A Gauss-Bonnet formula for discrete arithmetically defined groups

Harder, Günter

doi:10.24033/asens.1217

Cited by 122 publications

(101 citation statements)

References 4 publications

Supporting

Mentioning

100

Contrasting

Order By: Relevance

“…By using the results of [7], it can be shown, exactly as in [5], that in our normalization of measures, the volume vol (G/Γ) is a rational number. In fact the manifold K\GjΓ has the homotopy type of a compact manifold M, and vol (G/Γ) is a rational multiple of the Euler-Poincare characteristic E of M. Moreover, as observed in [5], the numbers iά k , k>l, are all rational, with denominator depending only on (G, K) and not on k. Thus there exists an integer K > 0 such that i vol (G/Γ)d k = e k E/κ, where e k is an integer, and e k E and id fc are of like sign.…”

Section: -Lj^mentioning

confidence: 99%

Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one

Gangolli

Warner

1980

Nagoya Mathematical Journal

View full text Add to dashboard Cite

IntroductionIn a previous paper [5], one of the present authors has worked out a theory of zeta functions of Selberg's type for compact quotients of symmetric spaces of rank one. In the present paper, we consider the analogues of those results when G/K is a noncompact symmetric space of rank one and Γ is a discrete subgroup of G such that G/Γ is not compact but such that vol(GjΓXoo. Thus, Γ is a non-uniform lattice. Certain mild restrictions, which are fulfilled in many arithmetic cases, will be put on Γ, and we shall consider how one can define a zeta function Z Γ of Selberg's type attached to the data (G, K, Γ).This will be attempted as in [5], by first defining, via the trace formula, the logarithmic derivative z Γ of Z Γ . The main reason why we get somewhere is the fact that in the case rank (GjK) = 1, the parabolic terms in the trace formula can be fully evaluated, for a spherical admissible function /, in terms of the spherical Fourier transform /. The computations necessary for this were performed by the present authors, and were reported in [21].For technical reasons, presently to be explained, we shall eventually exclude the case G = SU(2n, 1). Except for this case, our results for Z Γ are pretty satisfactory. We shall see that Z Γ is a meromorphic function. The location and orders of its zeroes and poles will turn out to contain topological and/or spectral information. It will also be seen that Z Γ enjoys a functional equation which involves not only the Harish-Chandra c-function (which determines the Plancherel measure of GjK) but also the determinant Ψ(s) of the intertwining operator M(s) which occurs in the

show abstract

Section: -Lj^mentioning

confidence: 99%

Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one

Gangolli

Warner

1980

Nagoya Mathematical Journal

View full text Add to dashboard Cite

show abstract

“…We define q H := dim H s (R)/K s H . The following is an easy consequence of Harder's Gauß-Bonnet Theorem: If χ(Γ H ) = 0 for some torsion-free arithmetic subgroup Γ H ⊂ H(Q), then q H is even and χ(Γ H ) has sign (−1) qH /2 for all Γ H (see [15]). …”

Section: Cohomology With Rational Coefficientsmentioning

confidence: 99%

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

Kionke

2013

Math. Z.

View full text Add to dashboard Cite

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 )).

show abstract

“…IX, § §6-7], and we follow it closely. The main theorem underlying this computation is due to Harder [16].…”

Section: A Mass Formula For the Voronoi Complexmentioning

confidence: 99%

On the cohomology of linear groups over imaginary quadratic fields

Sikirić

Gangl

Gunnells

et al. 2016

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

Abstract. Let Γ be the group GL N (O D ), where O D is the ring of integers in the imaginary quadratic field with discriminant D < 0. In this paper we investigate the cohomology of Γ for N = 3, 4 and for a selection of discriminants: D ≥ −24 when N = 3, and D = −3, −4 when N = 4. In particular we compute the integral cohomology of Γ up to p-power torsion for small primes p. Our main tool is the polyhedral reduction theory for Γ developed by Ash [4, Ch. II] and Koecher [18]. Our results extend work of Staffeldt [29], who treated the case n = 3, D = −4. In a sequel [11] to this paper, we will apply some of these results to the computations with the K-groups K 4 (O D ), when D = −3, −4.

show abstract

A Gauss-Bonnet formula for discrete arithmetically defined groups

Cited by 122 publications

References 4 publications

Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one

Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one

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

On the cohomology of linear groups over imaginary quadratic fields

Contact Info

Product

Resources

About