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].

Three-dimensional maps and subgroup growth

In this paper we derive a generating series for the number of cellular complexes known as pavings or three-dimensional maps, on n darts, thus solving an analogue of Tutte’s problem in dimension three. The generating series we derive also counts free subgroups of index n in $$\Delta ^+ = {\mathbb {Z}}_2*{\mathbb {Z}}_2*{\mathbb {Z}}_2$$ Δ + = Z 2 ∗ Z 2 ∗ Z 2 via a simple bijection between pavings and finite index subgroups which can be deduced from the action of $$\Delta ^+$$ Δ + on the cosets of a given subgroup. We then show that this generating series is non-holonomic. Furthermore, we provide and study the generating series for isomorphism classes of pavings, which correspond to conjugacy classes of free subgroups of finite index in $$\Delta ^+$$ Δ + . Computational experiments performed with software designed by the authors provide some statistics about the topology and combinatorics of pavings on $$n\le 16$$ n ≤ 16 darts.

BIG MAPPING CLASS GROUPS WITH HYPERBOLIC ACTIONS: CLASSIFICATION AND APPLICATIONS

We address the question of determining which mapping class groups of infinite-type surfaces admit nonelementary continuous actions on hyperbolic spaces. More precisely, let $\Sigma $ be a connected, orientable surface of infinite type with tame endspace whose mapping class group is generated by a coarsely bounded subset. We prove that ${\mathrm {Map}}(\Sigma )$ admits a continuous nonelementary action on a hyperbolic space if and only if $\Sigma $ contains a finite-type subsurface which intersects all its homeomorphic translates. When $\Sigma $ contains such a nondisplaceable subsurface K of finite type, the hyperbolic space we build is constructed from the curve graphs of K and its homeomorphic translates via a construction of Bestvina, Bromberg and Fujiwara. Our construction has several applications: first, the second bounded cohomology of ${\mathrm {Map}}(\Sigma )$ contains an embedded $\ell ^1$ ; second, using work of Dahmani, Guirardel and Osin, we deduce that ${\mathrm {Map}} (\Sigma )$ contains nontrivial normal free subgroups (while it does not if $\Sigma $ has no nondisplaceable subsurface of finite type), has uncountably many quotients and is SQ-universal.

Unique product groups and congruence subgroups

We prove that a uniform pro-[Formula: see text] group with no non-abelian free subgroups has a normal series with torsion-free abelian factors. We discuss this in relation to unique product groups. We also consider generalizations of Hantzsche–Wendt groups.

THE SET OF 2-GENERETED $C^*$-SIMPLE RELATIVELY FREE GROUPS HAS THE CARDINALITY OF THE CONTINUUM

In this paper we prove that the set of non-isomorphic 2-generated $C^*$-simple relatively free groups has the cardinality of the continuum. A non-trivial identity is satisfied in any (not absolutely free) relatively free group. Hence, they cannot contain a non-abelian absolutely free subgroups. The question of the existence of $C^*$-simple groups without free subgroups of rank 2 was posed by de la Harpe in 2007.

Free subgroups in maximal subgroups of SLn(D)

Given a non-commutative finite-dimensional [Formula: see text]-central division ring [Formula: see text], [Formula: see text] a subnormal subgroup of [Formula: see text] and [Formula: see text] a non-abelian maximal subgroup of [Formula: see text], then either [Formula: see text] contains a non-cyclic free subgroup or there exists a non-central maximal normal abelian subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] is a subfield of [Formula: see text], [Formula: see text] is Galois and [Formula: see text], also [Formula: see text] is a finite simple group with [Formula: see text].