论文标题
纤维产品和非准静电嵌入的cartan预测
Cartan projections of fiber products and non quasi-isometric embeddings
论文作者
论文摘要
令$γ$为有限生成的组，$ n $为$γ$的普通亚组。相对于$ n $，$γ$的纤维产物是$γ\times_nγ= \ {（γ，γw）：γ\ inγ，w \ in n \} $ in n \} $ in n \} $。对于每个表示形式，$ρ：γ\times_nγ\ rightarrow \ mathsf {gl} _d（k）$，其中$ k $是本地字段，我们在$γ$上固定单词长度函数的cartan投影的范围建立了$ρ$ $ρ$的上限。作为应用程序，我们展示了有限生成且有限呈现的纤维产品的示例$ p =γ\times_nγ$，其中$γ$是线性和gromov双曲线的，因此$ p $不接收是准杂种嵌入的线性表示。
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.
