Related papers: Corners over quasirandom groups

We define and study a class of subshifts of finite type (SFTs) defined by a family of allowed patterns of the same shape where, for any contents of the shape minus a corner, the number of ways to fill in the corner is the same. The main…

The genus spectrum of a finite group $G$ is the set of all $g\geq 2$ such that $G$ acts faithfully and orientation-preserving on a closed compact orientable surface of genus $g$. This article is an overview of some results relating the…

A graph $G$ is $\textit{universal}$ for a (finite) family $\mathcal{H}$ of graphs if every $H \in \mathcal{H}$ is a subgraph of $G$. For a given family $\mathcal{H}$, the goal is to determine the smallest number of edges an…

May the $\mathit{triforce}$ be the 3-uniform hypergraph on six vertices with edges $\{123',12'3,1'23\}$. We show that the minimum triforce density in a 3-uniform hypergraph of edge density $\delta$ is $\delta^{4-o(1)}$ but not…

Let $1 \to K \longrightarrow G \stackrel{\pi}\longrightarrow Q$ be an exact sequence of hyperbolic groups. Let $Q_1 < Q$ be a quasiconvex subgroup and let $G_1=\pi^{-1}(Q_1)$. Under relatively mild conditions (e.g. if $K$ is a closed…

This survey paper concerns mainly with some asymptotic topological properties of finitely presented discrete groups: quasi-simple filtration (QSF), geometric simple connectivity (GSC), topological inverse-representations, and the notion of…

Let $G$ be a group such that any non-trivial representation has dimension at least $d$. Let $X=(X_{1},X_{2},\ldots,X_{t})$ and $Y=(Y_{1},Y_{2},\ldots,Y_{t})$ be distributions over $G^{t}$. Suppose that $X$ is independent from $Y$. We show…

Let $\alpha$ be an approximately inner flow on a $C^*$ algebra $A$ with generator $\delta$ and let $\delta_n$ denote the bounded generators of the approximating flows $\alpha^{(n)}$. We analyze the structure of the set \cd=\{x\in D(\delta):…

Let $\N$ denote the set of positive integers. The asymptotic density of the set $A \subseteq \N$ is $d(A) = \lim_{n\to\infty} |A\cap [1,n]|/n$, if this limit exists. Let $ \mathcal{AD}$ denote the set of all sets of positive integers that…

Given a finite point set $P\subset\mathbb{R}^d$, we call a multiset $A$ a one-sided weak $\varepsilon$-approximant for $P$ (with respect to convex sets), if $|P\cap C|/|P|-|A\cap C|/|A|\leq\varepsilon$ for every convex set $C$. We show…

A subset $\{g_1, \ldots , g_d\}$ of a finite group $G$ is said to invariably generate $G$ if the set $\{g_1^{x_1}, \ldots, g_d^{x_d}\}$ generates $G$ for every choice of $x_i \in G$. The Chebotarev invariant $C(G)$ of $G$ is the expected…

A finite group is said to be $n$-cyclic if it contains $n$ cyclic subgroups. For a finite group $G$, the ratio of the number of cyclic subgroups to the number of subgroups is known as the cyclicity degree of the group $G$ and is denoted by…

A $k-$quasiperfect dominating set ($k\ge 1$) of a graph $G$ is a vertex subset $S$ such that every vertex not in $S$ is adjacent to at least one and at most k vertices in $S$. The cardinality of a minimum k-quasiperfect dominating set in…

Consider a one-ended word-hyperbolic group. If it is the fundamental group of a graph of free groups with cyclic edge groups then either it is the fundamental group of a surface or it contains a finitely generated one-ended subgroup of…

Let $o(G)$ be the average of the element orders of a finite group $G$. A research topic concerning this quantity is understanding the relation between $o(G)$ and $o(H)$, where $H$ is a subgroup of $G$. Let $\mathcal{N}$ be the class of…

Direction sets, recently introduced by Leonetti and Sanna, are generalization of ratio sets of subsets of positive integers. In this article, we generalize the notion of direction sets and define {\it $k$-generalized direction sets} and…

We study a characteristic subgroup of finitely generated groups, consisting of elements with uniform upper bound for word-lengths. For a group $G$, we denote this subgroup by $G_{bound}$. We give sufficient criteria for triviality and…

We say A is a quasi-normal subgroup of the group G if the commensurator of A in G is all of G. We develop geometric versions of commensurators in finitely generated groups. In particular, g is an element of the commensurator of A in G iff…

In any semigroup $S$ satisfying the Strong Folner Condition, there are three natural notions of density for a subset $A$ of $S$: Folner density $d(A)$, Banach density $d^*(A)$, and translation density $d_t(A)$. If $S$ is commutative or left…

Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built on the set of conjugacy class sizes of $G$: this is the simple undirected graph whose vertices are the prime numbers dividing some conjugacy class size of $G$, two…