We show that the class of amalgamated free products of two free groups over a cyclic subgroup admits amenable, faithful and transitive actions on infinite countable sets. This work generalizes the results on such actions for doubles of free group on 2 generators.
In 1930, Wilhelm Magnus introduced the so-called Freiheitssatz: Let $F$ be a free group with basis $\mathcal{X}$ and let $r$ be a cyclically reduced element of $F$ which contains a basis element $x \in \mathcal{X}$, then every non-trivial element of the normal closure of $r$ in $F$ contains the basis element $x$. Equivalently, the subgroup freely generated by $\mathcal{X} \backslash \{x\}$ embeds canonically into the quotient group $F / \langle \! \langle r \rangle \! \rangle_{F}$. In this article, we want to introduce a Freiheitssatz for amalgamated products $G=A \ast_{U} B$ of free groups $A…
We discuss the linearity and discreteness of amalgamated products of linear word-hyperbolic groups. In particular, we prove that the double of an Anosov group along a maximal cyclic subgroup is always linear, and we construct examples of such groups which do not admit any discrete and faithful representation in rank 1. We also build new examples of non-linear word-hyperbolic groups, elaborating on a previous work of Canary--Stover--Tsouvalas.
Let C be a class of groups. We give sufficient conditions ensuring that a free product of residually C groups is again residually C, and analogous conditions are given for locally embeddable into C groups. As a corollary, we obtain that the class of residually amenable groups and the one of LEA groups (or initially subamenable groups in the terminology of Gromov) are closed under taking free products. Moreover, we consider the pro-C topology and we characterize special HNN extensions and amalgamated free products that are residually C, where C is a suitable class of groups. In this way, we des…
We study Cartan subalgebras in the context of amalgamated free product II$_1$ factors and obtain several uniqueness and non-existence results. We prove that if $Γ$ belongs to a large class of amalgamated free product groups (which contains the free product of any two infinite groups) then any II$_1$ factor $L^{\infty}(X)\rtimesΓ$ arising from a free ergodic probability measure preserving action of $Γ$ has a unique Cartan subalgebra, up to unitary conjugacy. We also prove that if $\mathcal R=\mathcal R_1*\mathcal R_2$ is the free product of any two non-hyperfinite countable ergodic probability …
Garside groups are a natural lattice-theoretic generalisation of the braid groups and spherical type Artin--Tits groups. Here we show that the class of Garside groups is closed under some free products with cyclic amalgamated subgroups. We deduce that every tree product of infinite cyclic groups is a Garside group. Moreover, we study those cyclic HNN extensions of Garside groups that are Garside groups as well. Using a theorem of Pietrowski, we conclude this paper by stating that a non-cyclic one-relator group is Garside if and only if its centre is nontrivial.
We construct the type-preserving panel-regular lattices in buildings of type M, for M with entries two, three or infinity, which have cyclic stabilizers of spherical 2-residues. We obtain these lattices as fundamental groups of their associated quotient buildings, which are constructed by amalgamating quotients of generalized digons and triangles by actions of Singer cyclic groups. Each quotient construction has an associated gluing matrix which encodes how the quotient generalized polygons are amalgamated. We show that these gluing matrices also encode presentations of the panel-regular latti…
It is an old and challenging topic to investigate for which discrete groups G the full group C*-algebra C*(G) is residually finite-dimensional (RFD). In particular not much is known about how the RFD property behaves under fundamental constructions, such as amalgamated free products and HNN-extensions. In [CS19] it was proved that central amalgamated free products of virtually abelian groups are RFD. In this paper we prove that this holds much beyond this case. Our method is based on showing a certain approximation property for characters induced from central subgroups. In particular it allows…
A finitely generated group G is termed parafree if it is residually nilpotent and it has the same isomorphism types of nilpotent quotients as some free group. The two main results of this MSc. Thesis characterise the parafreeness of two constructions, namely amalgamated products with cyclic amalgams and cyclic HNN extensions.
In various occasions the conjugacy problem in finitely generated amalgamated products and HNN extensions can be decided efficiently for elements which cannot be conjugated into the base groups. This observation asks for a bound on how many such elements there are. Such bounds can be derived using the theory of amenable graphs: In this work we examine Schreier graphs of amalgamated products and HNN extensions. For an amalgamated product $G = H *_A K $ with $[H:A] \geq [K:A] \geq 2$, the Schreier graph with respect to $H$ or $K$ turns out to be non-amenable if and only if $[H:A] \geq 3$. Moreove…
We classify all subgroups of $SO(3)$ that are generated by two elements, each a rotation of finite order, about axes separated by an angle that is a rational multiple of $π$. In all cases we give a presentation of the subgroup. In most cases the subgroup is the free product, or the amalgamated free product, of cyclic groups or dihedral groups. The relations between the generators are all simple consequences of standard facts about rotations by $π$ and $π/2$. Embedded in the subgroups are explicit free groups on 2 generators, as used in the Banach-Tarski paradox.
A group $G$ is called $W^*$-superrigid (resp. $C^*$-superrigid) if it is completely recognizable from its von Neumann algebra $L(G)$ (resp. reduced $C^*$-algebra $C_r^*(G)$). Developing new technical aspects in Popa's deformation/rigidity theory we introduce several new classes of $W^*$-superrigid groups which appear as direct products, semidirect products with non-amenable core and iterations of amalgamated free products and HNN-extensions. As a byproduct we obtain new rigidity results in $C^*$-algebra theory including additional examples of $C^*$-superrigid groups and explicit computations o…
Group measure space decomposition of II_1 factors and W*-superrigidity
本文证明了群测度空间II_1的一个由任意自由遍历概率测度保持（p. m. p.）群Γ在一个相当大的族G中的作用，该族G包含Kazhdan群和一个非平凡群的所有自由积，以及在一个顺从子群上的某些合并的自由积.本文证明了：若T_n表示PSL（n，Z）中的上三角形矩阵群，则T_n上PSL（n，Z）与自身的合并自由积的任意自由混合p. m. p.作用是W*-超刚性的，即L^ \infty（X）\rtimes Γ与任意群测度空间因子L^ \infty（Y）\rtimes Λ之间的同构都来自于作用的共轭性.我们还证明了对于族G中的许多群Γ，Γ的Bernoulli作用是W*-超刚性的.
We prove a Tits alternative for topological full groups of minimal actions of finitely generated groups. On the one hand, we show that topological full groups of minimal actions of virtually cyclic groups are amenable. By doing so, we generalize the result of Juschenko and Monod for $\mathbf{Z}$-actions. On the other hand, when a finitely generated group $G$ is not virtually cyclic, then we construct a minimal free action of $G$ on a Cantor space such that the topological full group contains a non-abelian free group.
A notion of geometric complexity and its application to topological rigidity
我们引入了一个称为有限分解复杂性（FDC）的几何不变量来研究流形的拓扑刚性。例如，我们证明了如果紧致非球面流形M的基本群具有FDC，并且如果N同伦等价于M，则M x R^n同胚于N x R^n，其中n足够大。这个命题被称为稳定的波莱尔猜想。另一方面，我们证明了FDC群的类包括GL（n，K）的所有可数子群，对于任何域K，所有初等顺从群，并且在取子群，扩展，自由合并产品，HNN扩展和直接工会下是封闭的。
In this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type $\operatorname{FP}_n$ over a profinite ring $R$, analogous to the Bieri-Eckmann criteria for abstract groups. We use these to prove that the class of groups of type $\operatorname{FP}_n$ is closed under extensions, quotients by subgroups of type $\operatorname{FP}_n$, proper amalgamated free products and proper $\operatorname{HNN}$-extensions, for each $n$. We show, as a consequence of this, that elementary amenable profinite groups of finite rank are of type…
