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Following previous work on congruence subgroups and crystallographic braid groups, we study the lower central series of congruence braid groups related to the braid group $B_3$, showing in particular that corresponding quotients are almost crystallographic., Comment: All comments are welcome

Let $n\geq 2$. Let $VB_n$ (resp. $VP_n$) denote the virtual braid group (resp. virtual pure braid group), let $WB_n$ (resp. $WP_n$) denote the welded braid group (resp. welded pure braid group) and let $UVB_n$ (resp. $UVP_n$) denote the unrestricted virtual braid group (resp. unrestricted virtual pure braid group). In the first part of this paper we prove that, for $n\geq 4$, the group $VP_n$ and for $n\geq 3$ the groups $WP_n$ and $UVP_n$ are characteristic subgroups of $VB_n$, $WB_n$ and $UVB_n$, respectively. In the second part of the paper we show that, for $n\geq 2$, the virtual braid group $VB_n$, the unrestricted virtual pure braid group $UVP_n$, and the unrestricted virtual braid group $UVB_n$ have the R$_\infty$-property. As a consequence of the technique used for few strings we also prove that, for $n=2,3,4$, the welded braid group $WB_n$ has the R$_\infty$-property and that for $n=2$ the corresponding pure braid groups have the R$_\infty$-property. On the other hand for $n\geq 3$ it is unknown if the R$_\infty$-property holds or not for the virtual pure braid group $VP_n$ and the welded pure braid group $WP_n$., Comment: 26 pages

In this paper, we study subgroup separability (also known as LERF) and related properties for surface braid groups and virtual (singular) braid groups., Comment: 18 pages

For classical links Ohyama proved an inequality involving the minimal crossing number and the braid index, then motivated from this Takeda showed an analogous inequality for virtual links. In this paper, we are interested in studying properties of links independent of the type of crossings, and for this reason, we introduce generalized crossings for diagrams and generalized Reidemeister-type moves. The aim of this work is to prove the same type of inequality mentioned above but now involving the total crossing number and the braid index of generalized knots and links. In particular, we show that the result holds for virtual singular links., Comment: 15 pages, 20 figures

In this paper, we show that almost-direct products of (reduced) free groups admit (reduced) Magnus ordering. Besides the application is already used to to well known groups, we use the main result to give an ordering of some subgroups of the automorphism group of free groups and also to give an ordering of pure monomial braid groups., Comment: 10 pages

Let $n\ge 2$. In this paper we study some properties of the virtual singular braid group $VSG_n$. We define some numerical invariants for virtual singular braids arising from exponents of words in $VSG_n$ and describe the kernel of these homomorphisms. We determine all possible homomorphisms, up to conjugation, from the virtual singular braid group $VSG_n$ to the symmetric group $S_n$, and for the particular case $n=2$ we give a presentation (and a description) for the kernel in each case. For all possible homomorphisms we obtain a decomposition of $VSG_n$ as a semi-direct product of the kernel of the homomorphism and the symmetric group. Finally, we show that there are some forbidden relations in $VSG_n$ and then we introduce some quotients of it (e.g. the welded singular braid group, the unrestricted virtual singular braid group, among other related groups) and then we state for them similar results obtained for $VSG_n$., Comment: 19 pages, 7 figures. Comments are welcome

Let $n\ge 2$. Let $VB_n$ (resp. $VP_n$) be the virtual braid group (resp. the pure virtual braid group), and let $VT_n$ (resp. $PVT_n$) be the virtual twin group (resp. the pure virtual twin group). Let $\Pi$ be one of the following quotients: $VB_n/\Gamma_2(VP_n)$ or $VT_n/\Gamma_2(PVT_n)$ where $\Gamma_2(H)$ is the commutator subgroup of $H$. In this paper, we show that $\Pi$ is a crystallographic group and we characterize the elements of finite order and the conjugacy classes of elements in $\Pi$. Furthermore, we realize explicitly some Bieberbach groups and infinite virtually cyclic groups in $\Pi$. Finally, we also study other braid-like groups (welded, unrestricted, flat virtual, flat welded and Gauss virtual braid group) module the respective commutator subgroup in each case., Comment: In this new version some general results were added in Section 2

Let $n\geq 3$. In this paper we deal with the conjugacy problem in the Artin braid group quotient $B_n/[P_n,P_n]$. To solve it we use systems of equations over the integers arising from the action of $B_n/[P_n,P_n]$ over the abelianization of the pure Artin braid group $P_n/[P_n,P_n]$. Using this technique we also realize explicitly infinite virtually cyclic subgroups in $B_n/[P_n,P_n]$., Comment: 16 pages. Comments are welcome

Let $M$ be a compact surface without boundary, and $n\geq 2$. We analyse the quotient group $B_n(M)/\Gamma_2(P_n(M))$ of the surface braid group $B_{n}(M)$ by the commutator subgroup $\Gamma_2(P_n(M))$ of the pure braid group $P_{n}(M)$. If $M$ is different from the $2$-sphere $\mathbb{S}^2$, we prove that $B_n(M)/\Gamma_2(P_n(M))$ is isomorphic rho $P_n(M)/\Gamma_2(P_n(M)) \rtimes_{\varphi} S_n$, and that $B_n(M)/\Gamma_2(P_n(M))$ is a crystallographic group if and only if $M$ is orientable. If $M$ is orientable, we prove a number of results regarding the structure of $B_n(M)/\Gamma_2(P_n(M))$. We characterise the finite-order elements of this group, and we determine the conjugacy classes of these elements. We also show that there is a single conjugacy class of finite subgroups of $B_n(M)/\Gamma_2(P_n(M))$ isomorphic either to $S_n$ or to certain Frobenius groups. We prove that crystallographic groups whose image by the projection $B_n(M)/\Gamma_2(P_n(M))\to S_n$ is a Frobenius group are not Bieberbach groups. Finally, we construct a family of Bieberbach subgroups $\tilde{G}_{n,g}$ of $B_n(M)/\Gamma_2(P_n(M))$ of dimension $2ng$ and whose holonomy group is the finite cyclic group of order $n$, and if $\mathcal{X}_{n,g}$ is a flat manifold whose fundamental group is $\tilde{G}_{n,g}$, we prove that it is an orientable K\"ahler manifold that admits Anosov diffeomorphisms.

In this paper we prove that all pure Artin braid groups $P_n$ ($n\geq 3$) have the $R_\infty$ property. In order to obtain this result, we analyse the naturally induced morphism $\operatorname{\text{Aut}}(P_n) \to \operatorname{\text{Aut}}(\Gamma_2 (P_n)/\Gamma_3(P_n))$ which turns out to factor through a representation $\rho\colon S_{n+1} \to \operatorname{\text{Aut}}(\Gamma_2 (P_n)/\Gamma_3(P_n))$. We can then use representation theory of the symmetric groups to show that any automorphism $\alpha$ of $P_n$ acts on the free abelian group $\Gamma_2 (P_n)/\Gamma_3(P_n)$ via a matrix with an eigenvalue equal to 1. This allows us to conclude that the Reidemeister number $R(\alpha)$ of $\alpha$ is $\infty$., Comment: 18 pages

Let $n\geq 3$. In this paper we show that for any finite abelian subgroup $G$ of $S_n$ the crystallographic group $B_n/[P_n,P_n]$ has Bieberbach subgroups $\Gamma_{G}$ with holonomy group $G$. Using this approach we obtain an explicit description of the holonomy representation of the Bieberbach group $\Gamma_{G}$. As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of $\Gamma_{G}$ and determine the existence of Anosov diffeomorphisms and K\"ahler geometry of the flat manifold ${\cal X}_{\Gamma_{G}}$ with fundamental group the Bieberbach group $\Gamma_{G}\leq B_n/[P_n,P_n]$., Comment: 16 pages

Let $n \geq 3$. In this paper, we study the problem of whether a given finite group $G$ embeds in a quotient of the form $B_n/\Gamma_k(P_n)$, where $B_n$ is the $n$-string Artin braid group, $k \in \{2, 3\}$, and $\{\Gamma_l(P_n)\}_{l\in \mathbb{N}}$ is the lower central series of the $n$-string pure braid group $P_n$. Previous results show that a necessary condition for such an embedding to exist is that $|G|$ is odd (resp. is relatively prime with $6$) if $k=2$ (resp. $k=3$), where $|G|$ denotes the order of $G$. We show that any finite group $G$ of odd order (resp. of order relatively prime with $6$) embeds in $B_{|G|}/\Gamma_2(P_{|G|})$ (resp. in $B_{|G|}/\Gamma_3(P_{|G|})$). The result in the case of $B_{|G|}/\Gamma_2(P_{|G|})$ has been proved independently by Beck and Marin. One may then ask whether $G$ embeds in a quotient of the form $B_n/\Gamma_k(P_n)$, where $n < |G|$ and $k \in \{2, 3\}$. If $G$ is of the form $\mathbb{Z}_{p^r} \rtimes_{\theta} \mathbb{Z}_d$, where the action $\theta$ is injective, $p$ is an odd prime (resp. $p \geq 5$ is prime) $d$ is odd (resp. $d$ is relatively prime with $6$) and divides $p-1$, we show that $G$ embeds in $B_{p^r}/\Gamma_2(P_{p^r})$ (resp. in $B_{p^r}/\Gamma_3(P_{p^r})$). In the case $k=2$, this extends a result of Marin concerning the embedding of the Frobenius groups in $B_n/\Gamma_2(P_n)$, and is a special case of another result of Beck and Marin. Finally, we construct an explicit embedding in $B_9/\Gamma_2(P_9)$ of the two non-Abelian groups of order $27$, namely the semi-direct product $\mathbb{Z}_9 \rtimes \mathbb{Z}_3$, where the action is given by multiplication by $4$, and the Heisenberg group mod $3$.

Let $n, k \geq 3$. In this paper, we analyse the quotient group $B\_n/\Gamma\_k(P\_n)$ of the Artin braid group $B\_n$ by the subgroup $\Gamma\_k(P\_n)$ belonging to the lower central series of the Artin pure braid group $P\_n$. We prove that it is an almost-crystallographic group. We then focus more specifically on the case $k=3$. If $n \geq 5$, and if $\tau \in N$ is such that $gcd(\tau, 6) = 1$, we show that $B\_n/\Gamma\_3 (P\_n)$ possesses torsion $\tau$ if and only if $S\_n$ does, and we prove that there is a one-to-one correspondence between the conjugacy classes of elements of order $\tau$ in $B\_n/\Gamma\_3 (P\_n)$ with those of elements of order $\tau$ in the symmetric group $S\_n$. We also exhibit a presentation for the almost-crystallographic group $B\_n/\Gamma\_3 (P\_n)$. Finally, we obtain some $4$-dimensional almost-Bieberbach subgroups of $B\_3/\Gamma\_3 (P\_3)$, we explain how to obtain almost-Bieberbach subgroups of $B\_4/\Gamma\_3(P\_4)$ and $B\_3/\Gamma\_4(P\_3)$, and we exhibit explicit elements of order $5$ in $B\_5/\Gamma\_3 (P\_5)$.

Let n be greater than or equal to 3. We study the quotient group B\_n/[P n,P\_n] of the Artin braid group B\_n by the commutator subgroup of its pure Artin braid group P\_n. We show that B\_n/[P n,P\_n] is a crystallographic group, and in the case n=3, we analyse explicitly some of its subgroups. We also prove that B\_n/[P n,P\_n] possesses torsion, and we show that there is a one-to-one correspondence between the conjugacy classes of the finite-order elements of B\_n/[P n,P\_n] with the conjugacy classes of the elements of odd order of the symmetric group S\_n, and that the isomorphism class of any Abelian subgroup of odd order of S\_n is realised by a subgroup of B\_n/[P n,P\_n]. Finally, we discuss the realisation of non-Abelian subgroups of S\_n of odd order as subgroups of B\_n/[P n,P\_n], and we show that the Frobenius group of order 21, which is the smallest non-Abelian group of odd order, embeds in B\_n/[P n,P\_n] for all n greater than or equal to 7.