Zeta functions of groups and rings – recent developments

Voll, Christopher

doi:10.1017/cbo9781316227343.031

Cited by 8 publications

(8 citation statements)

References 42 publications

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“…The current paper forms part of a series of papers on representation growth; see [3,4,5,6]. For surveys, see [36,57]. It was shown in [41] that the congruence completion H( O S ) has polynomial representation growth.…”

Section: Introduction and Discussion Of Main Resultsmentioning

confidence: 99%

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Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A₂

Avni¹,

Klopsch²,

Onn³

et al. 2016

Proc. London Math. Soc.

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Abstract. We compute explicitly Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of various p-adic analytic and adèlic profinite groups of type A2. This has consequences for the representation zeta functions of arithmetic groups Γ ⊂ H(k), where k is a number field and H a k-form of SL3: assuming that Γ possesses the strong Congruence Subgroup Property, we obtain precise, uniform estimates for the representation growth of Γ. Our results are based on explicit, uniform formulae for the representation zeta functions of the p-adic analytic groups SL3(o) and SU3(o), where o is a compact discrete valuation ring of characteristic 0. These formulae build on our classification of similarity classes of integral p-adic 3 × 3 matrices in gl 3 (o) and gu 3 (o), where o is a compact discrete valuation ring of arbitrary characteristic. Organising the similarity classes by invariants which we call their shadows allows us to combine the Kirillov orbit method with Clifford theory to obtain explicit formulae for representation zeta functions. In a different direction we introduce and compute certain similarity class zeta functions.Our methods also yield formulae for representation zeta functions of various finite subquotients of groups of the form SL3(o), SU3(o), GL3(o), and GU3(o), arising from the respective congruence filtrations; these formulae are valid in case that the characteristic of o is either 0 or sufficiently large. Analysis of some of these formulae leads us to observe p-adic analogues of 'Ennola duality'.

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Section: Introduction and Discussion Of Main Resultsmentioning

confidence: 99%

“…The current paper forms part of a series of papers on representation growth; see [3,4,5,6]. For surveys, see [36,57].…”

Section: Introduction and Discussion Of Main Resultsmentioning

confidence: 99%

Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A₂

Avni¹,

Klopsch²,

Onn³

et al. 2016

Proc. London Math. Soc.

Self Cite

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show abstract

“…Example 2•1 (Subalgebra and submodule zeta functions [6]). The following goes back to [9,18], see [21] for a survey; we use the formalism from [13, section 2•1].…”

Section: •1 Local Maps Of Denef Typementioning

confidence: 99%

Stability results for local zeta functions of groups algebras, and modules

Rossmann

2017

Math. Proc. Camb. Phil. Soc.

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Various types of local zeta functions studied in asymptotic group theory admit two natural operations: (1) change the prime and (2) perform local base extensions. Often, the effects of both of these operations can be expressed simultaneously in terms of a single explicit formula. We show that in these cases, the behaviour of local zeta functions under variation of the prime in a set of density 1 in fact completely determines these functions for almost all primes and, moreover, it also determines their behaviour under local base extensions. We discuss applications to topological zeta functions, functional equations, and questions of uniformity.

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“…In the following, we are primarily interested in the case that R is finite and Linial and Weitz [65] gave the following formula for ask(M d×e (F q )); the same result appeared (with a different proof) in a recent paper by Fulman and Goldstein [36,Lemma 3.2] which also contains further examples of ask(M ). surveys of this area, see [60,92,93]. Conversely, our study of ask zeta functions contributes to asymptotic group theory: we will see that orbit-counting and conjugacy class zeta functions of suitable groups are instances of ask zeta functions.…”

Section: Introductionmentioning

confidence: 96%

“…In contrast to the probabilistic flavour of the work on the numbers

ask (M ∣ V)

cited above, our investigations of the functions

{sans-serifZ}_{M}^{sans-serifask} false(T false)

draw upon results and techniques that have been previously applied in asymptotic group theory and, specifically, the theory of zeta functions (representation zeta functions, in particular) of groups and other algebraic structures; for recent surveys of this area, see . Conversely, our study of ask zeta functions contributes to asymptotic group theory: we will see that orbit‐counting and conjugacy class zeta functions of suitable groups are instances of ask zeta functions.…”

Section: Introductionmentioning

confidence: 99%

The average size of the kernel of a matrix and orbits of linear groups

Rossmann

2018

Proc. London Math. Soc.

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Let frakturO be a compact discrete valuation ring of characteristic 0. Given a module M of matrices over frakturO, we study the generating function encoding the average sizes of the kernels of the elements of M over finite quotients of frakturO. We prove rationality and establish fundamental properties of these generating functions and determine them explicitly for various natural families of modules M. Using p‐adic Lie theory, we then show that special cases of these generating functions enumerate orbits and conjugacy classes of suitable linear pro‐p groups.

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Zeta functions of groups and rings – recent developments

Abstract: Abstract. I survey some recent developments in the theory of zeta functions associated to infinite groups and rings, specifically zeta functions enumerating subgroups and subrings of finite index or finite-dimensional complex representations.

Cited by 8 publications

References 42 publications

Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A₂

Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A₂

Stability results for local zeta functions of groups algebras, and modules

The average size of the kernel of a matrix and orbits of linear groups

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Zeta functions of groups and rings – recent developments

Abstract: Abstract. I survey some recent developments in the theory of zeta functions associated to infinite groups and rings, specifically zeta functions enumerating subgroups and subrings of finite index or finite-dimensional complex representations.

Cited by 8 publications

References 42 publications

Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A2

Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A2

Stability results for local zeta functions of groups algebras, and modules

The average size of the kernel of a matrix and orbits of linear groups

Contact Info

Product

Resources

About

Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A₂

Similarity classes of integral p-adic matrices and representation zeta functions of groups of type A₂