Geometric zeta functions of locally symmetric spaces

Abstract

Abstract. The theory of geometric zeta functions for locally symmetric spaces as initialized by Selberg and continued by numerous mathematicians is generalized to the case of higher rank spaces. We show analytic continuation, describe the divisor in terms of tangential cohomology and in terms of group cohomology which generalizes the Patterson conjecture. We also extend the range of zeta functions in considering higher dimensional flats.

A prime geodesic theorem for SL4

“…Proof: Noting that 2 dim n + dim k = 14 we see that this follows from Proposition 2.5 of [8]. This proposition was proved in the case that the function f is an Euler-Poincaré function for some finite dimensional representation of M .…”

“…Further, we require that at each point ab ∈ A − B the function takes the value l j+1 a e −sla . In [8] a function g j s is constructed with these properties, with the one difference that there the positive Weyl chamber A + is used instead of the negative Weyl chamber A − . With only very minor modification the construction in [8] will yield a function with our required properties, which we shall also call g j s .…”

“…In [8] a function g j s is constructed with these properties, with the one difference that there the positive Weyl chamber A + is used instead of the negative Weyl chamber A − . With only very minor modification the construction in [8] will yield a function with our required properties, which we shall also call g j s . Let η : N → R be a smooth, non-negative function of compact support, which is invariant under conjugation by elements of K M and satisfies…”

“…In [8] the following equation from [7], proved under the assumption that Γ is torsion free, is used:…”

“…A subgroup of G is weakly neat if every element is. The results of [8] use the assumptions that the group G has trivial centre and the group Γ is weakly neat. Note that, if G has trivial centre then Γ weakly neat implies Γ torsion free, since any non weakly neat torsion elements must be central.…”