Related papers: Representation growth and representation zeta func…

The representation zeta function of a profinite group $G$ encodes the distribution of continuous irreducible complex representations of $G$ as a function of the dimension. Its abscissa of convergence $\alpha(G)$ describes the polynomial…

We prove that the numbers of irreducible n-dimensional complex continuous representations of the special linear groups over p-adic integers grow slower than the square of n. We deduce that the abscissas of convergence of the representation…

We investigate the rate of growth of the function of n which counts the number of complex irreducible representations of a fixed group of degree less than or equal to n. The emphasis is on linear groups, especially compact real and p-adic…

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.

Inspired by work surrounding Igusa's local zeta function, we introduce topological representation zeta functions of unipotent algebraic groups over number fields. These group-theoretic invariants capture common features of established…

We prove two conjectures regarding the representation growth of groups of type $A_2$. The first, conjectured by Avni, Klopsch, Onn and Voll, regards the uniformity of representation zeta functions over local complete discrete valuation…

We present a conjectured formula for the representation zeta function of the Heisenberg group over $\mathcal{O}[x]/(x^n)$ where $\mathcal{O}$ is the ring of integers of some number field. We confirm the conjecture for $n\leq 3$ and raise…

We study the finite-dimensional continuous complex representations of $\mathrm{SL}_2$ over the ring of integers of non-Archimedean local fields of even residual characteristic. We prove that for characteristic two, the abscissa of…

The main results in this thesis deal with the representation growth of certain classes of groups. In chapter $1$ we present the required preliminary theory. In chapter $2$ we introduce the Congruence Subgroup Problem for an algebraic group…

We compute the representation zeta functions of some finitely generated nilpotent groups associated to unipotent group schemes over rings of integers in number fields. These group schemes are defined by Lie lattices whose presentations are…

We study zeta functions enumerating finite-dimensional irreducible complex linear representations of compact p-adic analytic and of arithmetic groups. Using methods from p-adic integration, we show that the zeta functions associated to…

We study analytic properties of the representation zeta functions of arithmetic groups of type $\mathsf{A}_2$, such as $\textrm{SL}_3(\mathbb{Z})$. In particular, we uncover further poles of these functions and determine a natural boundary…

We study the representation growth of simple compact Lie groups and of $\mathrm{SL}_n(\mathcal{O})$, where $\mathcal{O}$ is a compact discrete valuation ring, as well as the twist representation growth of $\mathrm{GL}_n(\mathcal{O})$. This…

This paper is a sequel to "Representation growth of maximal class groups: non-exceptional primes". We use a constructive method to calculate some exceptional cases of $p$-local representation zeta functions of a family of finitely generated…

We consider the representation growth of split extensions of $SL_2^m(O)$. We prove that the corresponding representation zeta functions factor as a product of the representation zeta function of $SL_2^m(O)$ and the relative representation…

We develop a practical method for computing local zeta functions of groups, algebras, and modules in fortunate cases. Using our method, we obtain a complete classification of generic local representation zeta functions associated with…

We introduce new methods from p-adic integration into the study of representation zeta functions associated to compact p-adic analytic groups and arithmetic groups. They allow us to establish that the representation zeta functions of…

We calculate the representation growth zeta function of the discrete Heisenberg group over the integers of a quadratic number field. This is done by forming equivalence classes of representations, called twist iso-classes, and explicitly…

Let $\Gamma$ be a group and $r_n(\Gamma)$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function $\calz_\Gamma(s) = \suml^\infty_{n=1} r_n(\Gamma)n^{-s}$. When…

We develop the representation theory of a finite semigroup over an arbitrary commutative semiring with unit, in particular classifying the irreducible and minimal representations. The results for an arbitrary semiring are as good as the…