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Register a new userAlgebraic and topological properties of Riordan groups over finite fields
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In this paper, we investigate algebraic and topological properties of the Riordan groups over finite fields. These groups provide a new class of topologically finitely generated profinite groups with finite width. We also introduce, characterize index-subgroups of our Riordan groups, and finally we show exactly the range of Hausdorff dimensions of these groups. The latter results are analogous to the work of Barnea and Klopsch for the Nottingham groups.
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We show that the algebraic fundamental group of a smooth projective curve over a finite field admits a finite topological presentation where the number of relations does not exceed the number of generators.
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.
Lie groups over local fields furnish prime examples of totally disconnected, locally compact groups. We discuss the scale, tidy subgroups and further subgroups (like contraction subgroups) for analytic endomorphisms of such groups. The text is both a research article and a worked out set of lecture notes for a mini-course held June 27-July 1, 2016 at the MATRIX research center in Creswick (Australia) as part of the Winter of Disconnectedness. The text can be read in parallel to the earlier lecture notes arXiv:0804.2234 which are devoted to automorphisms, with sketches of proof. Complementary aspects are emphasized.
Let G be a Lie group over a local field of positive characteristic which admits a contractive automorphism f (i.e., the forward iterates f^n(x) of each group element x converge to the neutral element 1). We show that then G is a torsion group of finite exponent and nilpotent. We also obtain results concerning the interplay between contractive automorphisms of Lie groups over local fields, contractive automorphisms of their Lie algebras, and positive gradations thereon. Some of the results even extend to Lie groups over arbitrary complete ultrametric fields.
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