paper

Short hierarchically hyperbolic groups II: quotients and the Hopf property for Artin groups

arXiv:2412.04364 · doi:10.1016/j.aim.2025.110736

Abstract

We prove that most Artin groups of large and hyperbolic type are Hopfian, meaning that every self-epimorphism is an isomorphism. The class covered by our result is generic, in the sense of Goldsborough-Vaskou. Moreover, assuming the residual finiteness of certain hyperbolic groups with an explicit presentation, we get that all large and hyperbolic type Artin groups are residually finite. We also show that most quotients of the five-holed sphere mapping class group are hierarchically hyperbolic, up to taking powers of the normal generators of the kernels. The main tool we use to prove both results is a Dehn-filling-like procedure for short hierarchically hyperbolic groups (these also include e.g. non-geometric 3-manifolds, and triangle- and square-free RAAGs).

arXiv commentV4: Expanded list of residually finite Artin groups in Remark 6.19. Also, some of the Artin groups from Figure 11 were known to be residually finite (thanks to Qiang Zhang for pointing this out), so we replaced them to make sure all known tools don't apply. V3: Improved exposition. V2: Careful proofs of lemmas involving relative hyperbolicity. 61 pages, 11 figures. All comments are welcome!

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