Related papers: Unbounded towers and products

We consider the preservation under products, finite powers, and forcing, of a selection principle based covering property of $T_0$ topological groups. Though the paper is in part a survey, it contributes some new information, including: 1.…

We prove that the alternating group of a topologically free action of a countably infinite group $\Gamma$ on the Cantor set has the property that all of its $\ell^2$-Betti numbers vanish and, in the case that $\Gamma$ is amenable, is stable…

We introduce the space of infinite volume ends of a locally compact second countable (lcsc) space that admits a Radon measure. In certain cases, this coincides with the classical space of ends. Consider a discrete subgroup $\Gamma$ of a…

Motivated by the model theory of higher order logics, a certain kind of topological spaces had been introduced on ultraproducts. These spaces are called ultratopologies. Ultratopologies provide a natural extra topological structure for…

Let $(X,T)$ be a Cantor minimal system, and let $\Gamma$ denote either its associated topological full group or the full group of a Bratteli diagram associated with $(X,T)$. In this paper we describe the structure of indecomposable…

Given a finite generating set $A$ for a group $\Gamma$, we study the map $W \mapsto WA$ as a topological dynamical system -- a continuous self-map of the compact metrizable space of subsets of $\Gamma$. If the set $A$ generates $\Gamma$ as…

We prove that every finitely generated, residually finite group $G$ embeds into a finitely generated perfect branch group $\Gamma$ such that many properties of $G$ are preserved under this embedding. Among those are the properties of being…

We introduce a very natural topology on the set of total orderings of monomials of any algebra having a countable basis over a field. This topological space and some notable subspaces are compact. This topological framework allows us to…

The normal covering number $\gamma(G)$ of a finite, non-cyclic group $G$ is the least number of proper subgroups such that each element of $G$ lies in some conjugate of one of these subgroups. We prove that there is a positive constant $c$…

Given a closed hyperbolic 3-manifold $M$, we construct a tower of covers with increasing Heegaard genus, and give an explicit lower bound on the Heegaard genus of such covers as a function of their degree. Using similar methods we prove…

In 1992, David Wright proved a remarkable theorem about which contractible open manifolds are covering spaces. He showed that if a one-ended open manifold M has pro-monomorphic fundamental group at infinity which is not pro-trivial and is…

We describe a flexible construction that produces triples of finitely generated, residually finite groups $M\hookrightarrow P \hookrightarrow \Gamma$, where the maps induce isomorphisms of profinite completions…

Let $\mathcal G$ denote the space of finitely generated marked groups. We give equivalent characterizations of closed subspaces $\mathcal S\subseteq \mathcal G$ satisfying the following zero-one law: for any sentence $\sigma$ in the…

We use hyperbolic towers to answer some model theoretic questions around the generic type in the theory of free groups. We show that all the finitely generated models of this theory realize the generic type $p_0$, but that there is a…

We present a simple approach to questions of topological orbit equivalence for actions of countable groups on topological and smooth manifolds. For example, for any action of a countable group $\Gamma$ on a topological manifold where the…

Let $\Gamma$ be the fundamental group of a closed orientable surface of genus at least two. Consider the composition of a uniformly random element of $\mathrm{Hom}(\Gamma,S_n)$ with the $(n-1)$-dimensional irreducible representation of…

We prove some finiteness results for discrete isometry groups $\Gamma$ of uniformly packed CAT$(0)$-spaces $X$ with uniformly bounded codiameter (up to group isomorphism), and for CAT$(0)$-orbispaces $M = \Gamma \backslash X$ (up to…

Results of Sierpinski and others have shown that certain finite-dimensional product sets can be written as unions of subsets, each of which is "narrow" in a corresponding direction; that is, each line in that direction intersects the subset…

Let G be a finite group and let M be a G-manifold. We introduce the concept of generalized orbifold invariants of M/G associated to an arbitrary group Gamma, an arbitrary Gamma-set, and an arbitrary covering space of a connected manifold…

A subset of the Cantor cube is null-additive if its algebraic sum with any null set is null. We construct a set of cardinality continuum such that: all continuous images of the set into the Cantor cube are null-additive, it contains a…