Related papers: Free coarse groups
In this paper we identify different classes of free group extension using core graphs. We show that every free group extension $H\leq K\leq F$ has a base $B$ such that the associated pointed graph morphism…
A ballean (or coarse structure) is a set endowed with some family of subsets, the balls, is such a way that balleans with corresponding morphisms can be considered as asymptotic counterparts of uniform topological spaces. For a ballean…
A $P_4$-free graph is called a cograph. In this paper we partially characterize finite groups whose power graph is a cograph. As we will see, this problem is a generalization of the determination of groups in which every element has prime…
We investigate fixed point properties for isometric actions of topological groups on a wide class of metric spaces, with a particular emphasis on Hilbert spaces. Instead of requiring the action to be continuous, we assume that it is…
We classify all compact quantum groups whose C*-algebra sits inside that of the free unitary quantum groups $U_{N}^{+}$. In other words, we classify all discrete quantum subgroups of $\widehat{U}_{N}^{+}$, thereby proving a quantum variant…
Given a free-by-cyclic group $G = F_N \rtimes_\varphi \mathbb{Z}$ determined by any outer automorphism $\varphi \in \mathrm{Out}(F_N)$ which is represented by an expanding irreducible train-track map $f$, we construct a $K(G,1)$ $2$-complex…
We introduce the co-surface graph $\mathcal{CS}$ of a finitely generated free group $\mathbb{F}$ and use it to study the geometry of hyperbolic group extensions of $\mathbb{F}$. Among other things, we show that the Gromov boundary of the…
The theory of covering spaces is often used to prove the Nielsen-Schreier theorem, which states that every subgroup of a free group is free. We apply the more general theory of semicovering spaces to obtain analogous subgroup theorems for…
Let $p$ be an odd prime and $F_\infty$ be a $\mathbb{Z}_p$-extension of a number field $F$. Given an elliptic curve $E$ over $F$, we study the structure of the fine Selmer group over $F_\infty$. It is shown that under certain conditions,…
The main result of the paper is a formula for the fundamental group of the coarse moduli space of a topological stack. As an application, we find simple general formulas for the fundamental group of the coarse quotient of a group action on…
Let $C$ be a smooth, projective and geometrically connected curve defined over a finite field $\mathbb{F}_q(C)$. Given a semisimple $C-S$-group scheme $\underline{G}$ where $S$ is a finite set of closed points of $C$, we describe the set of…
We show the existence of free dense subgroups, generated by 2 elements, in the holomorphic shear and overshear group of complex-Euklidean space and extend this result to the group of holomorphic automorphisms of Stein manifolds with Density…
The topological fundamental group $\pi_{1}^{top}$ is a homotopy invariant finer than the usual fundamental group. It assigns to each space a quasitopological group and is discrete on spaces which admit universal covers. For an arbitrary…
Let $G$ be a hyperbolic group that splits as a graph of free groups with cyclic edge groups. We prove that, unless $G$ is isomorphic to a free product of free and surface groups, every finite abelian group $M$ appears as a direct summand in…
We consider the space of all configurations of finitely many (potentially nested) circles in the plane. We prove that this space is aspherical, and compute the fundamental group of each of its connected components. It turns out these…
We give a systematic definition of the fundamental groups of gropes, which we call grope groups. We show that there exists a nontrivial homomorphism from the minimal grope group M to another grope group G only if G is the free product of M…
The (proper) power graph of a group is a graph whose vertex set is the set of all (nontrivial) elements of the group and two distinct vertices are adjacent if one is a power of the other. Various kinds of planarity of (proper) power graphs…
Symmetry is at the heart of much of mathematics, physics, and art. Traditional geometric symmetry groups are defined in terms of isometries of the ambient space of a shape or pattern. If we slightly generalize this notion to allow the…
The problem of the existence of non-pseudo-$\aleph_1$-compact $\mathbb R$-factorizable groups is studied. It is proved that any such group is submetrizable and has weight larger than $\omega_1$. Closely related results concerning the…
Given a Tychonoff space $X$, let $F(X)$ and $A(X)$ be respectively the free topological group and the free Abelian topological group over $X$ in the sense of Markov. In this paper, we consider two topological properties of $F(X)$ or $A(X)$,…