Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

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Metric Spaces Of Non Positive Curvature

Lectures on the Combinatorics of Free Probability: The free commutator

This is an almost self-contained monograph (containing some new results) on left-orderable groups which mostly rely on dynamical and probabilistic aspects, but also on geometric, combinatorial, analytic, and topological ones.

/pdf/groups-orders-and-dynamics-5cwgqvq1tr.pdf

Groups, Orders, and Dynamics

This article investigates structural, geometrical, and topological characterizations and properties of weakly modular graphs and of cell complexes derived from them. The unifying themes of our investigation are various `nonpositive curvature' and `local-to-global' properties and characterizations of weakly modular graphs and their subclasses. Weakly modular graphs have been introduced as a far-reaching common generalization of median graphs (and more generally, of modular and orientable modular graphs), Helly graphs, bridged graphs, and dual polar graphs occurring under different disguises in several seemingly-unrelated fields of mathematics: Metric graph theory, Geometric group theory, Incidence geometries and buildings, Theoretical computer science and combinatorial optimization. We give a local-to-global characterization of weakly modular graphs and their subclasses in terms of simple connectedness of associated triangle-square complexes and specific local combinatorial conditions. In particular, we revisit characterizations of dual polar graphs by Cameron and by Brouwer-Cohen. We also show that (disk-)Helly graphs are precisely the clique-Helly graphs with simply connected clique complexes. With $l_1$-embeddable weakly modular and sweakly modular graphs we associate high-dimensional cell complexes, having several strong topological and geometrical properties (contractibility and the CAT(0) property). Their cells have a specific structure: they are basis polyhedra of even $\triangle$-matroids in the first case and orthoscheme complexes of gated dual polar subgraphs in the second case. We resolve some open problems concerning subclasses of weakly modular graphs: we prove a Brady-McCammond conjecture about CAT(0) metric on the orthoscheme complexes of modular lattices; we answer Chastand's question about prime graphs for pre-median graphs.

Weakly modular graphs and nonpositive curvature

We show that braid groups with at most 6 strands are CAT(0) using the close connection between these groups, the associated non-crossing partition complexes, and the embeddability of their diagonal links into spherical buildings of type A. Furthermore, we prove that the orthoscheme complex of any bounded graded modular complemented lattice is CAT(0), giving a partial answer to a conjecture of Brady and McCammond.

The 6-strand braid group is CAT(0)

We show that a finitely generated residually finite rationally solvable (or RFRS) group $G$ is virtually fibred, in the sense that it admits a virtual surjection to $\mathbb{Z}$ with a finitely generated kernel, if and only if the first $L^2$-Betti number of $G$ vanishes. This generalises (and gives a new proof of) the analogous result of Ian Agol for fundamental groups of $3$-manifolds.

Residually finite rationally solvable groups and virtual fibring

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 $$\mathtt {F}_{}$$ 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 $$L^2$$-torsion polytope of Friedl–Luck is invariant under homotopy. We prove the vanishing of this polytope in the presence of amenability, thus proving a conjecture of Friedl–Luck–Tillmann.

The Bieri–Neumann–Strebel invariants via Newton polytopes

We show that braid groups with at most 6 strands are CAT(0) using the close connection between these groups, the associated non-crossing partition complexes and the embeddability of their diagonal links into spherical buildings of type A. Furthermore, we prove that the orthoscheme complex of any bounded graded modular complemented lattice is CAT(0), giving a partial answer to a conjecture of Brady and McCammond.

We investigate Friedl-Luck's universal $L^2$-torsion for descending HNN extensions of finitely generated free groups, and so in particular for $F_n$-by-$\mathbb{Z}$ groups. This invariant induces a semi-norm on the first cohomology of the group which is an analogue of the Thurston norm for $3$-manifold groups. We prove that this Thurston semi-norm is an upper bound for the Alexander semi-norm defined by McMullen, as well as for the higher Alexander semi-norms defined by Harvey. The same inequalities are known to hold for $3$-manifold groups. We also prove that the Newton polytopes of the universal $L^2$-torsion of a descending HNN extension of $F_2$ locally determine the Bieri-Neumann-Strebel invariant of the group. We give an explicit means of computing the BNS invariant for such groups. As a corollary, we prove that the Bieri-Neumann-Strebel invariant of a descending HNN extension of $F_2$ has finitely many connected components. When the HNN extension is taken over $F_n$ along a polynomially growing automorphism with unipotent image in $GL(n, \mathbb{Z})$, we show that the Newton polytope of the universal $L^2$-torsion and the BNS invariant completely determine one another. We also show that in this case the Alexander norm, its higher incarnations, and the Thurston norm all coincide.

https://msp.org/gt/2018/22-5/gt-v22-n5-p03-s.pdf

Dawid Kielak

Papers

The 6-strand braid group is CAT(0)

Residually finite rationally solvable groups and virtual fibring

The Bieri–Neumann–Strebel invariants via Newton polytopes

The 6-strand braid group is CAT(0)

Alexander and Thurston norms, and the Bieri-Neumann-Strebel invariants for free-by-cyclic groups