分层双曲性的组合观点及其在映射类群商中的应用

Pub Date : 2024-08-10 DOI:10.1112/topo.12351
Jason Behrstock, Mark Hagen, Alexandre Martin, Alessandro Sisto
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引用次数: 0

摘要

我们根据双曲简复上的作用给出了一个简单的组合标准,即一个群是层次双曲的。我们应用这一标准来证明,由 Dehn 扭矩的大幂构成的映射类群的商是层次双曲的(甚至在属 2 的情况下是相对双曲的)。在至少三属中,没有已知的映射类群的无限双曲商。然而,利用我们构建的层次双曲商,我们证明,在残余有限性假设下,映射类群有许多非元素双曲商。利用这些商,我们将里德和布里奇森-里德-维尔顿关于映射类群有限商的问题与特定双曲群的剩余有限性联系起来。
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A combinatorial take on hierarchical hyperbolicity and applications to quotients of mapping class groups

We give a simple combinatorial criterion, in terms of an action on a hyperbolic simplicial complex, for a group to be hierarchically hyperbolic. We apply this to show that quotients of mapping class groups by large powers of Dehn twists are hierarchically hyperbolic (and even relatively hyperbolic in the genus 2 case). In genus at least three, there are no known infinite hyperbolic quotients of mapping class groups. However, using the hierarchically hyperbolic quotients we construct, we show, under a residual finiteness assumption, that mapping class groups have many nonelementary hyperbolic quotients. Using these quotients, we relate questions of Reid and Bridson–Reid–Wilton about finite quotients of mapping class groups to residual finiteness of specific hyperbolic groups.

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