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Register a new userFiniteness of arithmetic hyperbolic reflection groups
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We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.
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We show that a relatively hyperbolic group quasi-isometrically embeds in a product of finitely many trees if the peripheral subgroups do, and we provide an estimate on the minimal number of trees needed. Applying our result to the case of 3-manifolds, we show that fundamental groups of closed 3-manifolds have linearly controlled asymptotic dimension at most 8. To complement this result, we observe that fundamental groups of Haken 3-manifolds with non-empty boundary have asymptotic dimension 2.
We address the question of determining which mapping class groups of infinite-type surfaces admit nonelementary continuous actions on hyperbolic spaces. More precisely, let $Sigma$ be a connected, orientable surface of infinite type with tame endspace whose mapping class group is generated by a coarsely bounded subset. We prove that $mathrm{Map}(Sigma)$ admits a continuous nonelementary action on a hyperbolic space if and only if $Sigma$ contains a finite-type subsurface which intersects all its homeomorphic translates. When $Sigma$ contains such a nondisplaceable subsurface $K$ of finite type, the hyperbolic space we build is constructed from the curve graphs of $K$ and its homeomorphic translates via a construction of Bestvina, Bromberg and Fujiwara. Our construction has several applications: first, the second bounded cohomology of $mathrm{Map}(Sigma)$ contains an embedded $ell^1$; second, using work of Dahmani, Guirardel and Osin, we deduce that $mathrm{Map}(Sigma)$ contains nontrivial normal free subgroups (while it does not if $Sigma$ has no nondisplaceable subsurface of finite type), has uncountably many quotients and is SQ-universal.
In the 1990s, J.H. Conway published a combinatorial-geometric method for analyzing integer-valued binary quadratic forms (BQFs). Using a visualization he named the topograph, Conway revisited the reduction of BQFs and the solution of quadratic Diophantine equations such as Pells equation. It appears that the crux of his method is the coincidence between the arithmetic group $PGL_2({mathbb Z})$ and the Coxeter group of type $(3,infty)$. There are many arithmetic Coxeter groups, and each may have unforeseen applications to arithmetic. We introduce Conways topograph, and generalizations to other arithmetic Coxeter groups. This includes a study of arithmetic flags and variants of binary quadratic forms.
We prove that any arithmetic hyperbolic $n$-manifold of simplest type can either be geodesically embedded into an arithmetic hyperbolic $(n+1)$-manifold or its universal $mathrm{mod}~2$ Abelian cover can.
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