Classification and Galois conjugacy of Hamming maps

We show that for each d>0 the d-dimensional Hamming graph H(d,q) has an orientably regular surface embedding if and only if q is a prime power p^e. If q>2 there are up to isomorphism \phi(q-1)/e such maps, all constructed as Cayley maps for a d-dimensional vector space over the field of order q. We show that for each such pair d, q the corresponding Belyi pairs are conjugate under the action of the absolute Galois group, and we determine their minimal field of definition. We also classify the orientably regular embedding of merged Hamming graphs for q>3

Self-avoiding walks and amenability

The connective constant $\mu(G)$ of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. The relationship between connective constants and amenability is explored in the current work. Various properties of connective constants depend on the existence of so-called 'graph height functions', namely: (i) whether $\mu(G)$ is a local function on certain graphs derived from $G$, (ii) the equality of $\mu(G)$ and the asymptotic growth rate of bridges, and (iii) whether there exists a terminating algorithm for approximating $\mu(G)$ to a given degree of accuracy. In the context of amenable groups, it is proved that the Cayley graphs of infinite, finitely generated, elementary amenable groups support graph height functions, which are in addition harmonic. In contrast, the Cayley graph of the Grigorchuk group, which is amenable but not elementary amenable, does not have a graph height function. In the context of non-amenable, transitive graphs, a lower bound is presented for the connective constant in terms of the spectral bottom of the graph. This is a strengthening of an earlier result of the same authors. Secondly, using a percolation inequality of Benjamini, Nachmias, and Peres, it is explained that the connective constant of a non-amenable, transitive graph with large girth is close to that of a regular tree. Examples are given of non-amenable groups without graph height functions, of which one is the Higman group.Comment: v2 differs from v1 in the inclusion of further material concerning non-amenable graphs, notably an improved lower bound for the connective constan

Extrinsic versus intrinsic diameter for Riemannian filling-discs and van Kampen diagrams

The diameter of a disc filling a loop in the universal covering of a Riemannian manifold may be measured extrinsically using the distance function on the ambient space or intrinsically using the induced length metric on the disc. Correspondingly, the diameter of a van Kampen diagram filling a word that represents the identity in a finitely presented group can either be measured intrinsically its 1-skeleton or extrinsically in the Cayley graph of the group. We construct the first examples of closed manifolds and finitely presented groups for which this choice -- intrinsic versus extrinsic -- gives rise to qualitatively different min-diameter filling functions.Comment: 44 pages, 12 figures, to appear in the Journal of Differential Geometr