Power quotients of surface groups and mapping class groups

2025/07/18 by Coulon, Rémi, Sisto, Alessandro, Wilton, Henry
#FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2507.13701

Abstract

Let Γ be the fundamental group of a closed, orientable, hyperbolic surface S. The n-power quotient, Γ(n), is the quotient of Γ by the nth powers of simple closed curves. We prove an analogue of the Dehn--Nielsen--Baer theorem for suitable large values of n: the outer automorphism group of Γ(n) is isomorphic to the quotient of the extended mapping class group of S by nth powers of Dehn twists. There is also a corresponding description of the automorphism group as the quotient of the extended mapping class group of the corresponding once-punctured surface, and we relate these groups via a Birman-type exact sequence. Along the way, and as consequences, we prove structural properties of Γ(n) for suitable large values of n, including: Γ(n) is virtually torsion-free, acylindrically hyperbolic, infinitely presented, with solvable word problem and finite asymptotic dimension.

Related

• Word hyperbolic extensions of surface groups
• Cubulating Surface-by-free Groups
• Groupoid-theoretical methods in the mapping class groups of surfaces
• Rigidity of mapping class groups mod powers of twists
• A Presentation of the Mapping Class Groups