A presentation for the baseleaf preserving mapping class group of the punctured solenoid. (English) Zbl 1181.57024

Summary: We give a presentation for the baseleaf preserving mapping class group MCG(\(\mathcal H\)) of the punctured solenoid \(\mathcal H\). The generators for our presentation were introduced previously, and several relations among them were derived. In addition, we show that MCG(\(\mathcal H)\) has no non-trivial central elements. Our main tool is a new complex of triangulations of the disk upon which MCG(\(\mathcal H\)) acts.


57M99 General low-dimensional topology
20F65 Geometric group theory
Full Text: DOI arXiv


[1] M A Armstrong, On the fundamental group of an orbit space, Proc. Cambridge Philos. Soc. 61 (1965) 639 · Zbl 0125.40002
[2] K S Brown, Presentations for groups acting on simply-connected complexes, J. Pure Appl. Algebra 32 (1984) 1 · Zbl 0545.20022
[3] A Douady, C J Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986) 23 · Zbl 0615.30005
[4] D B A Epstein, R C Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988) 67 · Zbl 0611.53036
[5] J L Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986) 157 · Zbl 0592.57009
[6] A Hatcher, On triangulations of surfaces, Topology Appl. 40 (1991) 189 · Zbl 0727.57012
[7] A Hatcher, W Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980) 221 · Zbl 0447.57005
[8] N V Ivanov, Automorphisms of complexes of curves and of Teichmüller spaces, Travaux en Cours 56, Hermann (1997) 113 · Zbl 0941.30027
[9] S Nag, D Sullivan, Teichmüller theory and the universal period mapping via quantum calculus and the \(H^{1/2}\) space on the circle, Osaka J. Math. 32 (1995) 1 · Zbl 0820.30027
[10] C Odden, Virtual automorphism group of the fundamental group of a closed surface, PhD thesis, Duke University, Durham (1997)
[11] C Odden, The baseleaf preserving mapping class group of the universal hyperbolic solenoid, Trans. Amer. Math. Soc. 357 (2005) 1829 · Zbl 1077.57017
[12] R C Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987) 299 · Zbl 0642.32012
[13] R C Penner, Universal constructions in Teichmüller theory, Adv. Math. 98 (1993) 143 · Zbl 0772.30040
[14] R C Penner, D , Teichmüller theory of the punctured solenoid · Zbl 1182.30076
[15] D Sullivan, Linking the universalities of Milnor-Thurston, Feigenbaum and Ahlfors-Bers, Publish or Perish (1993) 543 · Zbl 0803.58018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.