Extensions of finitely generated Veech groups

Bongiovanni, Eliot

doi:10.2140/agt.2026.26.989

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Extensions of finitely generated Veech groups

Eliot Bongiovanni

Algebraic & Geometric Topology 26 (2026) 989–1035

DOI: 10.2140/agt.2026.26.989

Abstract

Given a closed surface $S$ with finitely generated Veech group $G$ and its $π_{1} (S)$ -extension $Γ$ , there exists a hyperbolic space $\hat{E}$ on which $Γ$ acts isometrically and cocompactly. The space $\hat{E}$ is obtained by collapsing some regions of the surface bundle over the convex hull of the limit set of $G$ . Using the nice action of $Γ$ on the hyperbolic space $\hat{E}$ , it is shown that $Γ$ is hierarchically hyperbolic. These are generalizations of Dowdall–Durham–Leininger–Sisto, who assume in addition that $G$ is a lattice. Because finitely generated Veech groups are among the most basic examples of subgroups of mapping class groups which are expected to qualify as geometrically finite, this result is evidence for the development of a broader theory of geometric finiteness.

Keywords

Fuchsian group, Veech group, Veech dichotomy, mapping class group, hierarchical hyperbolicity, geometric finiteness, hyperbolicity, guessing geodesics, preferred paths, surface bundles

Mathematical Subject Classification

Primary: 20F65, 20F67, 57M60

Secondary: 30F40, 30F60

References

Publication

Received: 9 July 2024

Revised: 15 January 2025

Accepted: 9 February 2025

Published: 1 April 2026

Authors

Eliot Bongiovanni
Department of Mathematics University of Michigan Ann Arbor, MI United States

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Volume 26, issue 3 (2026)