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تسجيل مستخدم جديدInvariable generation of Thompson groups
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A subset $S$ of a group $G$ invariably generates $G$ if $G= langle s^{g(s)} | s in Srangle$ for every choice of $g(s) in G,s in S$. We say that a group $G$ is invariably generated if such $S$ exists, or equivalently if $S=G$ invariably generates $G$. In this paper, we study invariable generation of Thompson groups. We show that Thompson group $F$ is invariable generated by a finite set, whereas Thompson groups $T$ and $V$ are not invariable generated.
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Let $G$ be a simple algebraic group over an algebraically closed field $k$ and let $C_1, ldots, C_t$ be non-central conjugacy classes in $G$. In this paper, we consider the problem of determining whether there exist $g_i in C_i$ such that $langle g_1
, ldots, g_t rangle$ is Zariski dense in $G$. First we establish a general result, which shows that if $Omega$ is an irreducible subvariety of $G^t$, then the set of tuples in $Omega$ generating a dense subgroup of $G$ is either empty or dense in $Omega$. In the special case $Omega = C_1 times cdots times C_t$, by considering the dimensions of fixed point spaces, we prove that this set is dense when $G$ is an exceptional algebraic group and $t geqslant 5$, assuming $k$ is not algebraic over a finite field. In fact, for $G=G_2$ we only need $t geqslant 4$ and both of these bounds are best possible. As an application, we show that many faithful representations of exceptional algebraic groups are generically free. We also establish new results on the topological generation of exceptional groups in the special case $t=2$, which have applications to random generation of finite exceptional groups of Lie type. In particular, we prove a conjecture of Liebeck and Shalev on the random $(r,s)$-generation of exceptional groups.
Let $G$ be a simple algebraic group over an algebraically closed field and let $X$ be an irreducible subvariety of $G^r$ with $r geqslant 2$. In this paper, we consider the general problem of determining if there exists a tuple $(x_1, ldots, x_r) in
X$ such that $langle x_1, ldots, x_r rangle$ is Zariski dense in $G$. We are primarily interested in the case where $X = C_1 times cdots times C_r$ and each $C_i$ is a conjugacy class of $G$ comprising elements of prime order modulo the center of $G$. In this setting, our main theorem gives a complete solution to the problem when $G$ is a symplectic or orthogonal group. By combining our results with earlier work on linear and exceptional groups, this gives a complete solution for all simple algebraic groups. We also present several applications. For example, we use our main theorem to show that many faithful representations of symplectic and orthogonal groups are generically free. We also establish new asymptotic results on the probabilistic generation of finite simple groups by pairs of prime order elements, completing a line of research initiated by Liebeck and Shalev over 25 years ago.
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