A FINITE PRESENTATION FOR THE TWIST SUBGROUP OF THE MAPPING CLASS GROUP OF A NONORIENTABLE SURFACE

Title & Authors
A FINITE PRESENTATION FOR THE TWIST SUBGROUP OF THE MAPPING CLASS GROUP OF A NONORIENTABLE SURFACE
Stukow, Michal;

Abstract
Let $\small{N_{g,s}}$ denote the nonorientable surface of genus g with s boundary components. Recently Paris and Szepietowski [12] obtained an explicit finite presentation for the mapping class group $\small{\mathcal{M}(N_{g,s})}$ of the surface $\small{N_{g,s}}$, where $\small{s{\in}\{0,1\}}$ and g + s > 3. Following this work, we obtain a finite presentation for the subgroup $\small{\mathcal{T}(N_{g,s})}$ of $\small{\mathcal{M}(N_{g,s})}$ generated by Dehn twists.
Keywords
mapping class group;nonorientable surface;twist subgroup;presentation;
Language
English
Cited by
References
1.
J. S. Birman, Automorphisms of the fundamental group of a closed, orientable 2-manifold, Proc. Amer. Math. Soc. 21 (1969), no. 2, 351-354.

2.
D. R. J. Chillingworth, A finite set of generators for the homeotopy group of a non-orientable surface, Math. Proc. Cambridge Philos. Soc. 65 (1969), 409-430.

3.
S. Gervais, A finite presentation of the mapping class group of a punctured surface, Topology 40 (2001), no. 4, 703-725.

4.
S. P. Humphries, Generators for the mapping class group, In Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), pp. 44-47, Lecture Notes in Math., 722, Springer, Berlin, 1979.

5.
D. L. Johnson, Homeomorphisms of a surface which act trivially on homology, Proc. Amer. Math. Soc. 75 (1979), no. 1, 119-125.

6.
M. Korkmaz, Mapping class groups of nonorientable surfaces, Geom. Dedicata 89 (2002), 109-133.

7.
C. Labruere and L. Paris, Presentations for the punctured mapping class groups in terms of Artin groups, Algebr. Geom. Topol. 1 (2001), 73-114.

8.
W. B. R. Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. of Math. 76 (1962), 531-540.

9.
W. B. R. Lickorish, Homeomorphisms of non-orientable two-manifolds, Math. Proc. Cambridge Philos. Soc. 59 (1963), 307-317.

10.
W. B. R. Lickorish, A finite set of generators for the homeotopy group of a 2-manifold, Math. Proc. Cambridge Philos. Soc. 60 (1964), 769-778.

11.
M. Matsumoto, A presentation of mapping class groups in terms of Artin groups and geometric monodromy of singularities, Math. Ann. 316 (2000), no. 3, 401-418.

12.
L. Paris and B. Szepietowski, A presentation for the mapping class group of a nonori-entable surface, arXiv:1308.5856v1 [math.GT], 2013.

13.
M. Stukow, The twist subgroup of the mapping class group of a nonorientable surface, Osaka J. Math. 46 (2009), no. 3, 717-738.

14.
M. Stukow, Generating mapping class groups of nonorientable surfaces with boundary, Adv. Geom. 10 (2010), no. 2, 249-273.

15.
M. Stukow, A finite presentation for the mapping class group of a nonorientable surface with Dehn twists and one crosscap slide as generators, J. Pure Appl. Algebra 218 (2014), no. 12, 2226-2239.

16.
B. Wajnryb, A simple presentation for the mapping class group of an orientable surface, Israel J. Math. 45 (1983), no. 2-3, 157-174.