We study fixed point properties of the automorphism group of the universal Coxeter group Aut$\left(W_n\right)$. In particular, we prove that whenever Aut$\left(W_n\right)$ acts by isometries on complete $d$-dimensional CAT$\left(0\right)$ space with $d<\lfloor \frac{n}{2}\rfloor$, then it must fix a point. We also prove that Aut$\left(W_n\right)$ does not have Kazhdan’s property (T). Further, strong restrictions are obtained on homomorphisms of Aut$\left(W_n\right)$ to groups that do not contain a copy of $\mathrm{Sym}\left(n\right)$.

中文翻译：

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通用Coxeter群的自同构群

我们研究了通用Coxeter群Aut的自同构群的不动点性质$（{\mathrm{w\; ^}}_{ñ}）$。特别是，我们证明了$（{\mathrm{w\; ^}}_{ñ}）$ 完全按等距作用 $d$尺寸CAT$（0）$ 与空间 $d<\lfloor \frac{ñ}{\mathrm{2个}}\rfloor$，那么它必须定点。我们还证明了$（{\mathrm{w\; ^}}_{ñ}）$ 没有Kazhdan的财产（T）。此外，对Aut的同构性有严格的限制$（{\mathrm{w\; ^}}_{ñ}）$ 到不包含的副本的组 $\mathrm{象征}（ñ）$。