No Access

Some remarks on twin groups

Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, S. A. S. Nagar, P. O. Manauli, Punjab 140306, India

E-mail Address: tushar@iisermohali.ac.in

Search for more papers by this author

Neha Nanda

Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, S. A. S. Nagar, P. O. Manauli, Punjab 140306, India

E-mail Address: nehananda@iisermohali.ac.in

Search for more papers by this author

, and

Mahender Singh

Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, S. A. S. Nagar, P. O. Manauli, Punjab 140306, India

E-mail Address: mahender@iisermohali.ac.in

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0218216520420067Cited by:5 (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 twin group $T_{n}$ is a right angled Coxeter group generated by $n - 1$ involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this paper, we study some properties of twin groups whose analogues are well known for Artin braid groups. We give an algorithm for two twins to be equivalent under individual Markov moves. Further, we show that twin groups $T_{n}$ have $R_{\infty}$ -property and are not co-Hopfian for $n \geq 3$ .

Keywords:

AMSC: 20F55, 57M27, 20E45

References

1. V. G. Bardakov, M. V. Neshchadim and M. Singh, Automorphisms of pure braid groups, Monatsh. Math. 187(1) (2018) 1–19. Crossref, Web of Science, Google Scholar
2. V. Bardakov, M. Singh and A. Vesnin, Structural aspects of twin and pure twin groups, Geom. Dedicata 203 (2019) 135–154. Crossref, Web of Science, Google Scholar
3. A. Bartholomew, R. Fenn, N. Kamada and S. Kamada, Doodles on surfaces, J. Knot Theory Ramification 27(12) (2018) 1850071, 26 pp. Link, Web of Science, Google Scholar
4. A. Bartholomew, R. Fenn, N. Kamada and S. Kamada, Colorings and doubled colorings of virtual doodles, Topol. Appl. 264 (2019) 290–299. Crossref, Web of Science, Google Scholar
5. R. W. Bell and D. Margalit, Braid groups and the co-Hopfian property, J. Algebra 303(1) (2006) 275–294. Crossref, Web of Science, Google Scholar
6. K. S. Brown, Buildings (Springer-Verlag, New York, 1989), viii+215 pp. Crossref, Google Scholar
7. B. Cisneros, M. Flores, J. Juyumaya and C. Roque-Márquez, An Alexander type invariant for doodles, preprint (2020), arXiv:2005.06290. Google Scholar
8. C. G. Cox, Twisted conjugacy in Houghton’s groups, J. Algebra 490 (2017) 390–436. Crossref, Web of Science, Google Scholar
9. K. Dekimpe and D. Gonçalves, $R_{\infty}$ property for free groups, free nilpotent groups and free solvable groups, Bull. Lond. Math. Soc. 46(4) (2014) 737–746. Crossref, Web of Science, Google Scholar
10. A. Fel’shtyn and D. L. Gonçalves, Twisted conjugacy classes in symplectic groups, mapping class groups and braid groups, Geom. Dedicata 146 (2010) 211–223. Crossref, Web of Science, Google Scholar
11. A. Fel’shtyn, Y. Leonov and E. Troitsky, Twisted conjugacy classes in saturated weakly branch groups, Geom. Dedicata 134 (2008) 61–73. Crossref, Web of Science, Google Scholar
12. A. Fel’shtyn and T. Nasybullov, The $R_{\infty}$ and $S_{\infty}$ properties for linear algebraic groups, J. Group Theory 19(5) (2016) 901–921. Crossref, Web of Science, Google Scholar
13. R. Fenn and P. Taylor, Introducing doodles, Topology of Low-Dimensional Manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), Lecture Notes in Mathematics, Vol. 722 (Springer, Berlin, 1979), pp. 37–43. Crossref, Google Scholar
14. F. A. Garside, The braid groups and other groups, Quart. J. Math. 20 (1969) 235–254. Crossref, Web of Science, Google Scholar
15. D. L. Gonçalves and T. Nasybullov, On groups where the twisted conjugacy class of the unit element is a subgroup, Comm. Algebra 47(3) (2019) 930–944. Crossref, Web of Science, Google Scholar
16. D. Gonçalves and P. Sankaran, Sigma theory and twisted conjugacy, II: Houghton groups and pure symmetric automorphism groups, Pacific J. Math. 280(2) (2016) 349–369. Crossref, Web of Science, Google Scholar
17. D. L. Gonçalves and P. Sankaran, Twisted conjugacy in PL-homeomorphism groups of the circle, Geom. Dedicata 202 (2019) 311–320. Crossref, Web of Science, Google Scholar
18. J. González, J. L. León-Medina and C. Roque, Linear motion planning with controlled collisions and pure planar braids, preprint (2019), arXiv:1902.06190v2. Google Scholar
19. K. Gotin, Markov theorem for doodles on two-sphere, preprint (2018), arXiv:1807.05337. Google Scholar
20. S. P. Humphries, On reducible braids and composite braids, Glas. Math. J. 36 (1994) 197–199. Crossref, Web of Science, Google Scholar
21. A. Juhász, Twisted conjugacy in certain Artin groups, in Ischia Group Theory 2010 (World Scientific Publication, Hackensack, NJ, 2012), pp. 175–195. Google Scholar
22. C. Kassel and V. Turaev, Braid groups, in Graduate Texts in Mathematics, Vol. 247 (Springer, New York, 2008), xii+340 pp. Google Scholar
23. M. Khovanov, Doodle groups, Trans. Amer. Math. Soc. 349 (1997) 2297–2315. Crossref, Web of Science, Google Scholar
24. D. Krammer, The conjugacy problem for Coxeter groups, Groups Geom. Dyn. 3(1) (2009) 71–171. Crossref, Web of Science, Google Scholar
25. J. Mostovoy and C. Roque-Márquez, Planar pure braids on six strands, J. Knot Theory Ramification 29(1) (2020) 1950097, 11 pp. Link, Web of Science, Google Scholar
26. T. Mubeena and P. Sankaran, Twisted conjugacy and quasi-isometric rigidity of irreducible lattices in semisimple Lie groups, Indian J. Pure Appl. Math. 50(2) (2019) 403–412. Crossref, Web of Science, Google Scholar
27. T. Mubeena and P. Sankaran, Twisted conjugacy classes in abelian extensions of certain linear groups, Canad. Math. Bull. 57(1) (2014) 132–140. Crossref, Web of Science, Google Scholar
28. T. K. Naik, N. Nanda and M. Singh, Conjugacy classes and automorphisms of twin groups, to appear in Forum Math., arXiv:1906.06723v5, https://doi.org/10.1515/forum-2019-0321. Google Scholar
29. T. Nasybullov, Reidemeister spectrum of special and general linear groups over some fields contains $1$ , J. Algebra Appl. 18(8) (2019) 1950153, 12 pp. Link, Web of Science, Google Scholar
30. T. Nasybullov, Twisted conjugacy classes in unitriangular groups, J. Group Theory 22(2) (2019) 253–266. Crossref, Web of Science, Google Scholar
31. K. Reidemeister, Automorphismen von Homotopiekettenringen, Math. Ann. 112 (1936) 586–593. Crossref, Google Scholar
32. G. B. Shabat and V. A. Voevodsky, Drawing curves over number fields, The Grothendieck Festschrift, Vol. III, Progress in Mathematics, Vol. 88 (Birkhäuser Boston, Boston, MA, 1990), pp. 199–227. Crossref, Google Scholar