Michael Larsen, Alexander Lubotzky
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 Lie groups arithmetic groups.
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We announce the folowing result: Any finitely generated non virtually solvable linear group over a field of characteristic zero has uniform exponential growth.
We give a description of non-growing subsets in linear groups, which extends the Product theorem for simple groups of Lie type. We also give an account of various related aspects of growth in linear groups.
For a residually finite group $G$, its normal subgroups $G\supset G_1\supset G_2\cdots$ with $\cap_{n\in\mathbb N}G_n=\{e\}$ and for a growth function $γ$ we construct a unitary representation $π_γ$ of $G$. For the minimal growth, $π_γ$ is weakly equivalent to the regular representation, and for the maximal growth it is weakly equivalent to the direct sum of the quasiregular representations on the quotients $G/G_n$. In the case of intermediate growth we show two examples of different behaviour of $π_γ$.
We introduce a new method of proving upper estimates of growth of finitely generated groups and constructing groups of intermediate growth using graphs of their actions. These estimates are of the form $\exp(n^α)$ for some $α<1$, and provide the first examples of such bounds for simple groups of intermediate growth.
Residual finiteness growth measures how well-approximated a group is by its finite quotients. We prove that some related growth functions characterize linearity for a class of groups including all hyperbolic groups.
We show that every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) has linear growth. This implies that the the corresponding semigroup algebra is a PI algebra.
We investigate the conjugacy growth of finitely generated linear groups. We show that finitely generated non-virtually-solvable subgroups of GL_d have uniform exponential conjugacy growth and in fact that the number of distinct polynomials arising as characteristic polynomials of the elements of the ball of radius n for the word metric has exponential growth rate bounded away from 0 in terms of the dimension d only.
Representation growth of maximal class groups: various exceptional cases
本文作为极大类群表示增长研究的续作，通过构造性方法算出$M_n$族群$p$-局部表示zeta函数的例外情形，构造相关群的指定次数不可约表示，结合前序结果完整阐明$M_3$、$M_4$的不可约表示与全局zeta函数。
This article examines lower bounds for the representation growth of finitely generated (particularly profinite and pro-p) groups. It also considers the related question of understanding the maximal multiplicities of character degrees in finite groups, and in particular simple groups.
We give a short introduction to the subject of representation growth and representation zeta functions of groups, omitting all proofs. Our focus is on results which are relevant to the study of arithmetic groups in semisimple algebraic groups, such as the special linear group of degree n over the ring of integers. In the last two sections we state several results which were recently obtained in joint work with N. Avni, U. Onn and C. Voll.
We study the representation growth of alternating and symmetric groups in positive characteristic and restricted representation growth for the finite groups of Lie type. We show that the the number of representations of dimension at most n is bounded by a low degree polynomial in n. As a consequence, we show that the number of conjugacy classes of maximal subgroups of a finite almost simple group G is at most O(log|G|).
We prove that the residual girth of any finitely generated linear group is at most exponential. This means that the smallest finite quotient in which the $n$-ball injects has at most exponential size. If the group is also not virtually nilpotent, it follows that the residual girth is precisely exponential.
Let G be an arithmetic lattice in a semisimple algebraic group over a number field. We show that if G has the congruence subgroup property, then the number of n-dimensional irreducible representations of G grows like n^a, where a is a rational number.
Applications of $C^*$-Entropy to Representation Growth for Discrete Groups
该研究将拓扑1-有界熵赋予离散群全群$C^*$-代数的表示论解释，利用该熵框架强化相关表示增长结果，证明$SL_n(\mathbb{Z})$具有次指数表示增长。
Let $G$ be a real algebraic semi-simple Lie group and $Γ$ be the fundamental group of a compact negatively curved manifold. In this article we study the limit cone, introduced by Benoist, and the growth indicator function, introduced by Quint, for a class of representations $ρ:Γ\to G$ admitting a equivariant map from $\partialΓ$ to the Furstenberg boundary of $G$'s symmetric space together with a transversality condition. We then study how these objects vary with the representation.
Let $Γ$ be a group and $r_n(Γ)$ the number of its $n$-dimensional irreducible complex representations. We define and study the associated representation zeta function $\calz_Γ(s) = \suml^\infty_{n=1} r_n(Γ)n^{-s}$. When $Γ$ is an arithmetic group satisfying the congruence subgroup property then $\calz_Γ(s)$ has an ``Euler factorization". The "factor at infinity" is sometimes called the "Witten zeta function" counting the rational representations of an algebraic group. For these we determine precisely the abscissa of convergence. The local factor at a finite place counts the finite representati…
Methods from additive number theory are applied to construct families of finitely generated linear semigroups with intermediate growth.
We explain and correct a mistake in Section 2.6 and Appendix C of the first and second author's paper "Representation Growth and Rational Singularities of the Moduli Space of Local Systems" arXiv:1307.0371.
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 several questions.
We prove that the semistability growth of hyperbolic groups is linear, which implies that hyperbolic groups which are sci (simply connected at infinity) have linear sci growth. Based on the linearity of the end-depth of finitely presented groups we show that the linear sci is preserved under amalgamated products over finitely generated one-ended groups. Eventually one proves that most non-uniform lattices have linear sci.
We adapt methods from quiver representation theory and Hall algebra techniques to the counting of representations of virtually free groups over finite fields. This gives rise to the computation of the E-polynomials of $\mathbf{GL}_d(\mathbb{C})$-character varieties of virtually free groups. As examples we discuss the representation theory of $\mathbb{D}_\infty$ , $\mathbf{PSL}_2(\mathbb{Z})$ , $\mathbf{SL}_2(\mathbb{Z})$ , $\mathbf{GL}_2(\mathbb{Z})$ and $\mathbf{PGL}_2(\mathbb{Z})$ .
The deep theory of approximate subgroups establishes 3-step product growth for subsets of finite simple groups $G$ of Lie type of bounded rank. In this paper we obtain 2-step growth results for representations of such groups $G$ (including those of unbounded rank), where products of subsets are replaced by tensor products of representations. Let $G$ be a finite simple group of Lie type and $χ$ a character of $G$. Let $|χ|$ denote the sum of the squares of the degrees of all (distinct) irreducible characters of $G$ which are constituents of $χ$. We show that for all $δ>0$ there exists $ε>0$, in…
In this paper we give a bound to the number of conjugacy classes of maximal subgroups of any almost simple group whose socle is a classical group of Lie type. The bound is $2n^{5.2}+n\log_2\log_2 q$, where $n$ is the dimension of the classical socle and $q$ is the size of the defining field. To obtain the bound, we first bound the number of projective cross-characteristic representations of simple groups of Lie type as a function of the representation degree. These bounds are computed for different families of groups separately. In the computation, we use information on class numbers, minimal …
We prove that a linear growth graph has finitely many horofunctions. This provides a short and simple proof that any finitely generated infinite group of linear growth is virtually cyclic.
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 amounts to a study of the abscissae of convergence of the corresponding (twist) representation zeta functions. We determine the abscissae for a class of Mellin zeta functions which include the Witten zeta functions. As a special case, we obtain a new proof of the theorem of Larsen and Lubotzky that the abscissa of Witten zeta functions is $r/κ$, where $r$ is…
If a finitely generated monoid M is defined by a finite number of degree-preserving relations, then it has linear growth if and only if it can be decomposed into a finite disjoint union of subsets (which we call "sandwiches") of the form $a<w>b$, where $a,b,w$ are elements of $M$ and $<w>$ denotes the monogenic semigroup generated by $w$. Moreover, the decomposition can be chosen in such a way that the sandwiches are either singletons or "free" ones (meaning that all elements $a w^n b$ in each sandwich are pairwise different). So, the minimal number of free sandwiches in such a decomposition i…
Let $G$ be a virtually special group. Then the residual finiteness growth of $G$ is at most linear. This result cannot be found by embedding $G$ into a special linear group. Indeed, the special linear group $\text{SL}_k(\mathbb{Z})$, for $k > 2$, has residual finiteness growth $n^{k-1}$.
Sublinear growth of 1-cocycles and uniform convexity
研究1-余弦在有限生成群上的亚线性增长特性，证明其与Banach空间的均匀凸性相关联。
Consider an arithmetic group $\mathbf{G}(O_S)$, where $\mathbf{G}$ is an affine group scheme with connected, simply connected absolutely almost simple generic fiber, defined over the ring of $S$-integers $O_S$ of a number field $K$ with respect to a finite set of places $S$. For each $n \in \mathbb{N}$, let $R_n(\mathbf{G}(O_S))$ denote the number of irreducible complex representations of $\mathbf{G}(O_S)$ of dimension at most $n$. The degree of representation growth $α(\mathbf{G}(O_S)) = \lim_{n \rightarrow \infty} \log R_n(\mathbf{G}(O_S)) / \log n$ is finite if and only if $\mathbf{G}(O_S)$…
We develop a theory of levels for irreducible representations of symmetric groups of degree $n$ analogous to the theory of levels for finite classical groups. A key property of level is that the level of a character, provided it is not too big compared to $n$, gives a good lower bound on its degree, and, moreover, every character of low degree is either itself of low level or becomes so after tensoring with the sign character. Furthermore, if $l_1$ and $l_2$ satisfy a linear upper bound in $n$, then the maximal level of composition factors of the tensor product of representations of levels $l_…
On growth of cocycles of isometric representations on $L^p$-spaces
本文研究了$L^p$-空间上等距表示1-cocycles的渐近增长，证明了对于紧生成群G，要么所有1-cocycles有界（FL^p性质），要么存在增长较快的1-cocycle，并给出了调和1-cocycles的平均增长界及Liouville性质的条件。
In this work we study the structure of finitely generated groups for which a space of harmonic functions with fixed polynomial growth is finite dimensional. It is conjectured that such groups must be virtually nilpotent (the converse direction to Kleiner's theorem). We prove that this is indeed the case for solvable groups. The investigation is partly motivated by Kleiner's proof for Gromov's theorem on groups of polynomial growth.
We show that the space of harmonic functions on a finitely generated infinite group G is finite dimensional if, and only if, G has a finite-index subgroup isomorphic to the integers. A key tool is Wilkie and van den Dries's quantitative version of the linear-growth case of Gromov's theorem on groups of polynomial growth.
We relate the asymptotic representation theory of $SL(d,\mathbb{Z}_p)$ and the singularities of the moduli space of $SL(d)$-local systems on a smooth projective curve, proving new theorems about both. Regarding the former, we prove that, for every d, the number of n-dimensional representations of $SL(d,\mathbb{Z}_p)$ grows slower than $n^{22}$, confirming a conjecture of Larsen and Lubotzky. Regarding the latter, we prove that the moduli space of $SL(d)$-local systems on a smooth projective curve of genus at least 12 has rational singularities. Most of our results apply more generally to semi-…
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