Cartan projections of fiber products and non quasi-isometric embeddings
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913454326022144 |
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| author | Tsouvalas, Konstantinos |
| author_facet | Tsouvalas, Konstantinos |
| contents | Let $Γ$ be a finitely generated group and $N$ be a normal subgroup of $Γ$. The fiber product of $Γ$ with respect to $N$ is the subgroup $Γ\times_N Γ=\{(γ, γw): γ\in Γ, w \in N\}$ of the direct product $Γ\times Γ$. For every representation $ρ:Γ\times_N Γ\rightarrow \mathsf{GL}_d(k)$, where $k$ is a local field, we establish upper bounds for the norm of the Cartan projection of $ρ$ in terms of a fixed word length function on $Γ$. As an application, we exhibit examples of finitely generated and finitely presented fiber products $P=Γ\times_N Γ$, where $Γ$ is linear and Gromov hyperbolic, such that $P$ does not admit linear representations which are quasi-isometric embeddings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_11960 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Cartan projections of fiber products and non quasi-isometric embeddings Tsouvalas, Konstantinos Group Theory Let $Γ$ be a finitely generated group and $N$ be a normal subgroup of $Γ$. The fiber product of $Γ$ with respect to $N$ is the subgroup $Γ\times_N Γ=\{(γ, γw): γ\in Γ, w \in N\}$ of the direct product $Γ\times Γ$. For every representation $ρ:Γ\times_N Γ\rightarrow \mathsf{GL}_d(k)$, where $k$ is a local field, we establish upper bounds for the norm of the Cartan projection of $ρ$ in terms of a fixed word length function on $Γ$. As an application, we exhibit examples of finitely generated and finitely presented fiber products $P=Γ\times_N Γ$, where $Γ$ is linear and Gromov hyperbolic, such that $P$ does not admit linear representations which are quasi-isometric embeddings. |
| title | Cartan projections of fiber products and non quasi-isometric embeddings |
| topic | Group Theory |
| url | https://arxiv.org/abs/2205.11960 |