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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 note, 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 $n ge 3$.
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Study of certain 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 with the genus zero case corresponding to classical knot theory. Alexander and Markov theorems are known in this setting with the role of groups being played by a class of right-angled Coxeter groups called twin groups, denoted $T_n$, in the genus zero case. For the higher genus case, the role of groups is played by a new class of groups called virtual twin groups, denoted $VT_n$. A virtual twin group $VT_n$ contains the twin group $T_n$ and the pure virtual twin group $PVT_n$, an analogue of the pure braid group. The paper investigates in detail important structural aspects of these groups. We prove that the pure virtual twin group $PVT_n$ is an irreducible right-angled Artin group with trivial center and give its precise presentation. We show that $PVT_n$ has a decomposition as an iterated semidirect product of infinite rank free groups. We also give a complete description of the automorphism group of $PVT_n$ and establish splitting of some natural exact sequences of automorphism groups. As applications, we show that $VT_n$ is residually finite and $PVT_n$ has the $R_infty$-property. Along the way, we also obtain a presentation of $gamma_2(VT_n)$ and a freeness result on $gamma_2(PVT_n)$.
The twin group $T_n$ is a right angled Coxeter group generated by $n-1$ involutions and the pure twin group $PT_n$ is the kernel of the natural surjection from $T_n$ onto the symmetric group on $n$ symbols. In this paper, we investigate some structural aspects of these groups. We derive a formula for the number of conjugacy classes of involutions in $T_n$, which quite interestingly, is related to the well-known Fibonacci sequence. We also derive a recursive formula for the number of $z$-classes of involutions in $T_n$. We give a new proof of the structure of $Aut(T_n)$ for $n ge 3$, and show that $T_n$ is isomorphic to a subgroup of $Aut(PT_n)$ for $n geq 4$. Finally, we construct a representation of $T_n$ to $Aut(F_n)$ for $n ge 2$.
In various classes of infinite groups, we identify groups that are presentable by products, i.e. groups having finite index subgroups which are quotients of products of two commuting infinite subgroups. The classes we discuss here include groups of small virtual cohomological dimension and irreducible Zariski dense subgroups of appropriate algebraic groups. This leads to applications to groups of positive deficiency, to fundamental groups of three-manifolds and to Coxeter groups. For finitely generated groups presentable by products we discuss the problem of whether the factors in a presentation by products may be chosen to be finitely generated.
We prove that for geometrically finite groups cohomological dimension of the direct product of a group with itself equals 2 times the cohomological dimension dimension of the group.
Let $Gamma$ be the fundamental group of a surface of finite type and Comm$(Gamma)$ be its abstract commensurator. Then Comm$(Gamma)$ contains the solvable Baumslag--Solitar groups $langle a ,b : a b a^{-1} = b^n rangle$ for any $n > 1$. Moreover, the Baumslag--Solitar group $langle a ,b : a b^2 a^{-1} = b^3 rangle$ has an image in Comm$(Gamma)$ that is not residually finite. Our proofs are computer-assisted. Our results also illustrate that finitely-generated subgroups of Comm$(Gamma)$ are concrete objects amenable to computational methods. For example, we give a proof that $langle a ,b : a b^2 a^{-1} = b^3 rangle$ is not residually finite without the use of normal forms of HNN extensions.
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