Related papers: Limit set intersection theorems for Kleinian group…

We show that a finite collection of stable subgroups of a finitely generated group has finite height, finite width and bounded packing. We then use knowledge about intersections of conjugates to characterize finite families of…

The intersection graph of a group $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$…

We give a topological framework for the study of Sela's limit groups: limit groups are limits of free groups in a compact space of marked groups. Many results get a natural interpretation in this setting. The class of limit groups is known…

We consider two decision problems in infinite groups. The first problem is Subgroup Intersection: given two finitely generated subgroups $\langle \mathcal{G} \rangle, \langle \mathcal{H} \rangle$ of a group $G$, decide whether the…

Let $S$ and $T$ be subsets of $\mathbf{F}_q^n$. We show there are subsets $S'$ of $S$ and $T'$ of $T$ such that $S+T$ is the union of $S+T'$ and $S'+T$, with $|S'| + |T'|$ bounded by $c^n$ with $c < q$. The proof relies on the method of…

We propose an axiomatic approach towards studying unlikely intersections by introducing the framework of distinguished categories. This includes commutative algebraic groups and mixed Shimura varieties. It allows us to define all basic…

Let $k$ be a complete non-archimedean field (non trivially valued). Given a reductive $k$-group $G$, we prove that hyperspecial subgroups of $G(k)$ (i.e. those arising from reductive models of $G$) are maximal among bounded subgroups. The…

Let $n > k > t \geq j \geq 1$ be integers. Let $X$ be an $n$-element set, ${X\choose k}$ the collection of its $k$-subsets. A family $\mathcal F \subset {X\choose k}$ is called $t$-intersecting if $|F \cap F'| \geq t$ for all $F, F' \in…

It is a well-known fact that for any natural number $n$, there always exists a prime in $[n, 2n]$. Our aim in this note is to generalize this result to $[n, kn]$. A lower as well as an upper bound on the number of primes in $[n, kn]$ were…

Intersection growth concerns the asymptotic behavior of the index of the intersection of all subgroups of a group that have index at most n. In this note we show that the intersection growth of some groups may not be a nicely behaved…

In this short note, we address two problems in extremal set theory regarding intersecting families. The first problem is a question posed by Kupavskii: is it true that given two disjoint cross-intersecting families $\mathcal{A}, \mathcal{B}…

Two families $\mathcal{F}$ and $\mathcal{G}$ are called cross-intersecting if for every $F\in \mathcal{F}$ and $G\in \mathcal{G}$, the intersection $F\cap G$ is non-empty. For any positive integers $n$ and $k$, let $\binom{[n]}{k}$ denote…

Let $\Gamma<\mathrm{PSL}_2(\mathbb{C})\simeq \mathrm{Isom}^+(\mathbb{H}^3)$ be a finitely generated non-Fuchsian Kleinian group whose ordinary set $\Omega=\mathbb{S}^2-\Lambda$ has at least two components. Let $\rho : \Gamma \to…

We show that for any finitely generated subgroup $H$ of a limit group $L$ there exists a finite-index subgroup $K$ containing $H$, such that $K$ is a subgroup of a group obtained from $H$ by a series of extensions of centralizers and free…

Let $\Gamma$ be a countable discrete group, $H$ a lcsc totally disconnected group and $\rho : \Gamma \rightarrow H$ a homomorphism with dense image. We develop a general and explicit technique which provides, for every compact open subgroup…

In this paper, we present an alternative proof of a finiteness theorem due to Bombieri, Masser and Zannier concerning intersections of a curve in the multiplicative group of dimension n with algebraic subgroups of dimension n-2. The proof…

We prove the following conjecture of Furstenberg (1969): if $A,B\subset [0,1]$ are closed and invariant under $\times p \mod 1$ and $\times q \mod 1$, respectively, and if $\log p/\log q\notin \mathbb{Q}$, then for all real numbers $u$ and…

In this paper we show how some known weak forms of the Zilber--Pink conjecture can be strengthened by combining them with the Mordell--Lang conjecture or its variants. We illustrate this idea by proving some theorems on atypical…

A family $F$ of sets is said to be $t$-intersecting if $|A \cap B| \geq t$ for any $A,B \in F$. The seminal Complete Intersection Theorem of Ahlswede and Khachatrian (1997) gives the maximal size $f(n,k,t)$ of a $t$-intersecting family of…

For a normal subgroup $N$ of the free group $\F_d$ with at least two generators we introduce the radial limit set $\Lr(N,\Phi)$ of $N$ with respect to a graph directed Markov system $\Phi$ associated to $\F_d$. These sets are shown to…