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

The fundamental theorem of projective geometry states that every line-preserving bijection of a Desarguesian projective plane is induced from a semilinear transformation. This theorem extends to higher-dimensional projective spaces and has an affine version, possibly due to Darboux, which states that every line-preserving bijection of an affine space is composed of a linear transformation and a translation and a mapping induced by an automorphism of the underlying field. In the 1800s, properties of maps such as conformal and Möbius transformations were worked out and, in particular, Liouville showed that sufficiently smooth conformal mappings of the real plane arise from the action of the conformal group. (Liouville's contribution is discussed in [4].) In the 1950s and 1960s more work was done on conformal mappings and the regularity assumptions of Liouville's theorem were relaxed by analysts such as Gehring [5] and Rešetnjak [8]: theorems were proved about conformal mappings defined on open subsets of Euclidean space, rather than on the whole space or the one-point compactification of the whole space. Earlier, in the 1930s, Carathéodory [2] had considered maps of spheres preserving circles in the spheres, and maps of subsets of the sphere preserving arcs as well. Maps of subsets of the real plane preserving line segments do not seem to have been considered until quite recently (except that mappings of the disc that preserve line segments can be interpreted as mappings of the Klein model of the hyperbolic plane that preserve geodesics; it was apparently well known that these come from the isometry group of the hyperbolic plane).In the 1970s, Tits [9] extended the fundamental theorem of projective geometry to buildings, a combinatorial construction. He associated 'spherical buildings' to semisimple groups over arbitrary fields, and showed that certain bijections of the

In [5] Tait identifies a set of reflection principles which he calls Γ(2) n -reflection principles which Peter Koellner has shown to be consistent relative to κ(ω), the first ω-Erdös cardinal, in [2]. Tait also goes on in the same work to define a set of reflection principles which he calls Γ (m) n -reflection principles; however Koellner has shown that these are inconsistent when m > 2 in [3], but identifies restricted versions of them which he proves consistent relative to κ(ω). In this paper we introduce a new large-cardinal property with an ordinal parameter α > 0, calling those cardinals which satisfy it α-reflective cardinals. Its definition is motivated by the remarks Tait makes in [5] about why reflection principles must be restricted when parameters of third or higher order are introduced. We prove that if κ is κ+α+1 -supercompact and 0 < α < κ then κ is α-reflective. Furthermore we show that αreflective cardinals relativize to L, and that if κ(ω) exists then the set of cardinals λ < κ(ω), such that λ is α-refective for all α such that 0 < α < λ, is a stationary subset of κ(ω). We show that an ω-reflective cardinal satisfies some restricted versions of Γ (m) n -reflection, as well as all the reflection properties which Koellner proves consistent in [3].Part of this paper was written while I was a Research Intensive Academic at the Australian Catholic University. I made further revisions to it while I was an Adjunct Lecturer at the University of New South Wales, and while I held a post-doctoral position at the University of Münster. I am grateful to these institutions for their support. I am also thankful to Peter Koellner for giving me helpful feedback on an early draft.

We prove a prime geodesic theorem for compact quotients of affine buildings and apply it to get class number asymptotics for global fields of positive characteristic.

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