The conformal measures of a normal subgroup of a cocompact Fuchsian group

In this paper we study the conformal measures of a normal subgroup of a cocompact Fuchsian group. In particular, we relate the extremal conformal measures to the eigenmeasures of a suitable Ruelle operator. Using Ancona’s theorem, adapted to the Ruelle operator setting, we show that if the group of deck transformations G is hyperbolic then the extremal conformal measures and the hyperbolic boundary of G coincide. We then interpret these results in terms of the asymptotic behavior of cutting sequences of geodesics on a regular cover of a compact hyperbolic surface.

Edge Transitive Dihedral Covers of The Heawood Graph

A regular cover of a connected graph is called dihedral ifits transformation group is dihedral. In this paper, the authors clas-sify all dihedral coverings of the Heawood graph whose fibre-preservingautomorphism subgroups act edge-transitively.

Fully truncated simplices and their monodromy groups

We describe a simple way to manufacture faithful representations of the monodromy group of an n-polytope. This is used to determine the monodromy group for 𝓣n, the fully truncated n-simplex. As by-products, we get the minimal regular cover for 𝓣n, along with the analogous objects for a prism over a simplex.

ON GRAPHS OF PRIME VALENCY ADMITTING A SOLVABLE ARC-TRANSITIVE GROUP

Let $X$ be a simple, connected, $p$-valent, $G$-arc-transitive graph, where the subgroup $G\leq \text{Aut}(X)$ is solvable and $p\geq 3$ is a prime. We prove that $X$ is a regular cover over one of the three possible types of graphs with semi-edges. This enables short proofs of the facts that $G$ is at most 3-arc-transitive on $X$ and that its edge kernel is trivial. For pentavalent graphs, two further applications are given: all $G$-basic pentavalent graphs admitting a solvable arc-transitive group are constructed and an example of a non-Cayley graph of this kind is presented.