The group of order-preserving automorphisms of a finitely generated Archimedean ordered group of rank $2$ is either infinite cyclic or trivial according as the ratio in $\mathbb {R}$ of the generators of the subgroup is or is not quadratic over $\mathbb {Q}.$ In the case of an Archimedean ordered group of rank $2$ that is not finitely generated, the group of order-preserving automorphisms is free abelian. Criteria determining the rank of this free abelian group are established.

Let $H\le F$ be two finitely generated free groups. Given $g\in F$, we study the ideal $\mathfrak I_g$ of equations for g with coefficients in H, i.e. the elements $w(x)\in H*\langle x\rangle$ such that $w(g)=1$ in F. The ideal $\mathfrak I_g$ is a normal subgroup of $H*\langle x\rangle$, and it’s possible to algorithmically compute a finite normal generating set for $\mathfrak I_g$; we give a description of one such algorithm, based on Stallings folding operations. We provide an algorithm to find an equation in w(x)\in$\mathfrak I_g$ with minimum degree, i.e.  such that its cyclic reduction contains the minimum possible number of occurrences of x and x−1; this answers a question of A. Rosenmann and E. Ventura. More generally, we show how to algorithmically compute the set Dg of all integers d such that $\mathfrak I_g$ contains equations of degree d; we show that Dg coincides, up to a finite set, with either $\mathbb N$ or $2\mathbb N$. Finally, we provide examples to illustrate the techniques introduced in this paper. We discuss the case where ${\text{rank}}(H)=1$. We prove that both kinds of sets Dg can actually occur. We show that the equations of minimum possible degree aren’t in general enough to generate the whole ideal $\mathfrak I_g$ as a normal subgroup.

We show that when a finitely presented Bestvina–Brady group splits as an amalgamated product over a subgroup $H$, its defining graph contains an induced separating subgraph whose associated Bestvina–Brady group is contained in a conjugate of $H$.

Given a group G acting faithfully on a set S, we characterize precisely when the twisted Brin–Thompson group SVG is finitely presented. The answer is that SVG is finitely presented if and only if we have the following: G is finitely presented, the action of G on S has finitely many orbits of two-element subsets of S, and the stabilizer in G of any element of S is finitely generated. Since twisted Brin–Thompson groups are simple, a consequence is that any subgroup of a group admitting an action as above satisfies the Boone–Higman conjecture. In the course of proving this, we also establish a sufficient condition for a group acting cocompactly on a simply connected complex to be finitely presented, even if certain edge stabilizers are not finitely generated, which may be of independent interest.

To any free group automorphism, we associate a universal (cone of) limit tree(s) with three defining properties: first, the tree has a minimal isometric action of the free group with trivial arc stabilizers; second, there is a unique expanding dilation of the tree that represents the free group automorphism; and finally, the loxodromic elements are exactly the elements that weakly limit to dominating attracting laminations under forward iteration by the automorphism. So the action on the tree detects the automorphism’s dominating exponential dynamics.

As a corollary, our previously constructed limit pretree that detects the exponential dynamics is canonical. We also characterize all very small trees that admit an expanding homothety representing a given automorphism. In the appendix, we prove a variation of Feighn–Handel’s recognition theorem for atoroidal outer automorphisms.

For any prime p and S a p-group isomorphic to a Sylow p-subgroup of a rank $2$ simple group of Lie type in characteristic p, we determine all saturated fusion systems supported on S up to isomorphism.

We describe finitely generated and second countable prosoluble subgroups of free profinite products. We also give a description of relatively projective prosoluble groups.

We examine a cyclic order on the directed edges of a tree whose vertices have cyclically ordered links. We use it to show that a graph of groups with left-cyclically ordered vertex groups and convex left-ordered edge groups is left-cyclically orderable.

We study the vectorial length compactification of the space of conjugacy classes of maximal representations of the fundamental group $\Gamma$ of a closed hyperbolic surface $\Sigma$ in $\textrm{PSL}(2,{\mathbb{R}})^n$. We identify the boundary with the sphere ${\mathbb{P}}(({\mathcal{ML}})^n)$, where $\mathcal{ML}$ is the space of measured geodesic laminations on $\Sigma$. In the case $n=2$, we give a geometric interpretation of the boundary as the space of homothety classes of ${\mathbb{R}}^2$-mixed structures on $\Sigma$. We associate to such a structure a dual tree-graded space endowed with an ${\mathbb{R}}_+^2$-valued metric, which we show to be universal with respect to actions on products of two $\mathbb{R}$-trees with the given length spectrum.

Let V be a finite dimensional vector space over the field with p elements, where p is a prime number. Given arbitrary $\alpha ,\beta \in \mathrm {GL}(V)$, we consider the semidirect products $V\rtimes \langle \alpha \rangle $ and $V\rtimes \langle \beta \rangle $, and show that if $V\rtimes \langle \alpha \rangle $ and $V\rtimes \langle \beta \rangle $ are isomorphic, then $\alpha $ must be similar to a power of $\beta $ that generates the same subgroup as $\beta $; that is, if H and K are cyclic subgroups of $\mathrm {GL}(V)$ such that $V\rtimes H\cong V\rtimes K$, then H and K must be conjugate subgroups of $\mathrm {GL}(V)$. If we remove the cyclic condition, there exist examples of nonisomorphic, let alone nonconjugate, subgroups H and K of $\mathrm {GL}(V)$ such that $V\rtimes H\cong V\rtimes K$. Even if we require that noncyclic subgroups H and K of $\mathrm {GL}(V)$ be abelian, we may still have $V\rtimes H\cong V\rtimes K$ with H and K nonconjugate in $\mathrm {GL}(V)$, but in this case, H and K must at least be isomorphic. If we replace V by a free module U over ${\mathbb {Z}}/p^m{\mathbb {Z}}$ of finite rank, with $m>1$, it may happen that $U\rtimes H\cong U\rtimes K$ for nonconjugate cyclic subgroups of $\mathrm {GL}(U)$. If we completely abandon our requirements on V, a sufficient criterion is given for a finite group G to admit nonconjugate cyclic subgroups H and K of $\mathrm {Aut}(G)$ such that $G\rtimes H\cong G\rtimes K$. This criterion is satisfied by many groups.

We introduce and study two conditions on groups of homeomorphisms of Cantor space, namely the conditions of being vigorous and of being flawless. These concepts are dynamical in nature, and allow us to study a certain interplay between the dynamics of an action and the algebraic properties of the acting group. A group $G\leq \operatorname {Homeo}(\mathfrak {C})$ is vigorous if for any clopen set A and proper clopen subsets B and C of A, there is $\gamma \in G$ in the pointwise stabiliser of $\mathfrak {C}\backslash A$ with $B\gamma \subseteq C$. A nontrivial group $G\leq \operatorname {Homeo}(\mathfrak {C})$ is flawless if for all k and w a nontrivial freely reduced product expression on k variables (including inverse symbols), a particular subgroup $w(G)_\circ $ of the verbal subgroup $w(G)$ is the whole group. We show: 1) simple vigorous groups are either two-generated by torsion elements, or not finitely generated, 2) flawless groups are both perfect and lawless, 3) vigorous groups are simple if and only if they are flawless, and, 4) the class of vigorous simple subgroups of $\operatorname {Homeo}(\mathfrak {C})$ is fairly broad (the class is closed under various natural constructions and contains many well known groups, such as the commutator subgroups of the Higman–Thompson groups $G_{n,r}$, the Brin-Thompson groups $nV$, Röver’s group $V(\Gamma )$, and others of Nekrashevych’s ‘simple groups of dynamical origin’).

In this note we investigate the centraliser of a linearly growing element of $\mathrm{Out}(F_n)$ (that is, a root of a Dehn twist automorphism), and show that it has a finite index subgroup mapping onto a direct product of certain “equivariant McCool groups” with kernel a finitely generated free abelian group. In particular, this allows us to show it is VF and hence finitely presented.

We find an upper bound for the number of groups of order n up to isomorphism in the variety ${\mathfrak {S}}={\mathfrak {A}_p}{\mathfrak {A}_q}{\mathfrak {A}_r}$, where p, q and r are distinct primes. We also find a bound on the orders and on the number of conjugacy classes of subgroups that are maximal amongst the subgroups of the general linear group that are also in the variety $\mathfrak {A}_q\mathfrak {A}_r$.

We prove that centralisers of elements in [finitely generated free]-by-cyclic groups are computable. As a corollary, given two conjugate elements in a [finitely generated free]-by-cyclic group, the set of conjugators can be computed and the conjugacy problem with context-free constraints is decidable. We pose several problems arising naturally from this work.

To any free group automorphism, we associate a real pretree with several nice properties. First, it has a rigid/non-nesting action of the free group with trivial arc stabilizers. Secondly, there is an expanding pretree-automorphism of the real pretree that represents the free group automorphism. Finally and crucially, the loxodromic elements are exactly those whose (conjugacy class) length grows exponentially under iteration of the automorphism; thus, the action on the real pretree is able to detect the growth type of an element.

This construction extends the theory of metric trees that has been used to study free group automorphisms. The new idea is that one can equivariantly blow up an isometric action on a real tree with respect to other real trees and get a rigid action on a treelike structure known as a real pretree. Topology plays no role in this construction as all the work is done in the language of pretrees (intervals).

We introduce two families of two-generator one-relator groups called primitive extension groups and show that a one-relator group is hyperbolic if its primitive extension subgroups are hyperbolic. This reduces the problem of characterizing hyperbolic one-relator groups to characterizing hyperbolic primitive extension groups. These new groups, moreover, admit explicit decompositions as graphs of free groups with adjoined roots. In order to obtain this result, we characterize $2$-free one-relator groups with exceptional intersection in terms of Christoffel words, show that hyperbolic one-relator groups have quasi-convex Magnus subgroup, and build upon the one-relator tower machinery developed in previous work of the author.

The Basilica group is a well-known 2-generated weakly branch, but not branch, group acting on the binary rooted tree. Recently, a more general form of the Basilica group has been investigated by Petschick and Rajeev, which is an $s$-generated weakly branch, but not branch, group that acts on the $m$-adic tree, for $s,m\ge 2$. A larger family of groups, which contains these generalised Basilica groups, is the family of iterated monodromy groups. With the new developments by Francoeur, the study of the existence of maximal subgroups of infinite index has been extended from branch groups to weakly branch groups. Here we show that a subfamily of iterated monodromy groups, which more closely resemble the generalised Basilica groups, have maximal subgroups only of finite index.

We prove a nonarchimedean analogue of Jørgensen’s inequality, and use it to deduce several algebraic convergence results. As an application, we show that every dense subgroup of ${\mathrm {SL}_2}(K)$, where K is a p-adic field, contains two elements that generate a dense subgroup of ${\mathrm {SL}_2}(K)$, which is a special case of a result by Breuillard and Gelander [‘On dense free subgroups of Lie groups’, J. Algebra 261(2) (2003), 448–467]. We also list several other related results, which are well known to experts, but not easy to locate in the literature; for example, we show that a nonelementary subgroup of ${\mathrm {SL}_2}(K)$ over a nonarchimedean local field K is discrete if and only if each of its two-generator subgroups is discrete.

We classify the irreducible unitary representations of closed simple groups of automorphisms of trees acting $2$-transitively on the boundary and whose local action at every vertex contains the alternating group. As an application, we confirm Claudio Nebbia’s CCR conjecture on trees for $(d_0,d_1)$-semi-regular trees such that $d_0,d_1\in \Theta $, where $\Theta $ is an asymptotically dense set of positive integers.

In this article, we explore the problem of determining isomorphisms between the twisted complex group algebras of finite $p$-groups. This problem bears similarity to the classical group algebra isomorphism problem and has been recently examined by Margolis-Schnabel. Our focus lies on a specific invariant, referred to as the generalized corank, which relates to the twisted complex group algebra isomorphism problem. We provide a solution for non-abelian $p$-groups with generalized corank at most three.