论文标题
矩阵组在具有可能的抛物线异构体和唯一弧形连接空间的CAT（0）空间上的矩阵组动作的刚度
Rigidity of matrix group actions on CAT(0) spaces with possible parabolic isometries and uniquely arcwise connected spaces
论文作者
论文摘要
众所周知，$ \ mathrm {sl} _ {n}（\ mathbf {q} _ {p}）$在$（n-1）$ - 尺寸$ \ mathrm {cat}}（cat}}（0）$ Space（Affine Building）上无固定点。我们证明，$ n-1 $是$ \ mathrm {cat}（0）$空间的最小维度，矩阵组在没有固定点的情况下进行。明确地，让$ r $为具有身份的关联环，$ e_ {n}^{\ prime}（r）$ extended基本子组。 $ e_ {n}^{\ prime}（r）$的任何等距操作在完整的$ \ mathrm {cat（0）} $ d <n-1 $的$ x^{d} $具有固定点。讨论了自由组的自身形态群体的类似结果。此外，我们证明，$ \ mathrm {aut}（f_ {n}）的任何动作，n \ geq 3，同构的独特弧形连接空间上的$都具有固定点。
It is well-known that $\mathrm{SL}_{n}(\mathbf{Q}_{p})$ acts without fixed points on an $(n-1)$-dimensional $\mathrm{CAT}(0)$ space (the affine building). We prove that $n-1$ is the smallest dimension of $\mathrm{CAT}(0)$ spaces on which matrix groups act without fixed points. Explicitly, let $R$ be an associative ring with identity and $E_{n}^{\prime }(R)$ the extended elementary subgroup. Any isometric action of $E_{n}^{\prime }(R)$ on a complete $\mathrm{CAT(0)}$ space $X^{d}$ of dimension $d<n-1$ has a fixed point. Similar results are discussed for automorphism groups of free groups. Furthermore, we prove that any action of $\mathrm{Aut}(F_{n}),n\geq 3,$ on a uniquely arcwise connected space by homeomorphisms has a fixed point.
