ISOPERIMETRIC FUNCTIONS OF GROUPS ACTING ON Lδ-SPACES

A finitely generated group acting properly, cocompactly, and by isometries on an Lδ-metric space is finitely presented and has a sub-cubic isoperimetric function.

SOME DUALITY CONJECTURES FOR FINITE GRAPHS AND THEIR GROUP THEORETIC CONSEQUENCES

We pose some graph theoretic conjectures about duality and the diameter of maximal trees in planar graphs, and we give innovations in the following two topics in geometric group theory, where the conjectures have applications.Central extensions. We describe an electrostatic model concerning how van Kampen diagrams change when one takes a central extension of a group. Modulo the conjectures, this leads to a new proof that finitely generated class $c$ nilpotent groups admit degree $c+1$ polynomial isoperimetric functions.Filling functions. We collate and extend results about interrelationships between filling functions for finite presentations of groups. We use the electrostatic model in proving that the gallery length filling function, which measures the diameter of the duals of diagrams, is qualitatively the same as a filling function DlogA, concerning the sum of the diameter with the logarithm of the area of a diagram. We show that the conjectures imply that the space-complexity filling function filling length essentially equates to gallery length. We give linear upper bounds on these functions for a number of classes of groups including fundamental groups of compact geometrizable 3-manifolds, certain graphs of groups, and almost convex groups. Also we define restricted filling functions which concern diagrams with uniformly bounded vertex valence, and we show that, assuming the conjectures, they reduce to just two filling functions—the analogues of non-deterministic space and time.

Outerplanar Crossing Numbers, the Circular Arrangement Problem and Isoperimetric Functions

We extend a lower bound due to Shahrokhi, Sýkora, Székely and Vrťo for the outerplanar crossing number (in other terminologies also called convex, circular and one-page book crossing number) to a more general setting. In this setting we can show a better lower bound for the outerplanar crossing number of hypercubes than the best lower bound for the planar crossing number. We exhibit further sequences of graphs, whose outerplanar crossing number exceeds by a factor of $\log n$ the planar crossing number of the graph. We study the circular arrangement problem, as a lower bound for the linear arrangement problem, in a general fashion. We obtain new lower bounds for the circular arrangement problem. All the results depend on establishing good isoperimetric functions for certain classes of graphs. For several graph families new near-tight isoperimetric functions are established.

ℓ1-HOMOLOGY OF COMBABLE GROUPS AND 3-MANIFOLD GROUPS

S. Gersten asked whether the reduced ℓ1-homology of a group of type ℱn vanishes in all dimensions up to n - 1. We prove this for combable groups and for the fundamental groups of closed 3-manifolds. This means that the ℓ1-homology in these cases can be interpreted as "an amount of non-linearity" for the isoperimetric functions.