On Groups in Which Many Automorphisms Are Cyclic

Let G be a group. An automorphism α of G is said to be a cyclic automorphism if the subgroup ⟨x,xα⟩ is cyclic for every element x of G. In [F. de Giovanni, M.L. Newell, A. Russo: On a class of normal endomorphisms of groups, J. Algebra and its Applications 13, (2014), 6pp] the authors proved that every cyclic automorphism is central, namely, that every cyclic automorphism acts trivially on the factor group G/Z(G). In this paper, the class FW of groups in which every element induces by conjugation a cyclic automorphism on a (normal) subgroup of finite index will be investigated.

A few remarks on Pimsner–Popa bases and regular subfactors of depth 2

We prove that a finite index regular inclusion of $II_1$ -factors with commutative first relative commutant is always a crossed product subfactor with respect to a minimal action of a biconnected weak Kac algebra. Prior to this, we prove that every finite index inclusion of $II_1$ -factors which is of depth 2 and has simple first relative commutant (respectively, is regular and has commutative or simple first relative commutant) admits a two-sided Pimsner–Popa basis (respectively, a unitary orthonormal basis).

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].

Normal subgroups of SimpHAtic groups

A group is SimpHAtic if it acts geometrically on a simply connected simplicially hereditarily aspherical (SimpHAtic) complex. We show that finitely presented normal subgroups of the SimpHAtic groups are either: finite, or of finite index, or virtually free. This result applies, in particular, to normal subgroups of systolic groups. We prove similar strong restrictions on group extensions for other classes of asymptotically aspherical groups. The proof relies on studying homotopy types at infinity of groups in question. We also show that non-uniform lattices in SimpHAtic complexes (and in more general complexes) are not finitely presentable and that finitely presented groups acting properly on such complexes act geometrically on SimpHAtic complexes. In Appendix we present the topological two-dimensional quasi-Helly property of systolic complexes.

Planar algebras

We introduce a notion of planar algebra, the simplest example of which is a vector space of tensors, closed under planar contractions. A planar algebra with suitable positivity properties produces a finite index subfactor of a II1 factor, and vice versa.

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.

Multidimensional Multiplicative Combinatorial Properties of Dynamical Syndetic Sets

In this paper, it is shown that for a minimal system (X, G), if H is a normal subgroup of G with finite index n, then X can be decomposed into n components of closed sets such that each component is minimal under H-action. Meanwhile, we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension, extending a previous result by Glasscock, Koutsogiannis and Richter.