On super-rigidity of Gromov's random monster group
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910831950692352 |
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| author | Das, Kajal |
| author_facet | Das, Kajal |
| contents | In this article, we show super-rigidity of Gromov's random monster group. We prove that any morphism $ϕ_α$ from Gromov's random monster group $Γ_α$ to the group $G$ has finite image for almost all $α$, where $G$ is any of the following types of groups: mapping class group $MCG(S_{g,b})$, braid group $B_n$, outer automorphism group of a free group $Out(F_N)$, automorphism group of a free group $Aut(F_N)$, hierarchically hyperbolic group, a-$L^p$-menable group or K-amenable group. We introduce another property called hereditary super-rigidity and prove that $Γ_α$ has hereditary super-rigidity with respect to an a-$L^p$-menable group or a K-amenable group. We also establish a stability theorem for the groups with respect to which $Γ_α$ has super-rigidity and hereditary super-rigidity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_06255 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On super-rigidity of Gromov's random monster group Das, Kajal Group Theory Geometric Topology Probability 20F65, 20F67, 20P05, 53C24, 57K20 In this article, we show super-rigidity of Gromov's random monster group. We prove that any morphism $ϕ_α$ from Gromov's random monster group $Γ_α$ to the group $G$ has finite image for almost all $α$, where $G$ is any of the following types of groups: mapping class group $MCG(S_{g,b})$, braid group $B_n$, outer automorphism group of a free group $Out(F_N)$, automorphism group of a free group $Aut(F_N)$, hierarchically hyperbolic group, a-$L^p$-menable group or K-amenable group. We introduce another property called hereditary super-rigidity and prove that $Γ_α$ has hereditary super-rigidity with respect to an a-$L^p$-menable group or a K-amenable group. We also establish a stability theorem for the groups with respect to which $Γ_α$ has super-rigidity and hereditary super-rigidity. |
| title | On super-rigidity of Gromov's random monster group |
| topic | Group Theory Geometric Topology Probability 20F65, 20F67, 20P05, 53C24, 57K20 |
| url | https://arxiv.org/abs/2302.06255 |