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Register a new userBNS Invariants and Algebraic Fibrations of Group Extensions
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Let $G$ be a finitely generated group that can be written as an extension [ 1 longrightarrow K stackrel{i}{longrightarrow} G stackrel{f}{longrightarrow} Gamma longrightarrow 1 ] where $K$ is a finitely generated group. By a study of the BNS invariants we prove that if $b_1(G) > b_1(Gamma) > 0$, then $G$ algebraically fibers, i.e. admits an epimorphism to $Bbb{Z}$ with finitely generated kernel. An interesting case of this occurrence is when $G$ is the fundamental group of a surface bundle over a surface $F hookrightarrow X rightarrow B$ with Albanese dimension $a(X) = 2$. As an application, we show that if $X$ has virtual Albanese dimension $va(X) = 2$ and base and fiber have genus greater that $1$, $G$ is noncoherent. This answers for a broad class of bundles a question of J. Hillman.
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Wreath products of finite groups have permutation representations that are constructed from the permutation representations of their constituents. One can envision these in a metaphoric sense in which a rope is made from a bundle of threads. In this way, subgroups and quotients are easily visualized. The general idea is applied to the finite subgroups of the special unitary group of $(2times 2)$-matrices. Amusing diagrams are developed that describe the unit quaternions, the binary tetrahedral, octahedral, and icosahedral group as well as the dicyclic groups. In all cases, the quotients as subgroups of the permutation group are readily apparent. These permutation representations lead to injective homomorphisms into wreath products.
The paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an abelian extension of quandles. A non-abelian counterpart of this sequence involving dynamical cohomology classes is also established, and some applications to lifting of quandle automorphisms are given. Viewing the construction of the conjugation, the core and the generalised Alexander quandle of a group as an adjoint functor of some appropriate functor from the category of quandles to the category of groups, we prove that these functors map extensions of groups to extensions of quandles. Finally, we construct some natural group homomorphisms from the second cohomology of a group to the second cohomology of its core and conjugation quandles.
In this paper, we compute the {Sigma}^n(G) and {Omega}^n(G) invariants when 1 rightarrow H rightarrow G rightarrow K rightarrow 1 is a short exact sequence of finitely generated groups with K finite. We also give sufficient conditions for G to have the R_{infty} property in terms of {Omega}^n(H) and {Omega}^n(K) when either K is finite or the sequence splits. As an application, we construct a group F rtimes? Z_2 where F is the R. Thompsons group F and show that F rtimes Z_2 has the R_{infty} property while F is not characteristic.
We investigate two categorified braid conjugacy class invariants, one coming from Khovanov homology and the other from Heegaard Floer homology. We prove that each yields a solution to the word problem but not the conjugacy problem in the braid group.
We examine the action of the fundamental group $Gamma$ of a Riemann surface with $m$ punctures on the middle dimensional homology of a regular fiber in a Lefschetz fibration, and describe to what extent this action can be recovered from the intersection numbers of vanishing cycles. Basis changes for the vanishing cycles result in a nonlinear action of the framed braid group $widetilde{mathcal B}$ on $m$ strings on a suitable space of $mtimes m$ matrices. This action is determined by a family of cohomologous 1-cocycles ${mathcal S}_c:widetilde{mathcal B}to GL_m({mathbb{Z}}[Gamma])$ parametrized by distinguished configurations $c$ of embedded paths from the regular value to the critical values. In the case of the disc, we compare this family of cocycles with the Magnus cocycles given by Fox calculus and consider some abelian reductions giving rise to linear representations of braid groups. We also prove that, still in the case of the disc, the intersection numbers along straight lines, which conjecturally make sense in infinite dimensional situations, carry all the relevant information.
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