Normal subgroups of SimpHAtic groups

A group is SimpHAtic if it acts geometrically on a simply connected simplicially hereditarily aspherical (SimpHAtic) complex. We show that finitely presented normal subgroups of the SimpHAtic groups are either: finite, or of finite index, or virtually free. This result applies, in particular, to normal subgroups of systolic groups. We prove similar strong restrictions on group extensions for other classes of asymptotically aspherical groups. The proof relies on studying homotopy types at infinity of groups in question. We also show that non-uniform lattices in SimpHAtic complexes (and in more general complexes) are not finitely presentable and that finitely presented groups acting properly on such complexes act geometrically on SimpHAtic complexes. In Appendix we present the topological two-dimensional quasi-Helly property of systolic complexes.

BNS INVARIANTS AND ALGEBRAIC FIBRATIONS OF GROUP EXTENSIONS

Let G be a finitely generated group that can be written as an extension $$ \begin{align*} 1 \longrightarrow K \stackrel{i}{\longrightarrow} G \stackrel{f}{\longrightarrow} \Gamma \longrightarrow 1 \end{align*} $$ where K is a finitely generated group. By a study of the Bieri–Neumann–Strebel (BNS) invariants we prove that if $b_1(G)> b_1(\Gamma ) > 0$ , then G algebraically fibres; that is, 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 fibre have genus greater that $1$ , G is noncoherent. This answers for a broad class of bundles a question of J. Hillman ([9, Question 11(4)]). Finally, we show that there exist surface bundles over a surface whose BNS invariants have a structure that differs from that of Kodaira fibrations, determined by T. Delzant.

Alexander invariants and cohomology jump loci in group extensions

We study the integral, rational, and modular Alexander invariants, as well as the cohomology jump loci of groups arising as extensions with trivial algebraic monodromy. Our focus is on extensions of the form 1→K→G→Q→1, where Q is an abelian group acting trivially on H1(K;ℤ), with suitable modifications in the rational and mod-p settings. We find a tight relationship between the Alexander invariants, the characteristic varieties, and the resonance varieties of the groups K and G. This leads to an inequality between the respective Chen ranks, which becomes an equality in degrees greater than 1 for split extensions.

Smooth 2-Group Extensions and Symmetries of Bundle Gerbes

We study bundle gerbes on manifolds M that carry an action of a connected Lie group G. We show that these data give rise to a smooth 2-group extension of G by the smooth 2-group of hermitean line bundles on M. This 2-group extension classifies equivariant structures on the bundle gerbe, and its non-triviality poses an obstruction to the existence of equivariant structures. We present a new global approach to the parallel transport of a bundle gerbe with connection, and use it to give an alternative construction of this smooth 2-group extension in terms of a homotopy-coherent version of the associated bundle construction. We apply our results to give new descriptions of nonassociative magnetic translations in quantum mechanics and the Faddeev–Mickelsson–Shatashvili anomaly in quantum field theory. We also propose a definition of smooth string 2-group models within our geometric framework. Starting from a basic gerbe on a compact simply-connected Lie group G, we prove that the smooth 2-group extensions of G arising from our construction provide new models for the string group of G.

Supercharacter theories for algebra group extensions

We construct a few supercharacter theories for finite semidirect products where the normal subgroup is of algebra group type. In the case of algebra groups, these supercharacter theories coincide with those of P. Diaconis and I. M. Isaacs. For the parabolic subgroups of \mathrm{GL}(n,\mathbb{F}_{q}), the supercharacters and superclasses are classified.