arXiv subjects

Martin Bridson

Publications and source records attributed to Martin Bridson.

4 recordsLinked to original sources

arXiv2012

On groups whose geodesic growth is polynomial

This note records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely generated group $G$ has an element whose normal closure is abelian and of finite index, then $G$ has a finite generating set with respect to which the geodesic growth is polynomial (this includes all virtually cyclic groups).

math.GR↗

arXiv2008

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 rational number r, we associate a finite, aspherical 2-complex X_{r,P} and calculate the Dehn function of its fundamental group G_{r,P} in terms of r and the Perron-Frobenius eigenvalue of P. The range of functions obtained includes x^s, where s is an arbitrary rational number greater than or equal to 2. By repeatedly forming multiple HNN extensions of the groups G_{r,P} we exhibit a similar range of behavior among higher-dimensional Dehn functions, proving in particular that for each positive integer k and rational s greater than or equal to (k+1)/k there exists a group with k-dimensional Dehn function x^s. Similar isoperimetric inequalities are obtained for arbitrary manifold pairs (M,\partial M) in addition to (B^{k+1},S^k).

math.GR↗

arXiv1996

Fractional Isoperimetric Inequalities and subgroup distortion

It is shown that there exist infinitely many non-integers $r>2$ such that the Dehn function of some finitely presented group is $\simeq n^r$. For each positive rational number $s$ we construct pairs of finitely presented groups $H\subset G$ such that the distortion of $H$ in $G$ is $\simeq n^s$. And for each $s\ge 1$ we also construct finitely presented groups whose isodiametric function is $\simeq n^s$.

math.GR↗

arXiv1996

Finitely presented subgroups of automatic groups and their isoperimetric functions

We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgroups satisfy, at best, an exponential isoperimetric inequality.

math.GR↗