Related papers: Finitely approximable groups and actions Part II: …

We prove that every finitely generated group $G$ discriminated by a locally quasi-convex torsion-free hyperbolic group $\Gamma$ is effectively coherent: that is, presentations for finitely generated subgroups can be computed from the…

We give a geometric characterization of finite rational groups. In particular, we prove that a finite group is rational if and only if there exists a finite geometry $\Gamma$ of type $I$ and action of $G$ on $\Gamma$ as a group of…

We show that every word hyperbolic, surface-by-(noncyclic) free group Gamma is as rigid as possible: the quasi-isometry group of Gamma equals the abstract commensurator group Comm(Gamma), which in turn contains Gamma as a finite index…

We prove the equivalence of the two important facts about finite metric spaces and universal Urysohn metric spaces $\Bbb U$, namely theorem A and theorem B below: Theorem A (Approximation): The group of isometry $ISO(\Bbb U)$ contains…

We introduce a quantitative notion of lawlessness for finitely generated groups, encoded by the "lawlessness growth function" $\mathcal{A}_{\Gamma} : \mathbb{N} \rightarrow \mathbb{N}$. We show that $\mathcal{A}_{\Gamma}$ is bounded iff…

As a contribution of the programme of Goswami and Mandal (2014), we carry out explicit computations of $\mathbb{Q}(\Gamma,S)$, the quantum isometry group of the canonical spectral triple on $C_{r}^{*}(\Gamma)$ coming from the word length…

Given a finitely generated linear group $G$ over $\mathbb{Q}$, we construct a simple group $\Gamma$ that has the same finiteness properties as $G$ and admits $G$ as a quasi-retract. As an application, we construct a simple group of type…

The end compactification |\Gamma| of the locally finite graph \Gamma is the union of the graph and its ends, endowed with a suitable topology. We show that \pi_1(|\Gamma|) embeds into a nonstandard free group with hyperfinitely many…

Every finitely generated self-similar group naturally produces an infinite sequence of finite $d$-regular graphs $\Gamma_n$. We construct self-similar groups, whose graphs $\Gamma_n$ can be represented as an iterated zig-zag product and…

Let $H^n$ be the hyperbolic n-space with $n\geq 2$. Suppose that $\Gamma<Isom H^n$ is a discrete, torsion free subgroup and $a$ is a point in the domain of discontinuity $\Omega(\Gamma)$. Let $p$ be the projection map from $H^n$ to the…

We show that, given a finitely generated group $G$ as the coordinate group of a finite system of equations over a torsion-free hyperbolic group $\Gamma$, there is an algorithm which constructs a cover of a canonical solution diagram. The…

In a series of papers we proposed a model unifying general relativity and quantum mechanics. The idea was to deduce both general relativity and quantum mechanics from a noncommutative algebra ${\cal A}_{\Gamma}$ defined on a transformation…

Let $\Gamma$ be a finitely generated group and $N$ be a normal subgroup of $\Gamma$. The fiber product of $\Gamma$ with respect to $N$ is the subgroup $\Gamma \times_N \Gamma=\{(\gamma, \gamma w): \gamma \in \Gamma, w \in N\}$ of the direct…

We present novel constructions concerning the homology of finitely generated groups. Each construction draws on ideas of Gilbert Baumslag. There is a finitely presented acyclic group $U$ such that $U$ has no proper subgroups of finite index…

We construct a universal action of a countable locally finite group (the Hall's group) on a separable metric space by isometries. This single action contains all actions of all countable locally finite groups on all separable metric spaces…

We study the large-scale geometry of graph braid groups $\mathbb{B}_n(\mathsf{\Gamma})$, viewed as the fundamental groups of discrete configuration spaces $UD_n(\mathsf{\Gamma})$, which are special cube complexes in the sense of…

The quantum completion of the space of connections in a manifold can be seen as the set of all morphisms from the groupoid of the edges of the manifold to the (compact) gauge group. This algebraic construction generalizes the analogous…

We show that many countable groups acting on trees, including free products of infinite countable groups and surface groups, are isomorphic to dense subgroups of isometry groups of bounded Urysohn spaces. This extends previous results of…

Given a countable group $\Gamma$, letting $\mathcal{K}_\Gamma$ denote the class of {\pmp} actions of $\Gamma$, we study the question of when the model companion of $\mathcal{K}_\Gamma$ exists. Berenstein, Henson, and Ibarluc\'ia showed that…

Let $\Gamma$ be a finite rank subgroup of $\overline{\mathbb{Q}}^*$. We prove that the multiplicative group of the field generated by all elements in the divisible hull of $\Gamma$, is free abelian modulo this divisible hull. This proves…