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An infinite family of braid group representations in automorphism groups of free groups

Wonjun Chang

Department of Mathematics, Inha University, Incheon 22212, Republic of Korea

E-mail Address: wonjun5@gmail.com

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Byung Chun Kim

National Institute for Mathematical Sciences, Daejeon 34047, Republic of Korea

E-mail Address: wizardbc@gmail.com

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, and

Yongjin Song

Department of Mathematics, Inha University, Incheon 22212, Republic of Korea

E-mail Address: yjsong@inha.ac.kr

Corresponding author.

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https://doi.org/10.1142/S0218216520420079Cited by:1 (Source: Crossref)

This article is part of the issue:

Special Issue: Proceedings of 6th Russian — Chinese Conference on Knot Theory and Related Topics (Part 1)
Guest Editors: N. Abrosimov, Z. Cheng, L. H. Kauffman, A. Vesnin and Z. Yang

Abstract

The $d$ -fold ( $d \geq 3$ ) branched coverings on a disk give an infinite family of nongeometric embeddings of braid groups into mapping class groups. We, in this paper, give new explicit expressions of these braid group representations into automorphism groups of free groups in terms of the actions on the generators of free groups. We also give a systematic way of constructing and expressing these braid group representations in terms of a new gadget, called covering groupoid. We prove that each generator ${\tilde{β}}_{i}$ of braid group inside mapping class group induced by $d$ -fold covering is the product of $d - 1$ Dehn twists on the surface.

Keywords:

AMSC: 57M12, 57M50

References

1. J. Birman and H. Hilden, Erratum to isotopies of homeomorphisms of riemann surfaces, Ann. Math. 185 (2017) 345–345. Crossref, Web of Science, Google Scholar
2. J. S. Birman and H. M. Hilden, On isotopies of homeomorphisms of riemann surfaces, The Ann. Math. 97 (1973) 424. Crossref, Web of Science, Google Scholar
3. F. Callegaro and M. Salvetti, Families of superelliptic curves, complex braid groups and generalized Dehn twists, arXiv:1805.11968. Google Scholar
4. J. Crisp and L. Paris, Representations of the braid group by automorphisms of groups, invariants of links, and garside groups, Pacific J. Math. 221 (2005) 1–27. Crossref, Web of Science, Google Scholar
5. T. Ghaswala and A. McLeay, Mapping class groups of covers with boundary and braid group embeddings, Algeb. Geom. Topol. 20 (2020) 239–278. Crossref, Web of Science, Google Scholar
6. C.-S. Jeong and Y. Song, The pillar switchings of mapping class groups of surfaces, Int. J. Math. 24 (2013) 1350103. Link, Web of Science, Google Scholar
7. B. C. Kim and Y. Song, Configuration spaces, moduli spaces and 3-fold covering spaces, Manuscripta Math. 160 (2018) 391–409. Crossref, Web of Science, Google Scholar
8. K. H. Ko, S. J. Lee, J. H. Cheon, J. W. Han, J. Sung Kang and C. Park, New public-key cryptosystem using braid groups, in Advances in Cryptology — CRYPTO 2000 (Springer, Berlin Heidelberg, 2000), pp. 166–183. Crossref, Google Scholar
9. D. Margalit and R. R. Winarski, The Birman–Hilden theory, arXiv:1703.03448. Google Scholar
10. G. Segal and U. Tillmann, Mapping configuration spaces to moduli spaces, in Groups of Diffeomorphisms: In Honor of Shigeyuki Morita on the Occasion of his 60th Birthday (Mathematical Society of Japan, 2008). Crossref, Google Scholar
11. B. Szepietowski, Embedding the braid group in mapping class groups, Public. Matemàtiques 54 (2010) 359–368. Crossref, Web of Science, Google Scholar
12. M. Wada, Group invariants of links, Topology 31 (1992) 399–406. Crossref, Web of Science, Google Scholar
13. B. Wajnryb, Relations in the mapping class group, in Problems on Mapping Class Groups and Related Topics, ed. B. Farb (American Mathematical Society, 2006), pp. 115–120. Crossref, Google Scholar