Teichm\"uller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables

We generalize a new class of cluster type mutations for which exchange transformations are given by reciprocal polynomials. In the case of second-order polynomials of the form $x+2\cos{\pi/n_o}+x^{-1}$ these transformations are related to triangulations of Riemann surfaces of arbitrary genus with at least one hole/puncture and with an arbitrary number of orbifold points of arbitrary integer orders $n_o$. We propose the dual graph description of the corresponding Teichm\"uller spaces, construct the Poisson algebra of the Teichm\"uller space coordinates, propose the combinatorial description of the corresponding geodesic functions and find the mapping class group transformations.Comment: 20 pages, notations and many essential typos corrected, most significantly, formulae 2.3, 2.5, proof of Lemmata 2.6 and 4.5. Journal reference is added (published version contains typos

Matrix Models and Geometry of Moduli Spaces

We give the description of discretized moduli spaces (d.m.s.) \Mcdisc introduced in \cite{Ch1} in terms of discrete de Rham cohomologies for moduli spaces \Mgn. The generating function for intersection indices (cohomological classes) of d.m.s. is found. Classes of highest degree coincide with the ones for the continuum moduli space \Mc. To show it we use a matrix model technique. The Kontsevich matrix model is the generating function in the continuum case, and the matrix model with the potential N\alpha \tr {\bigl(- \fr 14 \L X\L X -\fr12\log (1-X)-\fr12X\bigr)} is the one for d.m.s. In the latest case the effects of Deligne--Mumford reductions become relevant, and we use the stratification procedure in order to express integrals over open spaces \Mdisc in terms of intersection indices, which are to be calculated on compactified spaces \Mcdisc. We find and solve constraint equations on partition function $\cal Z$ of our matrix model expressed in times for d.m.s.: t^\pm_m=\tr \fr{\d^m}{\d\l^m}\fr1{\e^\l-1}. It appears that $\cal Z$ depends only on even times and {\cal Z}[t^\pm_\cdot]=C(\aa N) \e^{\cal A}\e^{F(\{t^{-}_{2n}\}) +F(\{-t^{+}_{2n}\})}, where $F(\{t^\pm_{2n}\})$ is a logarithm of the partition function of the Kontsevich model, $\cal A$ being a quadratic differential operator in \dd{t^\pm_{2n}}.Comment: 40pp., LaTeX, no macros needed, 8 figures in tex

Spectral problem on graphs and L-functions

The scattering process on multiloop infinite p+1-valent graphs (generalized trees) is studied. These graphs are discrete spaces being quotients of the uniform tree over free acting discrete subgroups of the projective group $PGL(2, {\bf Q}_p)$. As the homogeneous spaces, they are, in fact, identical to p-adic multiloop surfaces. The Ihara-Selberg L-function is associated with the finite subgraph-the reduced graph containing all loops of the generalized tree. We study the spectral problem on these graphs, for which we introduce the notion of spherical functions-eigenfunctions of a discrete Laplace operator acting on the graph. We define the S-matrix and prove its unitarity. We present a proof of the Hashimoto-Bass theorem expressing L-function of any finite (reduced) graph via determinant of a local operator $\Delta(u)$ acting on this graph and relate the S-matrix determinant to this L-function thus obtaining the analogue of the Selberg trace formula. The discrete spectrum points are also determined and classified by the L-function. Numerous examples of L-function calculations are presented.Comment: 39 pages, LaTeX, to appear in Russ. Math. Sur