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On the virtually cyclic dimension of mapping class groups of punctured spheres

Javier Aramayona, Daniel Juan-Pineda and Alejandra Trujillo-Negrete

Algebraic & Geometric Topology 18 (2018) 2471–2495
Abstract

We calculate the virtually cyclic dimension of the mapping class group of a sphere with at most six punctures. As an immediate consequence, we obtain the virtually cyclic dimension of the mapping class group of the twice-holed torus and of the closed genus-two surface.

For spheres with an arbitrary number of punctures, we give a new upper bound for the virtually cyclic dimension of their mapping class group, improving the recent bound of Degrijse and Petrosyan (2015).

Keywords
geometric dimension, mapping class group, vc-dimension
Mathematical Subject Classification 2010
Primary: 20F65, 55R35
Secondary: 20F36
References
Publication
Received: 22 August 2017
Revised: 4 January 2018
Accepted: 12 January 2018
Published: 26 April 2018
Authors
Javier Aramayona
ICMAT and Departamento de Matemáticas
Universidad Autónoma de Madrid
Ciudad Universitaria de Cantoblanco
Madrid
Spain
Daniel Juan-Pineda
Centro de Ciencias Matemáticas
Universidad Nacional Autónoma de México
Morelia
Mexico
Alejandra Trujillo-Negrete
Centro de Ciencias Matemáticas
Universidad Nacional Autónoma de México
Morelia
Mexico