Related papers: On subgroups with narrow Schreier graphs

We present new infinite families of expander graphs of vertex degree 4, which is the minimal possible degree for Cayley graph expanders. Our first family defines a tower of coverings (with covering indices equals 2) and our second family is…

We say that a group G is a cube group if it is generated by a set S of involutions such that the corresponding Cayley graph Cay(G,S) is isomorphic to a cube. Equivalently, G is a cube group if it acts on a cube such that the action is…

Algebraically fibering group is an algebraic generalization of the fibered 3-manifold group in higher dimensions. Let $M(\mathcal{P})$ and $M(\mathcal{E})$ be the cusped and compact hyperbolic real moment-angled manifolds associated to the…

Given any directed graph E one can construct a graph inverse semigroup G(E), where, roughly speaking, elements correspond to paths in the graph. In this paper we study the semigroup-theoretic structure of G(E). Specifically, we describe the…

The relative Cayley graph of a group $G$ with respect to its proper subgroup $H$, is a graph whose vertices are elements of $G$ and two vertices $h\in H$ and $g\in G$ are adjacent if $g=hc$ for some $c\in C$, where $C$ is an inversed-closed…

A group is tubular if it acts on a tree with $\mathbb{Z}^2$ vertex stabilizers and $\mathbb{Z}$ edge stabilizers. We prove that a tubular group is virtually special if and only if it acts freely on a locally finite CAT(0) cube complex.…

We show that certain right-angled Coxeter groups have finite index subgroups that quotient to $\mathbb Z$ with finitely generated kernels. The proof uses Bestvina-Brady Morse theory facilitated by combinatorial arguments. We describe a…

A finite group is called $\psi$-divisible iff $\psi(H)|\psi(G)$ for any subgroup $H$ of a finite group $G$. Here, $\psi(G)$ is the sum of element orders of $G$. For now, the only known examples of such groups are the cyclic ones of…

Let $\Gamma$ be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph $\Gamma$ is $\mathcal{CFS}$, we prove that the right-angled Coxeter group $G_\Gamma$ is virtually a…

In this paper, we study strongly quasiconvex subgroups in a finitely generated $3$--manifold group $\pi_1(M)$. We prove that if $M$ is a compact, orientable $3$--manifold that does not have a summand supporting the Sol geometry in its…

The cogrowth series of a group with respect to a finite generating set is an important combinatorial quantity that seems very difficult to compute exactly, as evidenced by the scarcity of known examples. In this paper, we give a particular…

We prove that a connected, locally finite, quasi-transitive graph which is quasi-isometric to a planar graph is necessarily accessible. This leads to a complete classification of the finitely generated groups which are quasi-isometric to…

A connected graph is called \emph{geodetic} if there is a unique geodesic between each pair of vertices. In this paper we prove that if a finitely generated group admits a Cayley graph which is geodetic, then the group must be virtually…

A hereditary class H of graphs is $\chi$-bounded if there is a $\chi$-binding function f such that for every $G$ in $H$, $\chi(G)$ less than or equal to $f(\omega(G))$. Here we prove that if a graph $G$ is free of 1. {Chair; P$_4$+K$_1$} or…

We investigate pairs $(G,Y)$, where $G$ is a reductive algebraic group and $Y$ a purely-odd $G$-superscheme, asking when a pair corresponds to a quasi-reductive algebraic supergroup $\mathbb{G}$, that is, $\mathbb{G}_{\text{ev}}$ is…

In \cite{Oh22}, the second author defined a complex of groups decomposition of the fundamental group of a finitely generated 2-dimensional special group, called an \emph{intersection complex}, which is a quasi-isometry invariant. In this…

In the present paper, the structure of a finite group $G$ having a nonnormal T.I. subgroup $H$ which is also a Hall $\pi$-subgroup is studied. As a generalization of a result due to Gow, we prove that $H$ is a Frobenius complement whenever…

For $i= 1,2$, let $G_i$ be cocompact groups of isometries of hyperbolic space $\Hyp^n$ of real dimension $n$, $n \geq 3$. Let $H_i \subset G_i$ be infinite index quasiconvex subgroups satisfying one of the following conditions: 1) limit set…

Given a finite simplicial graph ${\cal G}$, the graph group $G{\cal G}$" is the group with generators in one-to-one correspondence with the vertices of ${\cal G}$ and with relations stating two generators commute if their associated…

A subgroup $H$ of a group $G$ is $commensurated$ in $G$ if for each $g\in G$, $gHg^{-1}\cap H$ has finite index in both $H$ and $gHg^{-1}$. If there is a sequence of subgroups $H=Q_0\prec Q_1\prec ...\prec Q_{k}\prec Q_{k+1}=G$ where $Q_i$…