Hyperbolic Groups Which Fiber in Infinitely Many Ways

高维双曲Coxeter群的虚拟纤维化

High dimensional hyperbolic Coxeter groups that virtually fiber

本文提出一种迭代程序构造在$\mathbb{Z}$上虚拟纤维化的双曲Coxeter群，结合前人结果与单纯加厚过程，在每个vcd $n\geq 2$中生成无限多同构类，并证明每次迭代vcd恰好增加一，覆盖所有vcd。

Words, Hausdorff dimension and randomly free groups

We study fibers of word maps in finite, profinite, and residually finite groups. Our main result is that, for any word w in the free group on d generators, there exists $ε> 0$ such that if G is a residually finite group with infinitely many non-isomorphic non-abelian upper composition factors, then all fibers of the word map $w\colon G^d \to G$ have Hausdorff dimension at most $d-ε$. We conclude that the profinite completion of a group G as above satisfies no probabilistic identity. It is therefore randomly free; namely, for any d > 0, the probability that d randomly chosen elements freely gen…