On lower bounds for cohomology growth in $$p$$ p -adic analytic towers

Abstract

Abstract. Let p and ℓ be two distinct prime numbers and let Γ be a group. We study the asymptotic behaviour of the mod-ℓ Betti numbers in p-adic analytic towers of finite index subgroups. If Θ is a finite ℓ-group of automorphisms of Γ, our main theorem allows to lift lower bounds for the mod-ℓ cohomology growth in the fixed point group Γ Θ to lower bounds for the growth in Γ. We give applications to S-arithmetic groups and we also obtain a similar result for cohomology with rational coefficients. IntroductionThe study of (co)homology growth of a group Γ is concerned with the question how the (co)homology and associated invariants change in towers of finite index (normal) subgroups. In recent years (co)homology growth has become a rather active topic with contributions by Lück [21,22], Lackenby [19,20] and CalegariEmerton [11,12]. In particular, Calegari and Emerton proved upper bound results for mod-p cohomology growth in p-adic analytic towers (see also Bergeron, Linnell, Lück and Sauer [3] for a different perspective).Throughout the article p and ℓ denote two distinct prime numbers. The main result in this article provides a simple way to prove asymptotic lower bounds for the mod-ℓ Betti number growth in p-adic analytic towers of subgroups.1.1. The main result. Let Γ be a group of type (VFP), by definition this means that every torsion-free finite index subgroup is of type (FP). Recall that a group Γ ′ is of type (FP) if the trivial group module Z admits a finite length resolution by finitely generated projective Z[Γ ′ ]-modules (see [10, p.199]). In particular, every torsion-free finite index subgroup of Γ has finite mod-ℓ cohomology. Assume there is an injective group homomorphism ι : Γ → G, where G is a compact p-adic analytic group. We will assume further that the image of ι is dense in G. We fix a finite ℓ-group Θ of continuous automorphisms of G and we assume that the image ι(Γ) is stable under Θ. Thus Θ also acts by automorphisms on the group Γ. From now on we identify Γ with its image under ι. We fix an open, normal, Θ-stable, uniform pro-p subgroup U G (see Lemma 3 below) and we consider the associated lower p-series of U defined by P 1 (U ) := U and P i+1 (U ) := Φ(P i (U )).

Functoriality, Smith theory, and the Brauer homomorphism

“…The paper [33] of the first author, already mentioned, studies a similar story in the setting of the local geometric Satake correspondence, when σ is inner. L. Clozel has pointed out that the arguments of §8 resemble some of the constructions in the theory of twisted endoscopy, as in [24], but in our case these constructions are on the dual side and in characteristic p. The paper of Kionke [23] applies the Smith inequalities to p-adic analytic towers of locally symmetric spaces for G. Finally we draw attention to the paper of Ash [1], which uses Smith theory to produce homology for GL n over certain fields.…”