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Parabolics and Invariants

The goal of the project is to obtain new insights into certain asymptotic topological and homological invariants of groups, specifically

- the $$\Sigma$$-invariants introduced by Bieri-Neumann-Strebel and Bieri-Renz,

- universal $$L^2$$-torsion introduced by Friedl-Lück, and

- higher generation introduced by Abels-Holz.

In the course of studying these invariants a prominent role is played by parabolic subgroups (of algebraic groups and Coxeter groups) and generalizations of these (in automorphisms of right-angles Artin groups, braid groups, etc.).

## Publications

We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative Laurent polynomial rings are themselves polynomials, rather than rational functions. We also exhibit a relationship between the Newton polytopes and invertibility of the matrices over Novikov rings, thus establishing a connection with the invariants of Bieri-Neumann-Strebel (BNS) via a theorem of Sikorav.

We offer several applications: we reprove Thurston's theorem on the existence of a polytope controlling the BNS invariants of a 3-manifold group; we extend this result to free-by-cyclic groups, and the more general descending HNN extensions of free groups. We also show that the BNS invariants of Poincare duality groups of type F in dimension 3 and groups of deficiency one are determined by a polytope, when the groups are assumed to be agrarian, that is their integral group rings embed in skew-fields. The latter result partially confirms a conjecture of Friedl.

We also deduce the vanishing of the Newton polytopes associated to elements of the Whitehead groups of many groups satisfying the Atiyah conjecture. We use this to show that the L2-torsion polytope of Friedl-Lueck is invariant under homotopy. We prove the vanishing of this polytope in the presence of amenability, thus proving a conjecture of Friedl-Lueck-Tillmann.

Related project(s):
8Parabolics and invariants

We prove that if a quasi-isometry of warped cones is induced by a map between the base spaces of the cones, the actions must be conjugate by this map. The converse is false in general, conjugacy of actions is not sufficient for quasi-isometry of the respective warped cones. For a general quasi-isometry of warped cones, using the asymptotically faithful covering constructed in a previous work with Jianchao Wu, we deduce that the two groups are quasi-isometric after taking Cartesian products with suitable powers of the integers.

Secondly, we characterise geometric properties of a group (coarse embeddability into Banach spaces, asymptotic dimension, property A) by properties of the warped cone over an action of this group. These results apply to arbitrary asymptotically faithful coverings, in particular to box spaces. As an application, we calculate the asymptotic dimension of a warped cone and improve bounds by Szabo, Wu, and Zacharias and by Bartels on the amenability dimension of actions of virtually nilpotent groups.

In the appendix, we justify optimality of our result on general quasi-isometries by showing that quasi-isometric warped cones need not come from quasi-isometric groups, contrary to the case of box spaces.

Related project(s):
8Parabolics and invariants

We prove that the smallest non-trivial quotient of the mapping class group of a connected orientable surface of genus at least 3 without punctures is Sp_2g(2), thus confirming a conjecture of Zimmermann. In the process, we generalise Korkmaz's results on C-linear representations of mapping class groups to projective representations over any field.

Related project(s):
8Parabolics and invariants

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## Team Members

Benjamin Brück
Doctoral student
Universität Bielefeld
benjamin.brueck(at)uni-bielefeld.de

Prof. Dr. Kai-Uwe Bux
Universität Bielefeld
kaiuwe.bux(at)gmail.com

Dr. Dawid Kielak