Conditions on the monodromy for a surface group extension to be CAT(0)

2022/01/26 by Kejia Zhu, Zhu, Kejia
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2201.11010

openalex publication_date 2022/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In order to determine when surface-by-surface bundles are non-positively curved, Llosa Isenrich and Py give a necessary condition: given a surface-by-surface group G with infinite monodromy, if G is CAT(0) then the monodromy representation is injective. We extend this to a more general result: Let G be a group with a normal surface subgroup R. Assume G/R satisfies the property that for every infinite normal subgroup Λ of G/R, there is an infinite subgroup Λ₀<Λ so that the centralizer C_G/R(Λ₀) is finite. If G is CAT(0) with infinite monodromy, then the monodromy representation has a finite kernel. We prove that acylindrically hyperbolic groups satisfy this property.

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