It is shown that, for every prime number p, the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gp of all finite p-groups has the cardinality of the continuum. Furthermore, it is shown, in addition, that the complete lattice of all semidirectly closed pseudovarieties of finite monoids whose intersection with the pseudovariety G of all finite groups is equal to the pseudovariety Gsol of all finite solvable groups has also the cardinality of the continuum.

To every dynamical system $(X,\varphi )$ over a totally disconnected compact space, we associate a left-orderable group $T(\varphi )$. It is defined as a group of homeomorphisms of the suspension of $(X,\varphi )$ which preserve every orbit of the suspension flow and act by dyadic piecewise linear homeomorphisms in the flow direction. We show that if the system is minimal, the group is simple and, if it is a subshift, then the group is finitely generated. The proofs of these two statements are short and elementary, providing straightforward examples of finitely generated simple left-orderable groups. We show that if the system is minimal, every action of the corresponding group on the circle has a fixed point. These constitute the first examples of finitely generated left-orderable groups with this fixed point property. We show that for every system $(X,\varphi )$, the group $T(\varphi )$ does not have infinite subgroups with Kazhdan's property $(T)$. In addition, we show that for every minimal subshift, the corresponding group is never finitely presentable. Finally, if $(X,\varphi )$ has a dense orbit, then the isomorphism type of the group $T(\varphi )$ is a complete invariant of flow equivalence of the pair $\{\varphi , \varphi ^{-1}\}$.

In this paper we study a notion of HL-extension (HL standing for Herwig–Lascar) for a structure in a finite relational language $\mathcal {L}$ . We give a description of all finite minimal HL-extensions of a given finite $\mathcal {L}$ -structure. In addition, we study a group-theoretic property considered by Herwig–Lascar and show that it is closed under taking free products. We also introduce notions of coherent extensions and ultraextensive $\mathcal {L}$ -structures and show that every countable $\mathcal {L}$ -structure can be extended to a countable ultraextensive structure. Finally, it follows from our results that the automorphism group of any countable ultraextensive $\mathcal {L}$ -structure has a dense locally finite subgroup.

For a finite group $G$, let $d(G)$ denote the minimal number of elements required to generate $G$. In this paper, we prove sharp upper bounds on $d(H)$ whenever $H$ is a maximal subgroup of a finite almost simple group. In particular, we show that $d(H)\leqslant 5$ and that $d(H)\geqslant 4$ if and only if $H$ occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.

In their book Subgroup Growth, Lubotzky and Segal asked: What are the possible types of subgroup growth of the pro-$p$ group? In this paper, we construct certain extensions of the Grigorchuk group and the Gupta–Sidki groups, which have all possible types of subgroup growth between $n^{(\log n)^{2}}$ and $e^{n}$. Thus, we give an almost complete answer to Lubotzky and Segal’s question. In addition, we show that a class of pro-$p$ branch groups, including the Grigorchuk group and the Gupta–Sidki groups, all have subgroup growth type $n^{\log n}$.

Motivated by the Bruhat and Cartan decompositions of general linear groups over local fields, we enumerate double cosets of the group of label-preserving automorphisms of a label-regular tree over the fixator of an end of the tree and over maximal compact open subgroups. This enumeration is used to show that every continuous homomorphism from the automorphism group of a label-regular tree has closed range.

In 1945–1946, C. L. Siegel proved that an $n$-dimensional lattice $\unicode[STIX]{x1D6EC}$ of determinant $\text{det}(\unicode[STIX]{x1D6EC})$ has at most $m^{n^{2}}$ different sublattices of determinant $m\cdot \text{det}(\unicode[STIX]{x1D6EC})$. In 1997, the exact number of the different sublattices of index $m$ was determined by Baake. We present a systematic treatment for counting the sublattices and derive a formula for the number of the sublattice classes under unimodular equivalence.

We construct total orders on the vertex set of an oriented tree. The orders are based only on up-down counts at the interior vertices and the edges along the unique geodesic from a given vertex to another.

As an application, we provide a short proof (modulo Bass–Serre theory) of Vinogradov’s result that the free product of left-orderable groups is left-orderable.

We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C\wr F$, where $C$ is a finite group and $F$ a non-abelian free group.

We study lattice embeddings for the class of countable groups $\unicode[STIX]{x1D6E4}$ defined by the property that the largest amenable uniformly recurrent subgroup ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ is continuous. When ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ comes from an extremely proximal action and the envelope of ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ is coamenable in $\unicode[STIX]{x1D6E4}$, we obtain restrictions on the locally compact groups $G$ that contain a copy of $\unicode[STIX]{x1D6E4}$ as a lattice, notably regarding normal subgroups of $G$, product decompositions of $G$, and more generally dense mappings from $G$ to a product of locally compact groups.

The class of all monolithic (that is, subdirectly irreducible) groups belonging to a variety generated by a finite nilpotent group can be axiomatised by a finite set of elementary sentences.

Leighton’s graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton’s theorem that allows generalisations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups with line patterns, building on the work of Cashen–Macura and Hagen–Touikan. To illustrate the potential for future applications, we give a quasi-isometric rigidity result for a family of cyclic doubles of free groups.

Let γn = [x1,…,xn] be the nth lower central word. Denote by Xn the set of γn -values in a group G and suppose that there is a number m such that $|{g^{{X_n}}}| \le m$ for each g ∈ G. We prove that γn+1(G) has finite (m, n) -bounded order. This generalizes the much-celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite.

In this paper we continue the study of right-angled Artin groups up to commensurability initiated in [CKZ]. We show that RAAGs defined by different paths of length greater than 3 are not commensurable. We also characterise which RAAGs defined by paths are commensurable to RAAGs defined by trees of diameter 4. More precisely, we show that a RAAG defined by a path of length n > 4 is commensurable to a RAAG defined by a tree of diameter 4 if and only if n ≡ 2 (mod 4). These results follow from the connection that we establish between the classification of RAAGs up to commensurability and linear integer-programming.

We show that in general for a given group the structure of a maximal hyperbolic tower over a free group is not canonical: we construct examples of groups having hyperbolic tower structures over free subgroups which have arbitrarily large ratios between their ranks. These groups have the same first order theory as non-abelian free groups and we use them to study the weight of types in this theory.

We study lattices in a product $G=G_{1}\times \cdots \times G_{n}$ of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor is quasi just-non-compact, meaning that $G_{i}$ is non-compact and every closed normal subgroup of $G_{i}$ is discrete or cocompact (e.g. $G_{i}$ is topologically simple). We show that the set of discrete subgroups of $G$ containing a fixed cocompact lattice $\unicode[STIX]{x1D6E4}$ with dense projections is finite. The same result holds if $\unicode[STIX]{x1D6E4}$ is non-uniform, provided $G$ has Kazhdan’s property (T). We show that for any compact subset $K\subset G$, the collection of discrete subgroups $\unicode[STIX]{x1D6E4}\leqslant G$ with $G=\unicode[STIX]{x1D6E4}K$ and dense projections is uniformly discrete and hence of covolume bounded away from $0$. When the ambient group $G$ is compactly presented, we show in addition that the collection of those lattices falls into finitely many $\operatorname{Aut}(G)$-orbits. As an application, we establish finiteness results for discrete groups acting on products of locally finite graphs with semiprimitive local action on each factor. We also present several intermediate results of independent interest. Notably it is shown that if a non-discrete, compactly generated quasi just-non-compact tdlc group $G$ is a Chabauty limit of discrete subgroups, then some compact open subgroup of $G$ is an infinitely generated pro-$p$ group for some prime $p$. It is also shown that in any Kazhdan group with discrete amenable radical, the lattices form an open subset of the Chabauty space of closed subgroups.

Let $\unicode[STIX]{x1D6E4}\leqslant \text{Aut}(T_{d_{1}})\times \text{Aut}(T_{d_{2}})$ be a group acting freely and transitively on the product of two regular trees of degree $d_{1}$ and $d_{2}$. We develop an algorithm that computes the closure of the projection of $\unicode[STIX]{x1D6E4}$ on $\text{Aut}(T_{d_{t}})$ under the hypothesis that $d_{t}\geqslant 6$ is even and that the local action of $\unicode[STIX]{x1D6E4}$ on $T_{d_{t}}$ contains $\text{Alt}(d_{t})$. We show that if $\unicode[STIX]{x1D6E4}$ is torsion-free and $d_{1}=d_{2}=6$, exactly seven closed subgroups of $\text{Aut}(T_{6})$ arise in this way. We also construct two new infinite families of virtually simple lattices in $\text{Aut}(T_{6})\times \text{Aut}(T_{4n})$ and in $\text{Aut}(T_{2n})\times \text{Aut}(T_{2n+1})$, respectively, for all $n\geqslant 2$. In particular, we provide an explicit presentation of a torsion-free infinite simple group on 5 generators and 10 relations, that splits as an amalgamated free product of two copies of $F_{3}$ over $F_{11}$. We include information arising from computer-assisted exhaustive searches of lattices in products of trees of small degrees. In an appendix by Pierre-Emmanuel Caprace, some of our results are used to show that abstract and relative commensurator groups of free groups are almost simple, providing partial answers to questions of Lubotzky and Lubotzky–Mozes–Zimmer.

A tubular group G is a finite graph of groups with ℤ2 vertex groups and ℤ edge groups. We characterize residually finite tubular groups: G is residually finite if and only if its edge groups are separable. Methods are provided to determine if G is residually finite. When G has a single vertex group an algorithm is given to determine residual finiteness.

Quasi-Sturmian words, which are infinite words with factor complexity eventually $n+c$ share many properties with Sturmian words. In this article, we study the quasi-Sturmian colorings on regular trees. There are two different types, bounded and unbounded, of quasi-Sturmian colorings. We obtain an induction algorithm similar to Sturmian colorings. We distinguish them by the recurrence function.

On établit une décomposition de l’homologie stable des groupes d’automorphismes des groupes libres à coefficients polynomiaux contravariants en termes d’homologie des foncteurs. Elle permet plusieurs calculs explicites, qui recoupent des résultats établis de manière indépendante par O. Randal-Williams et généralisent certains d’entre eux. Nos méthodes reposent sur l’examen d’extensions de Kan dérivées associées à plusieurs catégories de groupes libres, la généralisation d’un critère d’annulation homologique à coefficients polynomiaux dû à Scorichenko, le théorème de Galatius identifiant l’homologie stable des groupes d’automorphismes des groupes libres à celle des groupes symétriques, la machinerie des $\unicode[STIX]{x1D6E4}$-espaces et le scindement de Snaith.