An exactly solvable model of the Calogero type for the icosahedral group

We construct a quantum mechanical model of the Calogero type for the icosahedral group as the structural group. Exact solvability is proved and the spectrum is derived explicitly.Comment: 13 pages, no figures, latex 2epsilo

On the complete perturbative solution of one-matrix models

We summarize the recent results about complete solvability of Hermitian and rectangular complex matrix models. Partition functions have very simple character expansions with coefficients made from dimensions of representation of the linear group $GL(N)$, and arbitrary correlators in the Gaussian phase are given by finite sums over Young diagrams of a given size, which involve also the well known characters of symmetric group. The previously known integrability and Virasoro constraints are simple corollaries, but no vice versa: complete solvability is a peculiar property of the matrix model (hypergeometric) $\tau$-functions, which is actually a combination of these two complementary requirements.Comment: 8 page

Tadpole Labelled Oriented Graph Groups and Cyclically Presented Groups

We study a class of Labelled Oriented Graph (LOG) group where the underlying graph is a tadpole graph. We show that such a group is the natural HNN extension of a cyclically presented group and investigate the relationship between the LOG group and the cyclically presented group. We relate the second homotopy groups of their presentations and show that hyperbolicity of the cyclically presented group implies solvability of the conjugacy problem for the LOG group. In the case where the label on the tail of the LOG spells a positive word in the vertices in the circuit we show that the LOGs and groups coincide with those considered by Szczepa�nski and Vesnin. We obtain new presentations for these cyclically presented groups and show that the groups of Fibonacci type introduced by Johnson and Mawdesley are of this form. These groups generalize the Fibonacci groups and the Sieradski groups and have been studied by various authors. We continue these investigations, using small cancellation and curvature methods to obtain results on hyperbolicity, automaticity, SQ-universality, and solvability of decision problems