Related papers: Lower Bounds for Representation Growth
We give some background on uniform pro-p groups and the model theory of profinite NIP groups.
We give a survey on the theory of representation-finite and certain minimal representation-infinite algebras.The main goals are the existence of multiplicative bases and of coverings with good properties. Both are attained via…
We establish a sufficient condition for a finitely generated pro-$p$ group to be accessible in terms of finite generation of the module of ends.
We study the set of normalized multi-lengths for representations of closed surface groups and free groups into $(\mathrm{PSL}_2\mathbf{R})^d$ whose projections to $\mathrm{PSL}_2\mathbf{R}$ are all convex cocompact. These multi-lengths…
We establish an equality between two multiplicities: one in the restriction of tempered representations of a $p$-adic group to its closed subgroup with the same derived group; and one occurring in their corresponding component groups in…
In this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is $0$ for all…
In this paper we introduce a notion of Poincar\'e exponent for isometric representations of discrete groups on Hilbert spaces. Similarly as growth exponents control the geometry this exponent is shown to control the size of spectral gaps.…
We survey recent work ranging around the question in how far a group, or a property of a group, is determined by the set of finite quotient groups. Our focus lies on $S$-arithmetic groups, branch groups, and their relatives.
We give upper bounds for triples of subsets of a finite group such that the triples of elements that multiply to 1 form a perfect matching. Our bounds are the first to give exponential savings in powers of an arbitrary finite group.…
The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this…
We investigate the Jacobian decomposition of some algebraic curves over finite fields with genus $4$, $5$ and $10$. As a corollary, explicit equations for curves that are either maximal or minimal over the finite field with $p^2$ elements…
This article surveys recent advances and future challenges in the $2$-representation theory of finitary $2$-categories with a particular emphasis on problems related to classification of various classes of $2$-representations.
We determine the lower rank of the direct product of finitely many hereditarily just infinite profinite groups of finite lower rank.
The nonnegative and positive semidefinite (PSD-) ranks are closely connected to the nonnegative and positive semidefinite extension complexities of a polytope, which are the minimal dimensions of linear and SDP programs which represent this…
We study finite dimensional representations of the projective modular group. Various explicit dimension formulas are given.
The point of view of these notes on the topic is to bring out the flavour that Representation Theory is an extension of the first course on Group Theory. We also emphasize the importance of the base field. These notes cover completely the…
The solvable Farb growth of a group quantifies how well-approximated the group is by its finite solvable quotients. In this note we present a new characterization of polycyclic groups which are virtually nilpotent. That is, we show that a…
Let $\mathbb{F}_q$ be a finite field of order $q$, where $q$ is a power of a prime. For a set $A \subset \mathbb{F}_q$, under certain structural restrictions, we prove a new explicit lower bound on the size of the product set $A(A + 1)$.…
We consider limits over categories of extensions and show how certain well-known functors on the category of groups turn out as such limits. We also discuss higher (or derived) limits over categories of extensions.
We discuss the possibility of very regular subgroups of a Lie group, in presence of an index figure. Further, representations that reduce action to a very regular boundary.