Algebraic & Geometric Topology

The proper geometric dimension of the mapping class group

Javier Aramayona and Conchita Martínez-Pérez

Full-text: Open access

Abstract

We show that the mapping class group of a closed surface admits a cocompact classifying space for proper actions of dimension equal to its virtual cohomological dimension.

Article information

Source
Algebr. Geom. Topol., Volume 14, Number 1 (2014), 217-227.

Dates
Received: 20 February 2013
Accepted: 22 July 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715799

Digital Object Identifier
doi:10.2140/agt.2014.14.217

Mathematical Reviews number (MathSciNet)
MR3158758

Zentralblatt MATH identifier
1354.20025

Subjects
Primary: 20F34: Fundamental groups and their automorphisms [See also 57M05, 57Sxx] 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 20J05: Homological methods in group theory

Keywords
mapping class groups classifying space for proper actions cohomological dimension

Citation

Aramayona, Javier; Martínez-Pérez, Conchita. The proper geometric dimension of the mapping class group. Algebr. Geom. Topol. 14 (2014), no. 1, 217--227. doi:10.2140/agt.2014.14.217. https://projecteuclid.org/euclid.agt/1513715799


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References

  • J S Birman, H M Hilden, On isotopies of homeomorphisms of Riemann surfaces, Ann. of Math. 97 (1973) 424–439
  • N Brady, I J Leary, B E A Nucinkis, On algebraic and geometric dimensions for groups with torsion, J. London Math. Soc. 64 (2001) 489–500
  • M R Bridson, K Vogtmann, Automorphism groups of free groups, surface groups and free abelian groups, from: “Problems on mapping class groups and related topics”, (B Farb, editor), Proc. Sympos. Pure Math. 74, Amer. Math. Soc. (2006) 301–316
  • S A Broughton, Normalizers and centralizers of elementary abelian subgroups of the mapping class group, from: “Topology '90”, (B Apanasov, W D Neumann, A W Reid, L Siebenmann, editors), Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter, Berlin (1992) 77–89
  • K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer, New York (1982)
  • B Farb, D Margalit, A primer on mapping class groups, Princeton Mathematical Series 49, Princeton Univ. Press (2012)
  • 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 http://msp.org/idx/arx/1205.0513arXiv 1205.0513
  • L Ji, Well-rounded equivariant deformation retracts of Teichmüller spaces
  • L Ji, S A Wolpert, A cofinite universal space for proper actions for mapping class groups, from: “In the tradition of Ahlfors–Bers, V”, (M Bonk, J Gilman, H Masur, Y Minsky, M Wolf, editors), Contemp. Math. 510, Amer. Math. Soc. (2010) 151–163
  • S P Kerckhoff, The Nielsen realization problem, Ann. of Math. 117 (1983) 235–265
  • I J Leary, B E A Nucinkis, Some groups of type $VF$, Invent. Math. 151 (2003) 135–165
  • W Lück, Transformation groups and algebraic $K$–theory, Lecture Notes in Mathematics 1408, Springer, Berlin (1989)
  • 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
  • J Maher, Random walks on the mapping class group, Duke Math. J. 156 (2011) 429–468
  • C Martínez-Pérez, Subgroup posets, Bredon cohomology and equivariant Euler characteristics, Trans. Amer. Math. Soc. 365 (2013) 4351–4370
  • G Mislin, Classifying spaces for proper actions of mapping class groups, Münster J. Math. 3 (2010) 263–272
  • R C Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987) 299–339