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.

INTERSECTIONS OF SUBGROUPS IN VIRTUALLY FREE GROUPS AND VIRTUALLY FREE PRODUCTS

This note contains a (short) proof of the following generalisation of the Friedman–Mineyev theorem (earlier known as the Hanna Neumann conjecture): if $A$ and $B$ are nontrivial free subgroups of a virtually free group containing a free subgroup of index $n$, then $\text{rank}(A\cap B)-1\leq n\cdot (\text{rank}(A)-1)\cdot (\text{rank}(B)-1)$. In addition, we obtain a virtually-free-product analogue of this result.

Almost-automorphisms of trees, cloning systems and finiteness properties

We prove that the group of almost-automorphisms of the infinite rooted regular [Formula: see text]-ary tree [Formula: see text] arises naturally as the Thompson-like group of a so-called [Formula: see text]-ary cloning system. A similar phenomenon occurs for any Röver–Nekrashevych group [Formula: see text], for [Formula: see text] a self-similar group. We use this framework to expand on work of Belk and Matucci, who proved that the Röver group, using the Grigorchuk group for [Formula: see text], is of type [Formula: see text]. Namely, we find some natural conditions on subgroups of [Formula: see text] to ensure that [Formula: see text] is of type [Formula: see text] and, in particular, we prove this for all [Formula: see text] in the infinite family of Šunić groups. We also prove that if [Formula: see text] is itself of type [Formula: see text], then so is [Formula: see text], and that every finitely generated virtually free group is self-similar, so in particular every finitely generated virtually free group [Formula: see text] yields a type [Formula: see text] Röver–Nekrashevych group [Formula: see text].

On finitely generated submonoids of virtually free groups

We prove that it is decidable whether or not a finitely generated submonoid of a virtually free group is graded, introduce a new geometric characterization of graded submonoids in virtually free groups as quasi-geodesic submonoids, and show that their word problem is rational (as a relation). We also solve the isomorphism problem for this class of monoids, generalizing earlier results for submonoids of free monoids. We also prove that the classes of graded monoids, regular monoids and Kleene monoids coincide for submonoids of free groups.

Amenable minimal Cantor systems of free groups arising from diagonal actions

We study amenable minimal Cantor systems of free groups arising from the diagonal actions of the boundary actions and certain Cantor systems. It is shown that every virtually free group admits continuously many amenable minimal Cantor systems whose crossed products are mutually non-isomorphic Kirchberg algebras in the UCT class (with explicitly determined

GROUPS WITH CONTEXT-FREE CONJUGACY PROBLEMS

The conjugacy problem and the inverse conjugacy problem of a finitely generated group are defined, from a language theoretic point of view, as sets of pairs of words. An automaton might be obliged to read the two input words synchronously, or could have the option to read asynchronously. Hence each class of languages gives rise to four classes of groups; groups whose (inverse) conjugacy problem is an (a)synchronous language in the given class. For regular languages all these classes are identical with the class of finite groups. We show that the finitely generated groups with asynchronously context-free inverse conjugacy problem are precisely the virtually free groups. Moreover, the other three classes arising from context-free languages are shown all to coincide with the class of virtually cyclic groups, which is precisely the class of groups with synchronously one-counter (inverse) conjugacy problem. It is also proved that, for a δ-hyperbolic group and any λ ≥ 1, ϵ ≥ 0, the intersection of the inverse conjugacy problem with the set of pairs of (λ, ϵ)-quasigeodesics is context-free. Finally we show that the conjugacy problem of a virtually free group is an asynchronously indexed language.