Statistics of subgroups of the modular group

We count the finitely generated subgroups of the modular group [Formula: see text]. More precisely, each such subgroup [Formula: see text] can be represented by its Stallings graph [Formula: see text], we consider the number of vertices of [Formula: see text] to be the size of [Formula: see text] and we count the subgroups of size [Formula: see text]. Since an index [Formula: see text] subgroup has size [Formula: see text], our results generalize the known results on the enumeration of the finite index subgroups of [Formula: see text]. We give asymptotic equivalents for the number of finitely generated subgroups of [Formula: see text], as well as of the number of finite index subgroups, free subgroups and free finite index subgroups. We also give the expected value of the isomorphism type of a size [Formula: see text] subgroup and prove a large deviation statement concerning this value. Similar results are proved for finite index and for free subgroups. Finally, we show how to efficiently generate uniformly at random a size [Formula: see text] subgroup (respectively, finite index subgroup, free subgroup) of [Formula: see text].

Automorphic Schwarzian equations

This paper concerns the study of the Schwartz differential equation {\{h,\tau\}=s\operatorname{E}_{4}(\tau)}, where {\operatorname{E}_{4}} is the weight 4 Eisenstein series and s is a complex parameter. In particular, we determine all the values of s for which the solutions h are modular functions for a finite index subgroup of {\operatorname{SL}_{2}({\mathbb{Z}})}. We do so using the theory of equivariant functions on the complex upper-half plane as well as an analysis of the representation theory of {\operatorname{SL}_{2}({\mathbb{Z}})}. This also leads to the solutions to the Fuchsian differential equation {y^{\prime\prime}+s\operatorname{E}_{4}y=0}.

Contractible, hyperbolic but non-CAT(0) complexes

We prove that almost all arc complexes do not admit a CAT(0) metric with finitely many shapes, in particular any finite-index subgroup of the mapping class group does not preserve such a metric on the arc complex. We also show the analogous statement for all but finitely many disc complexes of handlebodies and free splitting complexes of free groups. The obstruction is combinatorial. These complexes are all hyperbolic and contractible but despite this we show that they satisfy no combinatorial isoperimetric inequality: for any n there is a loop of length 4 that only bounds discs consisting of at least n triangles. On the other hand we show that the curve complexes satisfy a linear combinatorial isoperimetric inequality, which answers a question of Andrew Putman.

Arithmeticity of discrete subgroups

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.

Derived length of zero entropy groups acting on projective varieties in arbitrary characteristic — A remark to a paper of Dinh-Oguiso-Zhang

Let [Formula: see text] be a projective variety of dimension [Formula: see text] over an algebraically closed field of arbitrary characteristic. We prove a Fujiki–Lieberman type theorem on the structure of the automorphism group of [Formula: see text]. Let [Formula: see text] be a group of zero entropy automorphisms of [Formula: see text] and [Formula: see text] the set of elements in [Formula: see text] which are isotopic to the identity. We show that after replacing [Formula: see text] by a suitable finite-index subgroup, [Formula: see text] is a unipotent group of the derived length at most [Formula: see text]. This result was first proved by Dinh et al. for compact Kähler manifolds.

Virtually Cocompactly Cubulated Artin–Tits Groups

We give a conjectural classification of virtually cocompactly cubulated Artin–Tits groups (i.e., having a finite index subgroup acting geometrically on a CAT(0) cube complex), which we prove for all Artin–Tits groups of spherical type, FC type, or two-dimensional type. A particular case is that for $n \geqslant 4$, the $n$-strand braid group is not virtually cocompactly cubulated.

Ordinary primes in Hilbert modular varieties

A well-known conjecture, often attributed to Serre, asserts that any motive over any number field has infinitely many ordinary reductions (in the sense that the Newton polygon coincides with the Hodge polygon). In the case of Hilbert modular cuspforms $f$ of parallel weight $(2,\ldots ,2)$, we show how to produce more ordinary primes by using the Sato–Tate equidistribution and combining it with the Galois theory of the Hecke field. Under the assumption of stronger forms of Sato–Tate equidistribution, we get stronger (but conditional) results. In the case of higher weights, we formulate the ordinariness conjecture for submotives of the intersection cohomology of proper algebraic varieties with motivic coefficients, and verify it for the motives whose $\ell$-adic Galois realisations are abelian on a finite-index subgroup. We get some results for Hilbert cuspforms of weight $(3,\ldots ,3)$, weaker than those for $(2,\ldots ,2)$.

VIRTUAL ALGEBRAIC FIBRATIONS OF KÄHLER GROUPS

This paper stems from the observation (arising from work of Delzant) that “most” Kähler groups $G$ virtually algebraically fiber, that is, admit a finite index subgroup that maps onto $\mathbb{Z}$ with finitely generated kernel. For the remaining ones, the Albanese dimension of all finite index subgroups is at most one, that is, they have virtual Albanese dimension $va(G)\leqslant 1$ . We show that the existence of algebraic fibrations has implications in the study of coherence and higher BNSR invariants of the fundamental group of aspherical Kähler surfaces. The class of Kähler groups with $va(G)=1$ includes virtual surface groups. Further examples exist; nonetheless, they exhibit a strong relation with surface groups. In fact, we show that the Green–Lazarsfeld sets of groups with $va(G)=1$ (virtually) coincide with those of surface groups, and furthermore that the only virtually RFRS groups with $va(G)=1$ are virtually surface groups.

Virtual Retraction Properties in Groups

If $G$ is a group, a virtual retract of $G$ is a subgroup, which is a retract of a finite index subgroup. Most of the paper focuses on two group properties: property (LR), that all finitely generated subgroups are virtual retracts; and property (VRC), that all cyclic subgroups are virtual retracts. We study the permanence of these properties under commensurability, amalgams over retracts, graph products, and wreath products. In particular, we show that (VRC) is stable under passing to finite index overgroups, while (LR) is not. The question whether all finitely generated virtually free groups satisfy (LR) motivates the remaining part of the paper, studying virtual free factors of such groups. We give a simple criterion characterizing when a finitely generated subgroup of a virtually free group is a free factor of a finite index subgroup. We apply this criterion to settle a conjecture of Brunner and Burns.

Tubular groups, 1-relator groups and nonpositive curvature

We show (using results of Wise and of Woodhouse) that a tubular group is always virtually special (meaning that it has a finite index subgroup embedding in a RAAG) if the underlying graph is a tree. We also adapt Gardam and Woodhouse’s argument on tubular groups which double cover 1-relator groups to show there exist 1-relator groups which are CAT(0) but not residually finite.