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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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We show that certain groups of piecewise linear homeomorphims of the interval are invariably generated.
We prove that R. Thompson groups F, T, V have linear divergence functions.
The groups G_{k,1} of Richard Thompson and Graham Higman can be generalized in a natural way to monoids, that we call M_{k,1}, and to inverse monoids, called Inv_{k,1}; this is done by simply generalizing bijections to partial functions or partial injective functions. The monoids M_{k,1} have connections with circuit complexity (studied in another paper). Here we prove that M_{k,1} and Inv_{k,1} are congruence-simple for all k. Their Green relations J and D are characterized: M_{k,1} and Inv_{k,1} are J-0-simple, and they have k-1 non-zero D-classes. They are submonoids of the multiplicative part of the Cuntz algebra O_k. They are finitely generated, and their word problem over any finite generating set is in P. Their word problem is coNP-complete over certain infinite generating sets. Changes in this version: Section 4 has been thoroughly revised, and errors have been corrected; however, the main results of Section 4 do not change. Sections 1, 2, and 3 are unchanged, except for the proof of Theorem 2.3, which was incomplete; a complete proof was published in the Appendix of reference [6], and is also given here.
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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