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Structure and automorphisms of pure virtual twin groups

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Abstract

The study of stable isotopy classes of a finite collection of immersed circles without triple or higher intersections on closed oriented surfaces is considered as a planar analogue of virtual knot theory, a far reaching generalisation of classical knot theory. Recent works have established Alexander and Markov theorems in the planar setting. In the classical case, the role of groups is played by twin groups, a class of right-angled Coxeter groups. A new class of groups called virtual twin groups, that extends twin groups in a natural way, plays the role of groups in the virtual case. The virtual twin group \(VT_n\) contains the pure virtual twin group \(PVT_n\), a planar analogue of the pure Artin braid group. In this paper, we prove that the pure virtual twin group \(PVT_n\) is an irreducible right-angled Artin group with trivial center and give it’s precise presentation. We show that \(PVT_n\) has a decomposition as an iterated semi-direct product of infinite rank free groups. We give a complete description of the automorphism group of \(PVT_n\) and establish splitting of natural exact sequences of automorphism groups. As applications, we show that \(VT_n\) is residually finite and \(PVT_n\) has the \(R_\infty \)-property.

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Acknowledgements

Tushar Kanta Naik acknowledges support from the NBHM via grant number 0204/3/2020/R &D-II/2475. Neha Nanda has received funding from European Union’s Horizon Europe research and innovation programme under the Marie Sklodowska Curie grant agreement no 101066588. Mahender Singh is supported by the Swarna Jayanti Fellowship grants DST/SJF/MSA-02/2018-19 and SB/SJF/2019-20.

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Authors and Affiliations

  1. School of Mathematical Sciences, National Institute of Science Education and Research, Bhubaneswar, An OCC of Homi Bhabha National Institute, P. O. Jatni, Khurda, Odisha, 752050, India

    Tushar Kanta Naik

  2. Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139, Université de Caen Normandie, 14000, Caen, France

    Neha Nanda

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

    Mahender Singh

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  1. Tushar Kanta Naik
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Correspondence to Neha Nanda.

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Communicated by John S. Wilson.

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Naik, T.K., Nanda, N. & Singh, M. Structure and automorphisms of pure virtual twin groups. Monatsh Math 202, 555–582 (2023). https://doi.org/10.1007/s00605-023-01851-0

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