Finite subgroups of arithmetic lattices in U(2,1)

Notes used for a course held in 2016 in the School of Advances in Group Theory and Applications, for some lectures given in 2018 for the students of the Master in Mathematics of the Vrije Universiteit Brussels, a course for master and Ph.D. students at the Universidade de S\~ao Paulo and at the conference Group algebras, representations and computations. We revise some problems on the study of finite subgroups of the group of units of integral group rings of finite groups and...

In this paper we introduce a particular lattice of subgroups called a "cyclic-diamond" and show that every finite non-cyclic group contains a cyclic-diamond as a sublattice of its lattice of subgroups. Turning to the infinite case, we show that an infinite abelian group does not contain a cyclic-diamond in its subgroup lattice if and only if all of its finitely generated subgroups are cyclic or isomorphic to $\mathbb{Z} \times \mathbb{Z}_{2^N}$ for some $N$.

We study certain lattices constructed from finite abelian groups. We show that such a lattice is eutactic, thereby confirming a conjecture by B\"ottcher, Eisenbarth, Fukshansky, Garcia, Maharaj. Our methods also yield simpler proofs of two known results: First, such a lattice is strongly eutactic if and only if the abelian group has odd order or is elementary abelian. Second, such a lattice has a basis of minimal vectors, except for the cyclic group of order 4.

Given two nonempty subsets $A, B$ of a group $G$, they are said to form a co-minimal pair if $A \cdot B = G$, and $A' \cdot B \subsetneq G$ for any $\emptyset \neq A' \subsetneq A$ and $A\cdot B' \subsetneq G$ for any $\emptyset \neq B' \subsetneq B$. In this article, we show several new results on co-minimal pairs in the integers and the integral lattices. We prove that for any $d\geq 1$, the group $\mathbb{Z}^{2d}$ admits infinitely many automorphisms such that for each suc...

We prove in a large number of cases, that a Zariski dense discrete subgroup of a simple real algebraic group $G$ which contains a higher rank lattice is a lattice in the group $G$. For example, we show that a Zariski dense subgroup of $SL_n({\mathbb R})$ which contains $SL_3({\mathbb Z})$ in the top left hand corner, is conjugate to $SL_n({\mathbb Z})$ .

In the 1940's Graham Higman initiated the study of finite subgroups of the unit group of an integral group ring. Since then many fascinating aspects of this structure have been discovered. Major questions such as the Isomorphism Problem and the Zassenhaus Conjectures have been settled, leading to many new challenging problems. In this survey we review classical and recent results, sketch methods and list questions relevant for the state of the art.

This paper contains both theoretical results and experimental data on the behavior of the dimensions of the cohomology spaces H^1(G,E_n), where Gamma is a lattice in SL(2,C) and E_n is one of the standard self-dual modules. In the case Gamma = SL(2,O) for the ring of integers O in an imaginary quadratic number field, we make the theory of lifting explicit and obtain lower bounds linear in n. We have accumulated a large amount of experimental data in this case, as well as for ...

We study a question of Greenberg-Shalom concerning arithmeticity of discrete subgroups of semisimple Lie groups with dense commensurators. We answer this question positively for normal subgroups of lattices. This generalizes a result of the second author and T. Koberda for certain normal subgroups of arithmetic lattices in SO(n,1) and SU(n,1).

We present detailed summaries of the talks that were given during a week-long workshop on Arithmetic Groups at the Banff International Research Station in April 2013. The vast majority of these reports are based on abstracts that were kindly provided by the speakers. Video recordings of many of the lectures are available online.

In \cite{Ghys} it is proved that any morphism from a subgroup of finite index of $\mathrm{SL}(n,\mathbb{Z})$ to the group of analytic diffeomorphisms of $\mathbb{S}^2$ has a finite image as soon as $n\geq 5$. The case $n=4$ is also claimed to follow along the same arguments; in fact this is not straightforward and this case indeed needs a modification of the argument. In this paper we recall the strategy for $n\geq 5$ and then focus on the case $n=4$.