Obtaining presentations from group actions without making choices

Putman, Andrew¹

¹Department of Mathematics, Rice University, 6100 Main St., Houston TX 77005, USA

Algebraic and Geometric Topology, Tome 11 (2011) no. 3, pp. 1737-1766

Résumé

Consider a group G acting nicely on a simply-connected simplicial complex X. Numerous classical methods exist for using this group action to produce a presentation for G. For the case that X∕G is 2–connected, we give a new method that has the novelty that one does not have to identify a fundamental domain for the action. Indeed, the resulting presentation is canonical in the sense that no arbitrary choices need to be made. It can be viewed as a nonabelian analogue of a simple result in the study of equivariant homology.

Détail

DOI : 10.2140/agt.2011.11.1737

Classification : 20F05, 57M07
Keywords: group presentations, group actions

Bibliographie

[1] M A Armstrong, On the fundamental group of an orbit space, Proc. Cambridge Philos. Soc. 61 (1965) 639

[2] M A Armstrong, A presentation for a group of automorphisms of a simplicial complex, Glasgow Math. J. 30 (1988) 331

[3] B Van Den Berg, On the Abelianization of the Torelli group, doctoral thesis, University of Utrecht (2003)

[4] R H Bing, Some aspects of the topology of 3–manifolds related to the Poincaré conjecture, from: "Lectures on modern mathematics, Vol II", Wiley (1964) 93

[5] A Björner, M Wachs, On lexicographically shellable posets, Trans. Amer. Math. Soc. 277 (1983) 323

[6] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften 319, Springer (1999)

[7] K S Brown, Presentations for groups acting on simply-connected complexes, J. Pure Appl. Algebra 32 (1984) 1

[8] K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1994)

[9] R E Goodrick, Non-simplicially collapsible triangulations of $I^{n}$, Proc. Cambridge Philos. Soc. 64 (1968) 31

[10] A Hatcher, W Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980) 221

[11] D Johnson, The structure of the Torelli group I: A finite set of generators for $\cal{I}$, Ann. of Math. $(2)$ 118 (1983) 423

[12] D Johnson, The structure of the Torelli group III: The abelianization of $\mathcal{T}$, Topology 24 (1985) 127

[13] H Maazen, Homology stability for the general linear group, doctoral thesis, University of Utrecht (1979)

[14] J Mccool, Some finitely presented subgroups of the automorphism group of a free group, J. Algebra 35 (1975) 205

[15] D Mccullough, A Miller, The genus 2 Torelli group is not finitely generated, Topology Appl. 22 (1986) 43

[16] G Mess, The Torelli groups for genus 2 and 3 surfaces, Topology 31 (1992) 775

[17] J Milnor, Introduction to algebraic $K$–theory, Annals of Mathematics Studies 72, Princeton University Press (1971)

[18] A Putman, Presentations for the level 2 and 3 congruence subgroups of $\mathrm{SL}_n(\mathbb{Z})$, in preparation

[19] A Putman, An infinite presentation of the Torelli group, Geom. Funct. Anal. 19 (2009) 591

[20] J P Serre, Trees, Springer (1980)

[21] E H Spanier, Algebraic topology, McGraw-Hill Book Co. (1966)