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We introduce the notion of a building lattice generalizing tree lattices. We give a Lefschetz formula and apply it to geometric zeta functions. We further generalize Bass's approach to Ihara zeta functions to the higher dimensional case of a building.MSC: 51E24, 11M36, 20E42, 20F65, 22D05, 22E40

A central issue in molecular orbital theory is to compute the HOMO-LUMO gap of a molecule, which measures the excitability of the molecule. Thus it would be of interest to learn how to construct a molecule with the prescribed HOMO-LUMO gap. In this paper, we classify all possible structures of fullerene Cayley graphs and compute their spectrum. For any natural number n not divisible by three, we show there exists an infinite family of fullerene graphs with the same HOMO-LUMO gap of size 2π √ 3n + O(n −2 ). Finally, we discuss how to realize those families in three dimensional space.

In the case where G =SL 2 (F ) for a non-archimedean local field F and Γ is a discrete torsion-free cocompact subgroup of G, there is a known relationship between the Ihara zeta function for the quotient of the Bruhat-Tits tree of G by the action of Γ, and an alternating product of determinants of twisted Poincaré series for parabolic subgroups of the affine Weyl group of G. We show how this can be generalised to other split simple algebraic groups of rank two over F , and formulate a conjecture about how this might be generalised to groups of higher rank. Twisted Poincaré series and Ihara zeta functions2.1. Poincaré series. Let (W, S) be a Coxeter system with a generating set S consisting of elements of order two. For w ∈ W , let ℓ(w) be the shortest length of a word consisting of elements

We generalize the theory of p-adic geometric zeta functions of Y. Ihara and K. Hashimoto to the higher rank case. We give the proof of rationality of the zeta function and the connection of the divisor to group cohomology, i.e. the p-adic analogue of the Patterson conjecture.Introduction. In [13] and [14] Y. Ihara defined geometric zeta functions for the group P SL 2 . This was the p-adic counterpart of the famous zeta functions of Selberg for Riemannian surfaces. Using his results on the structure of discrete subgroups of P SL 2 of a p-adic field he proved, among other things, the rationality of these zeta functions. By means of the geometric interpretation on the building, K. Hashimoto [10], [11], [12] extended this to arbitrary rank one groups. In the present paper we generalize Ihara's theory to the higher rank case. Our method uses harmonic analysis and is insofar not geometrical. We will provide geometric interpretations in subsequent work. We prove rationality of the zeta functions and show that the poles and zeroes are related to the cohomology of the discrete group, i.e. the p-adic counterpart of the Patterson-conjecture [6].