Related papers: Primitive stability and Bowditch's BQ-condition ar…

The equivalence of two conditions on the primitive elements in an $SL(2,\mathbb C)$ representation of the free group $F_2 = <a,b>$, namely Minsky's condition of primitive stability and the $BQ$-conditions introduced by Bowditch and…

The two largest known domains of discontinuity for the action of Out(F_2) on the PSL(2,C)-character variety of F_2 - defined by Minsky's primitive stability, and Bowditch's Q-conditions - were proven to be equal independently by Lee-Xu and…

We study a generalization of the $BQ$-conditions, introduced by Bowditch and further developed by Tan-Wong-Zhang, for representations of the free group of rank two into isometry groups of Gromov-hyperbolic spaces. We show the existence of…

We prove that the set of Bowditch representations (introduced by Bowditch in 1998, then generalized by Tan, Wong and Zhang in 2008) and the set of primitive-stable representations (introduced by Minsky in 2013) of the free group of rank two…

For the action of the outer automorphism group of the rank two free group on the corresponding variety of PSL(2,C) characters, two domains of discontinuity have been known to exist that are strictly larger than the set of Schottky…

Let $\mathbb{F}_{q^n}$ be a finite field with $q^n$ elements, and let $m_1$ and $m_2$ be positive integers. Given polynomials $f_1(x), f_2(x) \in \mathbb{F}_q[x]$ with $\textrm{deg}(f_i(x)) \leq m_i$, for $i = 1, 2$, and such that the…

The celebrated Primitive Normal Basis Theorem states that for any $n\ge 2$ and any finite field $\mathbb F_q$, there exists an element $\alpha\in \mathbb F_{q^n}$ that is simultaneously primitive and normal over $\mathbb F_q$. In this…

In a recent paper, Soner, Touzi and Zhang [20] have introduced a notion of second order backward stochastic differential equations (2BSDEs for short), which are naturally linked to a class of fully non-linear PDEs. They proved existence and…

In this paper, we study representations from the four-punctured sphere group into isometry groups of Gromov-hyperbolic spaces. We prove that the set of simple-stable representations (in analogy with Minsky's notion of primitive-stability)…

We first establish the existence of an unbounded solution to a backward stochastic differential equation (BSDE) with generator $g$ allowing a general growth in the state variable $y$ and a sub-quadratic growth in the state variable $z$,…

In this paper we generalize the results of Sharma, Awasthi and Gupta (see \cite{SAG}). We work over a field of any characteristic with $q = p^k$ elements and we give a sufficient condition for the existence of a primitive element $\alpha…

Let F_k be the free group on k generators. A word w \in F_k is called primitive if it belongs to some basis of F_k. We investigate two criteria for primitivity, and consider more generally, subgroups of F_k which are free factors. The first…

For $q=3^r$ ($r>0$), denote by $\mathbb{F}_q$ the finite field of order $q$ and for a positive integer $m\geq2$, let $\mathbb{F}_{q^m}$ be its extension field of degree $m$. We establish a sufficient condition for existence of a primitive…

In this paper, we give a complete criterion for a discrete faithful representation $\rho:F_n \ra \pslc$ to be primitive stable. This will answer Minsky's conjectures about geometric conditions on $\H^3/\rho(F_n)$ regarding the primitive…

An element $\alpha \in \mathbb F_{q^n}$ is \emph{normal} if $\mathcal{B} = \{\alpha, \alpha^q, \ldots, \alpha^{q^{n-1}}\}$ forms a basis of $\mathbb F_{q^n}$ as a vector space over $\mathbb F_{q}$; in this case, $\mathcal{B}$ is a normal…

We present different characterizations of the notion of irreducibility for Quantum Markov Semigroups (QMSs) and investigate its relationship with other relevant features of the dynamics, such as primitivity, positivity improvement and…

Let $\mathbb{F}_{q^n}$ be the extension of the field $\mathbb{F}_q$ of degree n, where $q$ is power of prime $p$, i.e $q=p^k$, where k is a positive integer. In this paper, we provide sufficient condition for the existence of a primitive…

We establish new characterizations of primitive elements and free factors in free groups, which are based on the distributions they induce on finite groups. For every finite group $G$, a word $w$ in the free group on $k$ generators induces…

We propose a quantitative notion of permutation stability for finitely generated groups. Our notion is related to, but distinct from, the ``stability rate'' introduced by Becker and Mosheiff (which is valid within the class of finitely…

In this paper we strengthen Kolchin's theorem ([1]) in the ordinary case. It states that if a differential field $E$ is finitely generated over a differential subfield $F \subset E$, $trdeg_F E < \infty$, and $F$ contains a nonconstant,…