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Register a new userGeneric stabilisers for actions of reductive groups
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Authors
Benjamin Martin
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Let $G$ be a reductive algebraic group over an algebraically closed field and let $V$ be a quasi-projective $G$-variety. We prove that the set of points $vin V$ such that ${rm dim}(G_v)$ is minimal and $G_v$ is reductive is open. We also prove some results on the existence of principal stabilisers in an appropriate sense.
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We prove an effective variant of the Kazhdan-Margulis theorem generalized to stationary actions of semisimple groups over local fields: the probability that the stabilizer of a random point admits a non-trivial intersection with a small $r$-neighborhood of the identity is at most $beta r^delta$ for some explicit constants $beta, delta > 0$ depending only the group. This is a consequence of a key convolution inequality. We deduce that vanishing at infinity of injectivity radius implies finiteness of volume. Further applications are the compactness of the space of discrete stationary random subgroups and a novel proof of the fact that all lattices in semisimple groups are weakly cocompact.
Let $R$ be a commutative unital ring. Given a finitely-presented affine $R$-group $G$ acting on a finitely-presented $R$-scheme $X$ of finite type, we show that there is a prime $p_0$ so that for any $R$-algebra $k$ which is a field of characteristic $p > p_0$, the centralisers in $G_k$ of all subsets $U subseteq X(k)$ are smooth. We prove this using the Lefschetz principle together with careful application of Gr{o}bner basis techniques.
Let $G$ be a reductive algebraic group---possibly non-connected---over a field $k$ and let $H$ be a subgroup of $G$. If $G= GL_n$ then there is a degeneration process for obtaining from $H$ a completely reducible subgroup $H$ of $G$; one takes a limit of $H$ along a cocharacter of $G$ in an appropriate sense. We generalise this idea to arbitrary reductive $G$ using the notion of $G$-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a $G$-completely reducible subgroup $H$ of $G$, unique up to $G(k)$-conjugacy, which we call a $k$-semisimplification of $H$. This gives a single unifying construction which extends various special cases in the literature (in particular, it agrees with the usual notion for $G= GL_n$ and with Serres $G$-analogue of semisimplification for subgroups of $G(k)$). We also show that under some extra hypotheses, one can pick $H$ in a more canonical way using the Tits Centre Conjecture for spherical buildings and/or the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf and Rousseau.
Let $G$ be a simple algebraic group of type $G_2$ over an algebraically closed field of characteristic $2$. We give an example of a finite group $Gamma$ with Sylow $2$-subgroup $Gamma_2$ and an infinite family of pairwise non-conjugate homomorphisms $rhocolon Gammarightarrow G$ whose restrictions to $Gamma_2$ are all conjugate. This answers a question of Burkhard Kulshammer from 1995. We also give an action of $Gamma$ on a connected unipotent group $V$ such that the map of 1-cohomologies ${rm H}^1(Gamma,V)rightarrow {rm H}^1(Gamma_p,V)$ induced by restriction of 1-cocycles has an infinite fibre.
Let $G$ be a transitive permutation group on a finite set $Omega$ and recall that a base for $G$ is a subset of $Omega$ with trivial pointwise stabiliser. The base size of $G$, denoted $b(G)$, is the minimal size of a base. If $b(G)=2$ then we can study the Saxl graph $Sigma(G)$ of $G$, which has vertex set $Omega$ and two vertices are adjacent if they form a base. This is a vertex-transitive graph, which is conjectured to be connected with diameter at most $2$ when $G$ is primitive. In this paper, we combine probabilistic and computational methods to prove a strong form of this conjecture for all almost simple primitive groups with soluble point stabilisers. In this setting, we also establish best possible lower bounds on the clique and independence numbers of $Sigma(G)$ and we determine the groups with a unique regular suborbit, which can be interpreted in terms of the valency of $Sigma(G)$.
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