## Algebraic & Geometric Topology

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

#### 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).

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 4 (2018), 2471-2495.

Dates
Revised: 4 January 2018
Accepted: 12 January 2018
First available in Project Euclid: 3 May 2018

https://projecteuclid.org/euclid.agt/1525312835

Digital Object Identifier
doi:10.2140/agt.2018.18.2471

Mathematical Reviews number (MathSciNet)
MR3797073

Zentralblatt MATH identifier
06867664

#### Citation

Aramayona, Javier; Juan-Pineda, Daniel; Trujillo-Negrete, Alejandra. On the virtually cyclic dimension of mapping class groups of punctured spheres. Algebr. Geom. Topol. 18 (2018), no. 4, 2471--2495. doi:10.2140/agt.2018.18.2471. https://projecteuclid.org/euclid.agt/1525312835

#### References

• J Aramayona, C Martínez-Pérez, The proper geometric dimension of the mapping class group, Algebr. Geom. Topol. 14 (2014) 217–227
• J S Birman, Braids, links, and mapping class groups, Annals of Mathematics Studies 82, Princeton Univ. Press (1974)
• C Bonatti, L Paris, Roots in the mapping class groups, Proc. Lond. Math. Soc. 98 (2009) 471–503
• P J Cameron, R Solomon, A Turull, Chains of subgroups in symmetric groups, J. Algebra 127 (1989) 340–352
• D Degrijse, N Petrosyan, Geometric dimension of groups for the family of virtually cyclic subgroups, J. Topol. 7 (2014) 697–726
• D Degrijse, N Petrosyan, Bredon cohomological dimensions for groups acting on $\rm CAT(0)$–spaces, Groups Geom. Dyn. 9 (2015) 1231–1265
• B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton Univ. Press (2012)
• R Flores, J González-Meneses, Classifying spaces for the family of virtually cyclic subgroups of braid groups, preprint (2016)
• M G Fluch, B E A Nucinkis, On the classifying space for the family of virtually cyclic subgroups for elementary amenable groups, Proc. Amer. Math. Soc. 141 (2013) 3755–3769
• J L Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986) 157–176
• S Hensel, D Osajda, P Przytycki, Realisation and dismantlability, Geom. Topol. 18 (2014) 2079–2126
• D Juan-Pineda, I J Leary, On classifying spaces for the family of virtually cyclic subgroups, from “Recent developments in algebraic topology” (A Ádem, J González, G Pastor, editors), Contemp. Math. 407, Amer. Math. Soc., Providence, RI (2006) 135–145
• D Juan-Pineda, A Trujillo-Negrete, On classifying spaces for the family of virtually cyclic subgroups in mapping class groups (2016) To appear in Pure Appl. Math. Q.
• A Karrass, A Pietrowski, D Solitar, Finite and infinite cyclic extensions of free groups, J. Austral. Math. Soc. 16 (1973) 458–466
• W Lück, The type of the classifying space for a family of subgroups, J. Pure Appl. Algebra 149 (2000) 177–203
• W Lück, Survey on classifying spaces for families of subgroups, from “Infinite groups: geometric, combinatorial and dynamical aspects” (L Bartholdi, T Ceccherini-Silberstein, T Smirnova-Nagnibeda, A Zuk, editors), Progr. Math. 248, Birkhäuser, Basel (2005) 269–322
• W Lück, On the classifying space of the family of virtually cyclic subgroups for $\rm CAT(0)$–groups, Münster J. Math. 2 (2009) 201–214
• W Lück, M Weiermann, On the classifying space of the family of virtually cyclic subgroups, Pure Appl. Math. Q. 8 (2012) 497–555
• C Martínez-Pérez, A bound for the Bredon cohomological dimension, J. Group Theory 10 (2007) 731–747
• J D McCarthy, Normalizers and centralizers of pseudo-Anosov mapping classes, PhD thesis, Columbia University (1982)
• S A Wolpert, Geometry of the Weil–Petersson completion of Teichmüller space, from “Surveys in differential geometry, VIII” (S-T Yau, editor), International, Somerville, MA (2003) 357–393