Eigenvalue-1 locus and abscissa of convergence for double-coset zeta functions

Determine, for a group G acting on a tree T that is either weakly locally ∞-transitive or (P)-closed and for any t in T such that the double-coset zeta function ζ_{G,G_t,G_t}(s) admits the determinant representation ζ_{G,G_t,G_t}(s) = det(I − E^•(s) + U^•_{t,t}(s)) / det(I − E^•(s)) with E^•(s) and U^•_{t,t}(s) explicit matrices of entire functions, the set of complex numbers s at which the matrices E^•(s) and E^•(s) − U^•_{t,t}(s) have eigenvalue 1. Provided ζ^•_{G,G_t,G_t}(s) is an infinite Dirichlet series, ascertain its abscissa of convergence, i.e., the maximal real r such that ζ^•_{G,G_t,G_t}(s) has a pole at s = r.

Background

The paper derives explicit determinant formulae for double-coset zeta functions associated with groups acting on trees in two settings: weakly locally ∞-transitive actions and (P)-closed actions. In this framework, the zeta function ζ{G,G{t_1},G_{t_2}}(s) is expressed as a quotient of determinants involving matrices E^•(s) (a weighted adjacency-type matrix depending on local data) and U^•_{t_1,t_2}(s) (a perturbation determined by the choice of stabilisers).

Corollary C shows that poles (respectively zeros) of ζ{G,G_t,G_t}^•(s) correspond precisely to values of s for which 1 is an eigenvalue of E^•(s) (respectively E^•(s) − U^•{t,t}(s)) but not of the other matrix. Motivated by this spectral characterisation, the authors raise the problem of identifying the complex parameters s where these matrices have eigenvalue 1 and, when the zeta function defines an infinite Dirichlet series, determining its abscissa of convergence.