title_full_unstemmed | Infinite groups with fixed point properties |
| title_short | Infinite groups with fixed point properties |
| title_sort | infinite groups with fixed point properties |
| work_keys_str_mv | AT arzhantsevag infinitegroupswithfixedpointproperties AT bridsonm infinitegroupswithfixedpointproperties AT januszkiewiczt infinitegroupswithfixedpointproperties AT learyi infinitegroupswithfixedpointproperties AT minasyana infinitegroupswithfixedpointproperties AT swiatkowskij infinitegroupswithfixedpointproperties |