Globally maximal arithmetic groups

Finiteness of arithmetic hyperbolic reflection groups

We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.

Finite arithmetic subgroups of GL_n

We discuss the following conjecture of Kitaoka: if a finite subgroup $G$ of $GL_{n}(O_{K})$ is invariant under the action of $Gal(K/\Bbb Q)$ then it is contained in $GL_{n}(K^{ab})$. Here $O_{K}$ is the ring of integers in a finite, Galois extension $K$ of $\Bbb Q$ and $K^{ab}$ is the maximal, abelian subextension of $K$. Our main result reduces this conjecture to a special case of elementary abelian $p-$groups $G$. Also, we construct some new examples which negatively answer a question of Kitaoka.

Effective bounds for Vinberg's algorithm for arithmetic hyperbolic lattices

A group of isometries of a hyperbolic $n$-space is called a reflection group if it is generated by reflections in hyperbolic hyperplanes. Vinberg gave a semi-algorithm for finding a maximal reflection sublattice in a given arithmetic subgroup of $O(n,1)$ of the simplest type. We provide an effective termination condition for Vinberg's semi-algorithm with which it becomes an algorithm for finding maximal reflection sublattices. The main new ingredient of the proof is an upper bound for the number of faces of an arithmetic hyperbolic Coxeter polyhedron in terms of its volume.

A classification of $\mathbb C$-Fuchsian subgroups of Picard modular groups

Given an imaginary quadratic extension $K$ of $\mathbb Q$, we give a classification of the maximal nonelementary subgroups of the Picard modular group $\operatorname{PSU}_{1,2}(\mathcal O_K)$ preserving a complex geodesic in the complex hyperbolic plane $\mathbb H^2_\mathbb C$. Complementing work of Holzapfel, Chinburg-Stover and Möller-Toledo, we show that these maximal $\mathbb C$-Fuchsian subgroups are arithmetic, arising from a quaternion algebra $\Big(\!\begin{array}{c} D\,,D_K\\\hline\mathbb QQ\end{array} \!\Big)$ for some explicit $D\in\mathbb N-\{0\}$ and $D_K$ the discriminant of $K$.…

群扩张中的算术

Arithmetic in Group Extensions

本文提出一种适用于具有非平凡可解根的群的 pc 表示推广，在内存和算术上更高效，并用于构造怪物群的极大子群及计算其未知特征标表。

Zariski Density and Computing in Arithmetic Groups

For $n > 2$, let $Γ$ denote either $SL(n, Z)$ or $Sp(n, Z)$. We give a practical algorithm to compute the level of the maximal principal congruence subgroup in an arithmetic group $H\leq Γ$. This forms the main component of our methods for computing with such arithmetic groups $H$. More generally, we provide algorithms for computing with Zariski dense groups in $Γ$. We use our GAP implementation of the algorithms to solve problems that have emerged recently for important classes of linear groups.

Counting arithmetic lattices and surfaces

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…

Arithmetic Monodromy in Sp(2n)

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.

双曲混合流形中极大测地子流形的有限性

Finiteness of Maximal Geodesic Submanifolds in Hyperbolic Hybrids

该研究证明大类非算术双曲n-流形（含Gromov等的混合流形）的有限体积浸入全测地超曲面有限，且高余维下维数≥2的极大测地子流形有限，证明结合算术群结构、动力系统与负曲率几何。

Outer automorphisms of algebraic groups and determining groups by their maximal tori

We give a cohomological criterion for existence of outer automorphisms of a semisimple algebraic group over an arbitrary field. This criterion is then applied to the special case of groups of type D_2n over a global field, which completes some of the main results from the paper "Weakly commensurable arithmetic groups and isospectral locally symmetric spaces" (Pub. Math. IHES, 2009) by Prasad and Rapinchuk and gives a new proof of a result from another paper by the same authors.

Trees, quaternion algebras and modular curves

We study the action on the Bruhat-Tits tree of unit groups of maximal orders in certain quaternion algebras over $\mathbb{F}_q(T)$ and discuss applications to arithmetic geometry and group theory.

格点在$S$-算术群中的流形作用

Actions of lattices in $p$-adic and $S$-arithmetic groups on manifolds

本文证明，当维数小于秩时，简单$p$-adic群中格点在紧流形上的$C^1$微分同胚作用是有限的，并将此结果推广到全不连通的$S$-算术群中的格点。

极大子群的指数几乎为奇数的有限群

Finite groups whose maximal subgroups have almost odd index

本文研究了由群不变量的算术性质决定有限群结构的问题，确定了极大子群的指数不被4整除的有限群的结构，并由此得到了一些新的2-幂零性准则。