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.

Formal language convexity in left-orderable groups

We propose a criterion for preserving the regularity of a formal language representation when passing from groups to subgroups. We use this criterion to show that the regularity of a positive cone language in a left-orderable group passes to its finite index subgroups, and to show that there exists no left order on a finitely generated acylindrically hyperbolic group such that the corresponding positive cone is represented by a quasi-geodesic regular language. We also answer one of Navas’ questions by giving an example of an infinite family of groups which admit a positive cone that is generated by exactly [Formula: see text] generators, for every [Formula: see text]. As a special case of our construction, we obtain a finitely generated positive cone for [Formula: see text].

Coarse properties of graphs

The objective of this work is to study large scale properties of graphs, graph sequences and groups. Firstly we consider the combinatorial cost and prove that having cost equal to 1 and hyperfiniteness are coarse invariants of graph sequences. We show that cost is multiplicative with respect to taking finite index subgroups. For an amenable group we investigate the properties of their Farber sequences, sofic approximations and vice versa. Secondly we consider the first uniformly finite homology group of graphs with coefficients in Z. We show that its non-vanishing depends on the ends, large circuits and (higher-dimensional) non-expansion of the graph. When the graph is transitive this is a full description. Finally we take a look at the Baumslag-Solitar group BS (2; 3). This group is non-Hopfian, meaning it has a quotient isomorphic to itself. We give a visual interpretation of this on the Cayley graph level.

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.

VIRTUALLY FIBERING RIGHT-ANGLED COXETER GROUPS

We show that certain right-angled Coxeter groups have finite index subgroups that quotient to $\mathbb{Z}$ with finitely generated kernels. The proof uses Bestvina–Brady Morse theory facilitated by combinatorial arguments. We describe a variety of examples where the plan succeeds or fails. Among the successful examples are the right-angled reflection groups in $\mathbb{H}^{4}$ with fundamental domain the $120$ -cell or the $24$ -cell.

On a family of groups generated by parabolic matrices

We study various aspects of the family of groups generated by the parabolic matrices A(t1 ζ), ... , A(tm ζ) where A(z) = ( 1 z0 1 ) and by the elliptic matrix ( 0 -1 1 0 ). The elements of the matrices W in such groups can be computed by a recursion formula. These groups are special cases of the generalized parametrized modular groups introduced in [16].We study the sets {z : tr W(z) ∈ [-2; +2]} [13] and their critical points and geometry, furthermore some finite index subgroups and the discretness of subgroups.

Isomorphisms Between Big Mapping Class Groups

We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these “big” mapping class groups and shows that each finite-index subgroup has finite outer automorphism group. As a key ingredient, we prove that all simplicial automorphisms between curve complexes of infinite-type orientable surfaces are induced by homeomorphisms.

The Johnson homomorphism and its kernel

We give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general result that also applies to “subsurface Torelli groups”. Using this, we extend Johnson’s calculation of the rational abelianization of the Torelli group not only to the subsurface Torelli groups, but also to finite-index subgroups of the Torelli group that contain the kernel of the Johnson homomorphism.