The topic of this course is the discrete subgroups of semisimple Lie groups. We discuss a criterion that ensures that such a subgroup is arithmetic. This criterion is a joint work with Sébastien Miquel, which extends previous work of Selberg and Hee Oh and solves an old conjecture of Margulis. We focus on concrete examples like the group $\mathrm {SL}(d,{\mathbb {R}})$ and we explain how classical tools and new techniques enter the proof: the Auslander projection theorem, the Bruhat decomposition, the Mahler compactness criterion, the Borel density theorem, the Borel–Harish-Chandra finiteness theorem, the Howe–Moore mixing theorem, the Dani–Margulis recurrence theorem, the Raghunathan–Venkataramana finite-index subgroup theorem and so on.

We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of $\mathbf{SL}_{n}(\mathbb{Z})$. This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group $\mathbf{SL}_{n}(K)$ for $K$ a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group $\mathbf{SL}_{n}(\mathbb{Z})$.

We prove two results about the width of words in $\operatorname{SL}_{n}(\mathbb{Z})$. The first is that, for every $n\geqslant 3$, there is a constant $C(n)$ such that the width of any word in $\operatorname{SL}_{n}(\mathbb{Z})$ is less than $C(n)$. The second result is that, for any word $w$, if $n$ is big enough, the width of $w$ in $\operatorname{SL}_{n}(\mathbb{Z})$ is at most 87.

Based on a simple object, an i.i.d. sequence of positive integer-valued random variables {an}n∊ℤ, we introduce and study two random structures and their connections. First, a population dynamics, in which each individual is born at time n and dies at time n + an. This dynamics is that of a D/GI/∞ queue, with arrivals at integer times and service times given by {an}n∊ℤ. Second, the directed random graph Tf on ℤ generated by the random map f(n) = n + an. Assuming only that E [a0] < ∞ and P [a0 = 1] > 0, we show that, in steady state, the population dynamics is regenerative, with one individual alive at each regeneration epoch. We identify a unimodular structure in this dynamics. More precisely, Tf is a unimodular directed tree, in which f(n) is the parent of n. This tree has a unique bi-infinite path. Moreover, Tf splits the integers into two categories: ephemeral integers, with a finite number of descendants of all degrees, and successful integers, with an infinite number. Each regeneration epoch is a successful individual such that all integers less than it are its descendants of some order. Ephemeral, successful, and regeneration integers form stationary and mixing point processes on ℤ.

We give several algorithms for finitely generated subgroups of the modular group PSL2(ℤ) given by sets of generators. First, we present an algorithm to check whether a finitely generated subgroup H has finite index in the full modular group. Then we discuss how to parametrise the right cosets of H in PSL2(ℤ), whether the index is finite or not. Further, we explain how an element in H can be written as a word in a given set of generators of H.

It is well known that if a convex hyperbolic polygon is constructed as a fundamental domain for a subgroup of SL(2,ℝ), then its translates by the group form a locally finite tessellation and its side-pairing transformations form a system of generators for the group. Such a hyperbolically convex fundamental domain for any discrete subgroup can be obtained by using Dirichlet’s and Ford’s polygon constructions. However, these two results are not well adapted for the actual construction of a hyperbolically convex fundamental domain due to their nature of construction. A third, and most important and practical, method of obtaining a fundamental domain is through the use of a right coset decomposition as described below. If Γ2 is a subgroup of Γ1 such that Γ1=Γ2⋅{L1,L2,…,Lm} and 𝔽 is the closure of a fundamental domain of the bigger group Γ1, then the set is a fundamental domain of Γ2. One can ask at this juncture, is it possible to choose the right coset suitably so that the set ℛ is a convex hyperbolic polygon? We will answer this question affirmatively for Hecke modular groups.

Let $K$ be a function field of genus $g$ with a finite constant field ${\mathbb{F}}_q$. Choose a place $\infty$ of $K$ of degree $\delta$ and let ${\mathbb{C}}$ be the arithmetic Dedekind domain consisting of all elements of $K$ that are integral outside $\infty$. An explicit formula is given (in terms of $q$, $g$ and $\delta$) for the minimum index of a non-congruence subgroup in SL$_2({\mathcal{C}})$. It turns out that this index is always equal to the minimum index of an arbitrary proper subgroup in SL$_2({\mathcal{C}})$. The minimum index of a normal non-congruence subgroup is also determined.

It is shown that the natural action of the absolute Galois group on the ideals defining principal congruence subgroups of certain nonarithmetic Fuchsian triangle groups is compatible with its action on the algebraic curves that these congruence groups uniformize.

We study a class of subgroups of $\text{PS}{{\text{L}}_{2}}\left( \mathbb{Z} \right)$ which can be characterized in different ways, such as congruence groups, modular forms, modular curves, elliptic surfaces, lattices and graphs.

The Modular Group M is PSL2(Z) the group of linear fractional transformations with integral entries and determinant one. M has been of great interest in many diverse fields of Mathematics, including Number Theory, Automorphic Function Theory and Group Theory. In this paper we give an effective algorithm to determine, for each integer d, a complete set of representatives for the trace classes in trace d. This algorithm depends on the combinatorial group theoretic structure of M. It has been subsequently extended by Sheingorn to the general Hecke groups. The number h(d) of trace classes in trace d is equal to the ideal class number of the field The algorithm mentioned above then provides a new straightforward computational procedure for determining h(d). Finally as an outgrowth of the algorithm we present a wide generalization of the Fermat Two-Square theorem. This last result can also be derived from classical work of Gauss.

Let R be a commutative ring with identity. A subgroup S of GLn(R), where n ≥ 2, is said to be standard if and only if S contains all the q-elementary matrices and all conjugates of those matrices by products of elementary matrices, where q is the ideal in R generated by Xij,xii — Xjj(i ≠ j), for all (xij) ∊ S. It is known that, when n ≧ 3, the standard subgroups of GLn(R) are precisely those normalized by the elementary matrices. To demonstrate how completely this result can break down for n = 2 we prove that GL2(Z), where Z is the ring of rational integers, has uncountably many non-normal, standard subgroups and uncountably many non-standard, normal subgroups.