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Charalampos Stylianakis

Publications and source records attributed to Charalampos Stylianakis.

8 recordsLinked to original sources
arXiv2026

Finiteness properties of the Torelli group of surfaces with 2 boundary components

In this paper we prove that the Torelli group of a surface of genus at least 3 with 2 boundary components is finitely generated. As a consequence, we answer Putman's question on the finite generation of the stabilizer subgroup of the Torelli group of a non separating simple closed curve. Furthermore, we prove that the Johnson kernel is finitely generated if the genus of the surface is at least 5 and has 2 boundary components.

math.GT↗
arXiv2026

Forbidden relations in universal virtual braid groups

We study natural automorphisms of the universal virtual braid group $UV_n(k)$. These automorphisms induce commuting involutions in the outer automorphism group and generate a subgroup isomorphic to $\mathbb{Z}_2^k\times\mathbb{Z}_2$. We then show that the two one-forbidden quotients of $UV_n(k)$ are isomorphic. Furthermore, we introduce the universal unrestricted virtual braid group $UUV_n(k)$ obtained by imposing simultaneously the two forbidden relations, and derive several structural properties inherited from the universal setting. Since the multi-virtual braid group $M_kVB_n$ is a quotient of $UV_n(k)$, the corresponding results for $M_kVB_n$ follow as consequences. In particular, for $k=1$ we prove that the quotients of $VB_n$ by the two forbidden relations are isomorphic and obtain structural properties for the unrestricted virtual braid group.

math.GR↗
arXiv2025

Powers of half-twists and congruence subgroups of braid groups

In this work, we study the relationship between congruence subgroups $B_n[m]$ and $\mathcal{N}_n(σ_1^m)$ the normal closure of $σ_1^m$, where $σ_1$ is the classical generator of $B_n$. We characterize the conditions under which $\mathcal{N}_n(σ_1^m)$ has finite index in $B_n[m]$ and provide explicit generators for these finite quotients. For the cases where the index is infinite, we show that $B_n[m]/\mathcal{N}_n(σ_1^m)$ contains a free subgroup. Additionally, we compute the Abelianisation of Coxeter braid subgroups in the finite index cases and construct new finite quotients using commutators of congruence subgroups.

math.GR↗
arXiv2024

Congruence subgroups of braid groups and crystallographic quotients. Part I

This paper is the first of a two part series devoted to describing relations between congruence and crystallographic braid groups. We recall and introduce some elements belonging to congruence braid groups and we establish some (iso)-morphisms between crystallographic braid groups and corresponding quotients of congruence braid groups.

math.GR↗
arXiv2023

Abelian quotients for groups generated by cubed half-twists

Below we construct non-cyclic and torsion-free abelian quotients for subgroups of braid groups generated by cube powers of half-twists. In the case of 3 and 4 strands we compute the abelianization of these groups. Also, we get abelianizations of level-3 congruence subgroups of braid groups on 3 and 4 strands. Moreover, we construct non-cyclic and torsion-free abelian quotients for level-3 congruence subgroups of braid groups. The constructions are based on the original construction of Dennis Johnson \cite{DJ}, who computed an abelian quotient of the Torelli group.

math.GR↗
arXiv2016

Congruence subgroups of braid groups

In this paper we give a description of the generators of the prime level congruence subgroups of braid groups. Also, we give a new presentation of the symplectic group over a finite field, and we calculate symmetric quotients of the prime level congruence subgroups of braid groups. Finally, we find a finite generating set for the level-3 congruence subgroup of the braid group on 3 strands.

math.GR↗
arXiv2015

The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere

In this paper we show that the normal closure of the mth power of a half-twist has infinite index in the mapping class group of a punctured sphere. Furthermore, in some cases we prove that the quotient of the mapping class group of the punctured sphere by the normal closure of a power of a half-twist contains a free abelian subgroup. As a corollary we prove that the quotient of the hyperelliptic mapping class group of a surface of genus at least two by the normal closure of the mth power of a Dehn twist has infinite order, and for some integers m the quotient contains a free nonabelian subgroup. As a second corollary we recover a result of Coxeter: the normal closure of the mth power of a half-twist in the braid group of at least five strands has infinite index if n is at least four. Our method is to reformulate the Jones representation of the mapping class group of a punctured sphere, using the action of Hecke algebras on W-graphs, as introduced by Kazhdan-Lusztig.

math.GR↗