On virtual singular braid groups
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917392608657408 |
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| author | Ocampo, Oscar |
| author_facet | Ocampo, Oscar |
| contents | The virtual singular braid group arises as a natural common generalization of classical singular braid groups and virtual braid groups. In this paper, we study several algebraic properties of the virtual singular braid group $VSG_n$. We introduce numerical invariants for virtual singular braids arising from exponent sums of words in $VSG_n$, and describe explicitly the kernels of the associated homomorphisms onto abelian groups. We then determine all group homomorphisms, up to conjugation, from $VSG_n$ to the symmetric group $S_n$, and obtain corresponding semi-direct product decompositions. In the particular case $n=2$, we provide explicit presentations and algebraic descriptions of the kernels. Moreover, we show that certain relations are forbidden in $VSG_n$, and we introduce and study natural quotients of the virtual singular braid group, including welded and unrestricted versions, for which analogous structural results are obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_13885 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On virtual singular braid groups Ocampo, Oscar Group Theory Geometric Topology 20F36, 20F05, 57K12 The virtual singular braid group arises as a natural common generalization of classical singular braid groups and virtual braid groups. In this paper, we study several algebraic properties of the virtual singular braid group $VSG_n$. We introduce numerical invariants for virtual singular braids arising from exponent sums of words in $VSG_n$, and describe explicitly the kernels of the associated homomorphisms onto abelian groups. We then determine all group homomorphisms, up to conjugation, from $VSG_n$ to the symmetric group $S_n$, and obtain corresponding semi-direct product decompositions. In the particular case $n=2$, we provide explicit presentations and algebraic descriptions of the kernels. Moreover, we show that certain relations are forbidden in $VSG_n$, and we introduce and study natural quotients of the virtual singular braid group, including welded and unrestricted versions, for which analogous structural results are obtained. |
| title | On virtual singular braid groups |
| topic | Group Theory Geometric Topology 20F36, 20F05, 57K12 |
| url | https://arxiv.org/abs/2207.13885 |