Tyakal N. Venkataramana
We sketch a simplification of proofs of old results on the arithmeticity of the group generated by opposing integral unipotent radicals in higher rank arithmetic groups
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Let F be a local field of positive characteristic, and let G be either a Heisenberg group over F, or a certain (nonabelian) two-dimensional unipotent group over F. If H is an arithmetic subgroup of G, we provide an explicit description of every automorphism of H. From this description, it follows that every automorphism of H virtually extends to a virtual automorphism of G.
Let G be a unipotent algebraic subgroup of some GL_m(C) defined over Q. We describe an algorithm for finding a finite set of generators of the subgroup G(Z) = G \cap GL_m(Z). This is based on a new proof of the result (in more general form due to Borel and Harish-Chandra) that such a finite generating set exists.
This is the first of two papers in which we introduce and study two bivariate zeta functions associated to unipotent group schemes over rings of integers of number fields. One of these zeta functions encodes the numbers of isomorphism classes of irreducible complex representations of finite dimensions of congruence quotients of the associated group and the other one encodes the numbers of conjugacy classes of each size of such quotients. In this paper, we show that these zeta functions satisfy Euler factorizations and almost all of their Euler factors are rational and satisfy functional equati…
Fixing an arithmetic lattice $Γ$ in an algebraic group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $Δ$ with $[Γ: Γ\cap Δ] [Δ: Γ\cap Δ] = n$. This growth function gives a new setting where methods of F. Grunewald, D. Segal, and G. C. Smith's "Subgroups of finite index in nilpotent groups" apply to study arithmetic lattices in an algebraic group. In particular, we show that for any unipotent algebraic $\mathbb{Z}$-group with arithmetic lattice $Γ$, the Dirichlet function associated to the commensurability growth function satisfies an Eule…
Arithmetic subgroups of Chevalley group schemes over function fields II: Conjugacy classes of maximal unipotent subgroups
本文研究函数域上 $\mathcal{S}$-算术子群的极大幂单子群共轭类，利用 Picard 群与群秩加以参数化，并通过 Bruhat–Tits 建筑扇形稳定子刻画其结构。
Let $G$ be a semisimple real algebraic Lie group of real rank at least two and $U$ be the unipotent radical of a non-trivial parabolic subgroup. We prove that a discrete Zariski dense subgroup of $G$ that contains an irreducible lattice of $U$ is an arithmetic lattice of $G$. This solves a conjecture of Margulis and extends previous work of Hee Oh.
Similar to the symplectic cases, there is a family of fourteen orthogonal hypergeometric groups with a maximally unipotent monodromy (cf. Table 1.1). We show that two of the fourteen orthogonal hypergeometric groups associated to the pairs of parameters $(0, 0, 0, 0, 0)$, $(\frac{1}{6}, \frac{1}{6}, \frac{5}{6}, \frac{5}{6}, \frac{1}{2})$; and $(0, 0, 0, 0, 0)$, $(\frac{1}{4}, \frac{1}{4}, \frac{3}{4}, \frac{3}{4}, \frac{1}{2})$ are arithmetic. We also give a table (cf. Table 2.1) which lists the quadratic forms $\mathrm{Q}$ preserved by these fourteen hypergeometric groups, and their two line…
The unipotent subgroup of a finite group of Lie type over a prime field Z/pZ comes equipped with a natural set of generators; the properties of the Cayley graph associated to this set of generators have been much studied. In the present paper, we show that the diameter of this Cayley graph is bounded above and below by constant multiples of np + n^2 log p, where n is the rank of the associated Lie group. This generalizes a result of the first author, which treated the case of SL_n(Z/pZ). (Keywords: diameter, Cayley graph, finite groups of Lie type. AMS classification: 20G40, 05C25)
We prove that any non-cocompact irreducible lattice in a higher rank semi-simple Lie group contains a subgroup of finite index, which has three generators.
Based on a result of Singh--Venkataramana, Bajpai--Dona--Singh--Singh gave a criterion for a discrete Zariski-dense subgroup of Sp(2n,Z) to be a lattice. We adapt this criterion so that it can be used in some situations that were previously excluded. We apply the adapted method to subgroups of Sp(6,Z) and Sp(4,Z) that arise as the monodromy groups of hypergeometric differential equations. In particular, we show that out of the 40 maximally unipotent Sp(6) hypergeometric groups more than half are arithmetic, answering a question of Katz in the negative.
The Bohr compactification of an arithmetic group
本文研究算术群的玻尔紧化结构，通过阿贝尔化、列维分解及高R-秩单群情形，将Bohr(Γ)分解为紧李因子与投射有限完备化的乘积或半直积形式。
$ $Abert, Gelander and Nikolov [AGN17] conjectured that the number of generators $d(Γ)$ of a lattice $Γ$ in a high rank simple Lie group $H$ grows sub-linearly with $v = μ(H / Γ)$, the co-volume of $Γ$ in $H$. We prove this for non-uniform lattices in a very strong form, showing that for $2-$generic such $H$'s, $d(Γ) = O_H(\log v / \log \log v)$, which is essentially optimal. While we can not prove a new upper bound for uniform lattices, we will show that for such lattices one can not expect to achieve a better bound than $d(Γ) = O(\log v)$.
The reverse decomposition of unipotents for bivectors
本文针对交换环上一般线性群的第二基本表示，将初等生成元构造为任意矩阵及其逆的初等共轭乘积的多项式表达，并得到了相关矩阵任意列的稳定化定理。
Groups Generated by Root Unipotents: Higher-rank and rank-one
研究由特定幂幺元生成的子群，对\(n\geq3\)，证明\(\Gamma(Q)\)是\(S -\)算术的并给出相关方法；研究一阶族\(\Gamma_q\)，将同余子群问题简化，还给出应用及验证。
We address two questions of Simon Thomas. First, we show that for any n>2 one can find a four generated free subgroup of SLn(Z) which is profinitely dense. More generally, we show that an arithmetic group Γwhich admits the congruence subgroup property, has a profinitely dense free subgroup with an explicit bound of its rank. Next, we show that the set of profinitely dense, locally free subgroups of such an arithmetic group Γis uncountable.
In the study of Fuchsian groups, it is a nontrivial problem to determine a set of generators. Using a dynamical approach we construct for any cocompact arithmetic Fuchsian group a fundamental region in $\mathbf{SL}_2(\mathbb{R})$ from which we determine a set of small generators.
The group U_n(F) of all nxn unipotent upper-triangular matrices over F has derived length d := Ceiling(log_2 (n)), equivalently 2^{d-1} < n <= 2^d. We prove that U_n(F) has a 3-generated subgroup of derived length d, and it has a 2-generated subgroup of derived length d if and only if (21/32)* 2^d < n <= 2^d.
Several musical scales, like the major scale, can be described as finite arithmetic sequences modulo octave, i.e. chunks of an arithmetic sequence in a cyclic group. Hence the question of how many different arithmetic sequences in a cyclic group will give the same support set. We prove that this number is always a totient number and characterize the different possible cases. In particular, there exists scales with an arbitrarily large number of different generators, but none with 14 generators. Some connex results and extensions are also given, for instance on characterization via a Discrete F…
Arithmetic groups are groups of matrices with integral entries. We shall first discuss their origin in number theory (Gauss, Minkowski) and their role in the "reduction theory of quadratic forms". Then we shall describe these groups by generators and relations. The next topic will be: are all subgroups of finite index given by congruence conditions? Finally, we shall discuss rigidity properties of arithmetic groups.
The main component of (constructive) recognition algorithms for black box groups of Lie type in computational group theory is the construction of unipotent elements. In the existing algorithms unipotent elements are found by random search and therefore the running time of these algorithms is polynomial in the underlying field size $q$ which makes them unfeasible for most practical applications \cite{guralnick01.169}. Meanwhile, the input size of recognition algorithms involves only $\log q$. The present paper introduces a new approach to construction of unipotent elements in which the running …
Small diameters and generators for arithmetic lattices in $\mathrm{SL}_2(\mathbb{R})$ and certain Ramanujan graphs
本文证明了 $\mathrm{SL}_{2}(\mathbb{R})$ 中特定算术格具有小覆盖集和小生成元，并在不依赖 Ramanujan 界限的情况下给出了某些 Ramanujan 图直径的 Ramanujan 强度界限。
Relative homology of arithmetic subgroups of $\mathrm{SU}(3)$
本文通过 Euler-Poincaré 特征描述了定义于函数环上的 $\mathrm{SU}(3)$ 算术子群模极大幂零子群共轭类代表系的相对同调群，度量了其同调群与幂零子群同调群余积的差异。
We initiate a new, computational approach to a classical problem: certifying non-freeness of ($2$-generator, parabolic) Möbius subgroups of $\mathrm{SL}(2,\mathbb{Q})$. The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of $\mathrm{SL}(2, R)$ for a localization $R= \mathbb{Z}[\frac{1}{b}]$ of $\mathbb{Z}$. We prove that a Möbius subgroup $G$ is not free by showing that it has finite index in the relevant $\mathrm{SL}(2, R)$. Further information about the structure of $G$ is obtained; for example, we compute the minimal subgroup of finite index …
Symplectic Hypergeometric Groups of Degree Six
本文研究了六次辛超几何群的算术性，通过计算多项式对并扩展搜索范围，证明了在具有极大幂单单值群的40个群中至少18个、以及剩余246个群中160个群在Sp(6)中是算术的。
Thin monodromy in $\mathrm{Sp}(4)$ and $\mathrm{Sp}(6)$
本文利用计算机辅助乒乓方法研究 Sp(4) 和 Sp(6) 型超几何群的薄性，完成了 Sp(4) 全部及 Sp(6) 中 40 个极大幂单情形的分类，并首次给出 63 个实秩三的非算术例子。
In order to enumerate the fake projective planes, as announced in~\cite{CS}, we found explicit generators and a presentation for each maximal arithmetic subgroup $\barΓ$ of~$PU(2,1)$ for which the (appropriately normalized) covolume equals~$1/N$ for some integer~$N\ge1$. Prasad and Yeung \cite{PY1,PY2} had given a list of all such $\barΓ$ (up to equivalence). The generators were found by a computer search which uses the natural action of $PU(2,1)$ on the unit ball $B(\C^2)$ in~$\C^2$. Our main results here give criteria which ensure that the computer search has found sufficiently many elements…
We give estimates on the number $AL_H(x)$ of arithmetic lattices $Γ$ of covolume at most $x$ in a simple Lie group $H$. In particular, we obtain a first concrete estimate on the number of arithmetic 3-manifolds of volume at most $x$. Our main result is for the classical case $H=PSL(2,R)$ where we compute the limit of $\log AL_H(x) / x\log x$ when $x\to\infty$. The proofs use several different techniques: geometric (bounding the number of generators of $Γ$ as a function of its covolume), number theoretic (bounding the number of maximal such $Γ$) and sharp estimates on the character values of th…
We describe a connection between the combinatorics of generators for certain groups and the combinatorics of Helly's 1913 theorem on convex sets. We use this connection to prove fixed point theorems for actions of these groups on nonpositively curved metric spaces. These results are encode d in a property that we introduce called ``property $\FA_r$'', which reduces to Serre's property $\FA$ when $r=1$. The method applies to $S$-arithmetic groups in higher $\Q$-rank, to simplex reflection groups (including some non-arithmetic ones), and to higher rank Chevalley groups over polynomial and other …
To each of the symmetry groups of the Platonic solids we adjoin a carefully designed involution yielding topological generators of PU(2) which have optimal covering properties as well as efficient navigation. These are a consequence of optimal strong approximation for integral quadratic forms associated with certain special quaternion algebras and their arithmetic groups. The generators give super efficient 1-qubit quantum gates and are natural building blocks for the design of universal quantum gates.
Let X be a group with identity e, let A be an infinite set of generators for X, and let (X,d_A) be the metric space with the word metric d_A induced by A. If the diameter of the space is infinite, then for every positive integer h there are infinitely many elements x in X with d_A(e,x)=h. It is proved that if P is a nonempty finite set of prime numbers and A is the set of positive integers whose prime factors all belong to P, then the diameter of the metric space (\Z,d_A) is infinite. Let λ_A(h) denote the smallest positive integer x with d_A(e,x)=h. It is an open problem to compute λ_A(h) and…
Let k be a global field and let k_v be the completion of k with respect to v, a non-archimedean place of k. Let \mathbf{G} be a connected, simply-connected algebraic group over k, which is absolutely almost simple of k_v-rank 1. Let G=\mathbf{G}(k_v). Let Γbe an arithmetic lattice in G and let C=C(Γ) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for \Gamm$ by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is \ha…
We study certain polynomial trace identities in the group $SL(2,\IC)$ and their application in the theory of discrete groups. We obtain canonical representations for two generator groups in §4 and then in §5 we give a new proof for Gehring and Martin's polynomial trace identities for good words, and extend that result to a larger class which is also closed under a semigroup operation inducing polynomial composition. This new approach is through the use of quaternion algebras over indefinites and an associated group of units. We obtain structure theorems for these quaternion algebras which appe…
We generalize an algorithm established in earlier work \cite{algebrapaper} to compute finitely many generators for a subgroup of finite index of an arithmetic group acting properly discontinuously on hyperbolic space of dimension $2$ and $3$, to hyperbolic space of higher dimensions using Clifford algebras. We hence get an algorithm which gives a finite set of generators of finite index subgroups of a discrete subgroup of Vahlen's group, i.e. a group of $2$-by-$2$ matrices with entries in the Clifford algebra satisfying certain conditions. The motivation comes from units in integral group ring…
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