Euler characteristic–zeta value identity for unimodular t.d.l.c. groups

Determine, for unimodular totally disconnected locally compact groups G that satisfy the double coset property with respect to a compact open subgroup O, whether the Dirichlet series ζ_{G,O}(s) admits a meromorphic continuation ζ_{G,O}: C → C ∪ {∞} satisfying the identity χ̃_G = ζ_{G,O}(-1)^{-1} · μ_O, where χ̃_G is the rational discrete Euler–Poincaré characteristic defined in the paper and μ_O is the Haar measure normalized by μ_O(O)=1.

Background

In several structured settings (buildings of uniform thickness, parahoric/pro-p radical subgroups in p-adic groups), the authors prove that ζ_{G,O}(s) is meromorphic and that its value at s = −1 is inversely related to the Euler–Poincaré characteristic measure χ̃_G. This mirrors classical identities linking growth/zeta functions to Euler characteristics.

The open problem asks for a general characterization of unimodular t.d.l.c. groups and compact open subgroups O where this identity holds.