## Journal of the Mathematical Society of Japan

### Commensurators of surface braid groups

#### Abstract

We prove that if g and n are integers at least two, then the abstract commensurator of the braid group with n strands on a closed orientable surface of genus g is naturally isomorphic to the extended mapping class group of a compact orientable surface of genus g with n boundary components.

#### Article information

Source
J. Math. Soc. Japan, Volume 63, Number 4 (2011), 1391-1435.

Dates
First available in Project Euclid: 27 October 2011

https://projecteuclid.org/euclid.jmsj/1319721145

Digital Object Identifier
doi:10.2969/jmsj/06341391

Mathematical Reviews number (MathSciNet)
MR2855817

Zentralblatt MATH identifier
1245.20041

#### Citation

KIDA, Yoshikata; YAMAGATA, Saeko. Commensurators of surface braid groups. J. Math. Soc. Japan 63 (2011), no. 4, 1391--1435. doi:10.2969/jmsj/06341391. https://projecteuclid.org/euclid.jmsj/1319721145

#### References

• P. Bellingeri, On automorphisms of surface braid groups, J. Knot Theory Ramifications, 17 (2008), 1–11.
• J. S. Birman, Braids, links, and mapping class groups, Ann. of Math. Stud., 82, Princeton Univ. Press, Princeton, N.J., 1974.
• J. S. Birman, A. Lubotzky and J. McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J., 50 (1983), 1107–1120.
• T. Brendle and D. Margalit, Commensurations of the Johnson kernel, Geom. Topol., 8 (2004), 1361–1384.
• T. Brendle and D. Margalit, Addendum to: Commensurations of the Johnson kernel, Geom. Topol., 12 (2008), 97–101.
• B. Farb and N. V. Ivanov, The Torelli geometry and its applications: research announcement, Math. Res. Lett., 12 (2005), 293–301.
• A. Fathi, F. Laudenbach, V. Poénaru et al., Travaux de Thurston sur les surfaces, Séminaire Orsay, Astérisque, 66–67, Soc. Math. France, Paris, 1979.
• S. Gervais, A finite presentation of the mapping class group of a punctured surface, Topology, 40 (2001), 703–725.
• W. J. Harvey, Boundary structure of the modular group, In: Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference, Ann. of Math. Stud., 97, Princeton Univ. Press, Princeton, N.J., 1981, pp.,245–251.
• E. Irmak, Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups, Topology, 43 (2004), 513–541.
• E. Irmak, N. V. Ivanov and J. D. McCarthy, Automorphisms of surface braid groups, preprint..
• N. V. Ivanov, Subgroups of Teichmüller modular groups, Transl. Math. Monogr., 115, Amer. Math. Soc., Providence, RI, 1992.
• N. V. Ivanov, Automorphisms of complexes of curves and of Teichmüller spaces, Int. Math. Res. Not., 1997 (1997), 651–666.
• N. V. Ivanov, Mapping class groups, In: Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp.,523–633.
• Y. Kida, Automorphisms of the Torelli complex and the complex of separating curves, J. Math. Soc. Japan, 63 (2011), 363–417.
• Y. Kida and S. Yamagata, The co-Hopfian property of surface braid groups, preprint, arXiv:1006.2599.
• M. Korkmaz, Automorphisms of complexes of curves on punctured spheres and on punctured tori, Topology Appl., 95 (1999), 85–111.
• F. Luo, Automorphisms of the complex of curves, Topology, 39 (2000), 283–298.
• J. D. McCarthy and W. R. Vautaw, Automorphisms of Torelli groups, preprint..
• A. Putman, A note on the connectivity of certain complexes associated to surfaces, Enseign. Math. (2), 54 (2008), 287–301.
• W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.), 19 (1988), 417–431.
• P. Zhang, Automorphisms of braid groups on closed surfaces which are not $S^2$, $T^2$, $P^2$ or the Klein bottle, J. Knot Theory Ramifications, 15 (2006), 1231–1244.