Snowflake groups, Perron–Frobenius eigenvalues and isoperimetric spectra

Snowflake Groups, Perron-frobenius Eigenvalues, and Isoperimetric Spectra

The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k-spheres mapped into k-connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior for such functions. First, to each non-negative integer matrix P and positive ...

Isoperimetric Inequalities and Eigenvalues

Isoperimetric Inequalities and Eigenvalues

An upper bound is given on the minimum distance between i subsets of the same size of a regular graph in terms of the i-th largest eigenvalue in absolute value. This yields a bound on the diameter in terms of the i-th largest eigenvalue, for any integer i. Our bounds are shown to be asymptotically tight. A recent result by Quenell relating the diameter, the second eigenvalue, and the girth of a...

Isoperimetric Inequalities and Eigenvalues of the Laplacian

Given a domain D in euclidean space or on a Riemannian manifold, one naturally associates with it such fundamental geometric quantities as its volume, the measure of its boundary, various curvature functions and their extremes or integrals. One can also associate with the domain D the set of eigenfunctions and eigenvalues of the Laplace operator, subject to given boundary conditions. Recent wor...

Density of isoperimetric spectra

Dehn functions of groups have been the subject of intense activity over the past two decades. The Dehn function δ(x) of a group G is a quasi-isometry invariant which describes the best possible isoperimetric inequality that holds in any geometric model for the group. Specifically, for a given x , δ(x) is the smallest number A such that every null-homotopic loop of length at most x bounds a disk...