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Given a Baumslag-Solitar group, we study its space of subgroups from a topological and dynamical perspective. We first determine its perfect kernel (the largest closed subset without isolated points). We then bring to light a natural partition of the space of subgroups into one closed subset and countably many open subsets that are invariant under the action by conjugation. One of our main results is that the restriction of the action to each piece is topologically transitive. This partition is described by an arithmetically defined function, that we call the phenotype, with values in the positive integers or infinity. We eventually study the closure of each open piece and also the closure of their union. We moreover identify in each phenotype a (the) maximal compact invariant subspace., Comment: 60 pages, companion webpage available at https://doi.org/10.5281/zenodo.7225585 . Comments welcome!

We show that if a right-angled Artin group [Formula: see text] has a non-trivial, minimal action on a tree [Formula: see text] with more than two ends, then [Formula: see text] contains a separating subgraph [Formula: see text] such that [Formula: see text] stabilizes an edge in [Formula: see text].

We consider HNN extensions [Formula: see text], where [Formula: see text] are normal subgroups of [Formula: see text] and [Formula: see text]. We show that class-preserving automorphisms of those HNN extensions are all inner if [Formula: see text] satisfies certain conditions. From Grossman’s result, it follows that outer automorphism groups of such conjugacy separable HNN extensions are residually finite.

We develop a version of the Bass-Serre theory for Lie algebras (over a field $k$) via a homological approach. We define the notion of fundamental Lie algebra of a graph of Lie algebras and show that this construction yields Mayer-Vietoris sequences. We extend some well known results in group theory to $\mathbb{N}$-graded Lie algebras: for example, we show that one relator $\mathbb{N}$-graded Lie algebras are iterated HNN extensions with free bases which can be used for cohomology computations and apply the Mayer-Vietoris sequence to give some results about coherence of Lie algebras., 27 pages

Bass-Serre theory was initiated in 1970 by Jean-Pierre Serre, in [Ser77]. The theory's main motivation was to study the structure of discrete and torsion-free subgroups of SL2(Qp), more precisely Ihara's theorem stating that all torsion-free subgroups of SL2(Qp) are free. Inspired by covering space theory in algebraic topology, J-P Serre explains that showing the freedom of a group by making it act freely on a tree is more natural. So, he deduces a simple and elegant proof of this theorem and allows to generalize several theorems of combinatorial group theory: Nielsen-Schreier, Kurosh and of Nagao theorems and so on. This theory shows more generally that a group acts on a tree without inversion if and only it is isomorphic to a non-trivial amalgam or to an HNN extension.In 2011 B. Deschamps and I. Suarez introduced in [DSA11] a combinatorial theory for profinite groups. They proved an analogue for profinite groups of Serre's theorem on freedom of a group : a profinite group has a dense free subgroup if and only this group acts profreely on a protree. The notion of profree action can be summarized to making the groups of the inverse system of finite groups associated with a profine group act freely on each floor of an inverse system of graphs with certain arithmetic conditions.The purpose of this thesis is to give a contribution of Deschamps-Suarez theory of prographs. Tools and techniques developed by Deschamps and Suarez, placed in a general context, allow to show an analog of the Deschamps-Suarez theorem for profinite groups with a dense amalgamated subgroup and a generalization of this result. Finally, these results are illustrated on well-known Galois situations., La théorie de Bass-Serre a été développée en 1970 par Jean-Pierre Serre, dans [Ser77]. Elle a pour motivation principale d'étudier la structure des sous-groupes discrets et sans torsion de SL2(Qp), plus précisément le théorème d'Ihara suivant lequel ces sous-groupes sont libres. S'inspirant de la topologie notamment de la théorie des revêtements, il devient alors plus naturel, explique J-P Serre de montrer la liberté d'un groupe en le faisant agir librement sur un arbre. Il en déduit ainsi une preuve simple et élégante de ce théorème jugé mystérieux et permet de généraliser plusieurs résultats de théorie combinatoire des groupes : les théorèmes de Nielsen-Schreier, de Kurosh et de Nagao entre autres. Cette théorie montre plus généralement qu'un groupe agit sur un arbre sans inversion si et seulement il est isomorphe à un amalgame non trivial ou à une extension HNN.En 2011 B. Deschamps et I. Suarez ont introduit dans [DSA11] une théorie combinatoire pour les groupes profinis et ont démontré un analogue pour les groupes profinis du théorème de Serre sur liberté d'un groupe : un groupe profini possède un sous-groupe libre dense si et seulement il agit prolibrement sur un poarbre. La notion d'action prolibre se résume moralement à faire agir librement les groupes du système projectif de groupes finis associés à un groupe profini sur chaque étage d’un système projectif de graphes avec certaines conditions arithmétiques.L'objet de cette thèse est de donner une contribution à cette théorie des prographes. Les outils et techniques développés par Deschamps et Suarez étant placé dans un cadre assez général nous permettent alors de montrer un analogue du théorème de DS pour les groupes profinis possédant un sous-groupe amalgamé dense ainsi qu'une généralisation de ce résultat. Enfin nous illustrons ces résultats sur des situations galoisiennes bien connues.

Let G = F ⋊ H be a semidirect product of a free group F and a finite group H. The H-module structure of the abelianization F a b is described in terms of splitting of G as the fundamental group of a graph of finite groups.

Minor changes in the exposition and small corrections on the previous version., Comment: 11 pages no figure

We obtain a generalization of Shirvani’s results on necessary conditions of residual finiteness of generalized free products and HNN-extensions of groups to the case of residually C groups, where C is an arbitrary class of groups.

To a large class of graphs of groups we associate a C*-algebra universal for generators and relations. We show that this C*-algebra is stably isomorphic to the crossed product induced from the action of the fundamental group of the graph of groups on the boundary of its Bass-Serre tree. We characterise when this action is minimal, and find a sufficient condition under which it is locally contractive. In the case of generalised Baumslag-Solitar graphs of groups (graphs of groups in which every group is infinite cyclic) we also characterise topological freeness of this action. We are then able to establish a dichotomy for simple C*-algebras associated to generalised Baumslag-Solitar graphs of groups: they are either a Kirchberg algebra, or a stable Bunce-Deddens algebra., Comment: 59 pages

Let VB$_n$ be the virtual braid group on $n$ strands and let $\mathfrak{S}_n$ be the symmetric group on $n$ letters. Let $n,m \in \mathbb{N}$ such that $n \ge 5$, $m \ge 2$ and $n \ge m$. We determine all possible homomorphisms from VB$_n$ to $\mathfrak{S}_m$, from $\mathfrak{S}_n$ to VB$_m$ and from VB$_n$ to VB$_m$. As corollaries we get that Out(VB$_n$) is isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ and that VB$_n$ is both Hopfian and co-Hofpian.

In 2011, while investigating fundamental groups of wild spaces, K.Eda [7] showed that the fundamental group of the Hawaiian earring (the Hawaiian earring group, in short) has the property that for any homomorphism h from it to a free product A*B, there exists a natural number N such that is contained in a conjugate subgroup to A or B. In the present article, we prove a corresponding property for certain HNN extensions and amalgamated free products. This allows us to show that some one-relator groups, including Baumslag–Solitar groups, are n-slender.

Let C be an arbitrary class of groups which has the root property, consists of finite groups only, and contains at least one nonidentity group. It is proved that every extension of a free group by a C-group is conjugacy C-separable. It is also proved that, if G is a free product of two conjugacy C-separable groups with finite amalgamated subgroup or an HNN-extension of a conjugacy C-separable group with finite associated subgroups, then the group G is residually C if and only if it is conjugacy C-separable.

The word problem of a finitely generated group is the set of words over the generators that are equal to the identity in the group. The word problem is therefore a formal language. If this language happens to be context-free, then the group is called context-free. Finitely generated virtually free groups are context-free. In the seminal paper Muller–Schupp [38] the converse was shown: every context-free group is virtually free. Over the past decades a wide range of other characterizations of context-free groups have been found. It underlines that context-free groups play a major role in combinatorial group theory.

We show that a right-angled Artin group, defined by a graph Γ that has at least three vertices, does not split over an infinite cyclic subgroup if and only if Γ is biconnected. Further, we compute JSJ-decompositions of 1-ended right-angled Artin groups over infinite cyclic subgroups.

We construct HNN extensions of discrete quantum groups, we study their representation theory and we show that an HNN extension of amenable discrete quantum groups is K-amenable.

A necessary and sufficient condition for the residual finiteness of a (generalized) free product of two residually finite solvable-by-finite minimax groups with cyclic amalgamated subgroups is obtained. This generalizes the well-known Dyer theorem claiming that every free product of two polycyclic-by-finite groups with cyclic amalgamated subgroups is a residually finite group.

We give a general structure theory for reconstructing non-trivial group actions on sets without any further assumptions on the group, the action, or the set on which the group acts. Using certain "local data" [Formula: see text] from the action we build a group [Formula: see text] of the data and a space [Formula: see text] with an action of [Formula: see text] on [Formula: see text] that arise naturally from the data [Formula: see text]. We use these to obtain an approximation to the original group G, the original space X and the original action G × X → X. The data [Formula: see text] is distinguished by the property that it may be chosen from the action locally. For a large enough set of local data [Formula: see text], our definition of [Formula: see text] in terms of generators and relations allows us to obtain a presentation for the group G. We demonstrate this on several examples. When the local data [Formula: see text] is chosen from a graph of groups, the group [Formula: see text] is the fundamental group of the graph of groups and the space [Formula: see text] is the universal covering tree of groups. For general non-properly discontinuous group actions our local data allows us to imitate a fundamental domain, quotient space and universal covering for the quotient. We exhibit this on a non-properly discontinuous free action on ℝ. For a certain class of non-properly discontinuous group actions on the upper half-plane, we use our local data to build a space on which the group acts discretely and cocompactly. Our combinatorial approach to reconstructing abstract group actions on sets is a generalization of the Bass–Serre theory for reconstructing group actions on trees. Our results also provide a generalization of the notion of developable complexes of groups by Haefliger.

A JSJ-splitting of a group $G$ over a certain class of subgroups is a graph of groups decomposition of $G$ which describes all possible decompositions of $G$ as an amalgamated product or an HNN extension over subgroups lying in the given class. Such decompositions originated in 3-manifold topology. In this paper we generalize the JSJ-splitting constructions of Sela, Rips-Sela and Dunwoody-Sageev and we construct a JSJ-splitting for any finitely presented group with respect to the class of all slender subgroups along which the group splits. Our approach relies on Haefliger's theory of group actions on CAT$(0)$ spaces.

In this paper, we study the quotient and “blow-up” of graph-of-groups [Formula: see text] and of their automorphisms [Formula: see text]. We show that the existence of such a blow-up of any [Formula: see text], relative to a given family of “local” graph-of-groups isomorphisms [Formula: see text] depends crucially on the [Formula: see text]-conjugacy class of the correction term [Formula: see text] for any edge [Formula: see text] of [Formula: see text], where [Formula: see text]-conjugacy is a new but natural concept introduced here. As an application, we obtain a criterion as to whether a partial Dehn twist can be blown up relative to local Dehn twists, to give an actual Dehn twist. The results of this paper are also used crucially in the follow-up papers [Lustig and Ye, Normal form and parabolic dynamics for quadratically growing automorphisms of free groups, arXiv:1705.04110v2; Ye, Partial Dehn twists of free groups relative to local Dehn twists — A dichotomy, arXiv:1605.04479 ; When is a polynomially growing automorphism of [Formula: see text] geometric, arXiv:1605.07390 ].

Let H i be a sublattice subgroup of a lattice-ordered group G i (i = 1, 2). Suppose that H 1 and H 2 are isomorphic as lattice-ordered groups, say by φ. In general, there is no lattice-ordered group in which G 1 and G 2 can be embedded (as lattice-ordered groups) so that the embeddings agree on the images of H 1 and H 1φ. In this article we prove that the group free product of G 1 and G 2 amalgamating H 1 and H 1φ is right orderable and so embeddable (as a group) in a lattice-orderable group. To obtain this, we use our necessary and sufficient conditions for the free product of right-ordered groups with amalgamated subgroup to be right orderable [BLUDOV, V. V.—GLASS, A. M. W.: Word problems, embeddings, and free products of right-ordered groups with amalgamated subgroup, Proc. London Math. Soc. (3) 99 (2009), 585–608]. We also provide new limiting examples to show that amalgamation can fail in the category of lattice-ordered groups even when the amalgamating sublattice subgroups are convex and normal (ℓ-ideals) and solve of Problem 1.42 from [KOPYTOV, V. M.—MEDVEDEV, N. YA.: Ordered groups. In: Selected Problems in Algebra. Collection of Works Dedicated to the Memory of N. Ya. Medvedev, Altaii State University, Barnaul, 2007, pp. 15–112 (Russian)].

We describe the structure of the automorphism groups of algebras Morita equivalent to the first Weyl algebra $ A_1(k) $. In particular, we give a geometric presentation for these groups in terms of amalgamated products, using the Bass-Serre theory of groups acting on graphs. A key role in our approach is played by a transitive action of the automorphism group of the free algebra $ k $ on the Calogero-Moser varieties $ \CC_n $ defined in [5]. In the end, we propose a natural extension of the Dixmier Conjecture for $ A_1(k) $ to the class of Morita equivalent algebras.

A valuated group with normal forms is a group with an integer-valued length function satisfying some of Lyndon's axioms (Lyndon, 1963 [Lyn63] ) and an additional axiom considered by Hurley (1980) [Hur80] . We prove a subgroup theorem for valuated groups with normal forms analogous to Grushko–Neumann's theorem. We also study the CSA property in such groups.

We use permutation groups to give necessary and sufficient conditions for the free product of right-ordered groups with amalgamated subgroup to be right orderable. We obtain several consequences answering previously posed problems and also prove the right-orderable analogues of the Higman Embedding Theorem and the Boone–Higman Theorem.

We prove that the amalgamated product of free groups with cyclic amalgamations satisfying certain conditions are virtually free-by-cyclic. In case the cyclic amalgamated subgroups lie outside the derived group such groups are free-by-cyclic. Similarly a one-relator HNN-extension in which the conjugated elements either coincide or are independent modulo the derived group is shown to be free-by-cyclic. In general, the amalgamated product of free groups with cyclic amalgamations is free-by-(torsion-free nilpotent). The special case of the double of a free group amalgamating a cyclic subgroup is shown to be virtually free-by-abelian. Analagous results are obtained for certain one-relator HNN-extensions.