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Register a new userEndomorphisms of Lie groups over local fields
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Helge Glockner
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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.
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Let $G$ be a Lie group over a totally disconnected local field and $alpha$ be an analytic endomorphism of $G$. The contraction group of $alpha$ ist the set of all $xin G$ such that $alpha^n(x)to e$ as $ntoinfty$. Call sequence $(x_{-n})_{ngeq 0}$ in $G$ an $alpha$-regressive trajectory for $xin G$ if $alpha(x_{-n})=x_{-n+1}$ for all $ngeq 1$ and $x_0=x$. The anti-contraction group of $alpha$ is the set of all $xin G$ admitting an $alpha$-regressive trajectory $(x_{-n})_{ngeq 0}$ such that $x_{-n}to e$ as $ntoinfty$. The Levi subgroup is the set of all $xin G$ whose $alpha$-orbit is relatively compact, and such that $x$ admits an $alpha$-regressive trajectory $(x_{-n})_{ngeq 0}$ such that ${x_{-n}colon ngeq 0}$ is relatively compact. The big cell associated to $alpha$ is the set $Omega$ of all all products $xyz$ with $x$ in the contraction group, $y$ in the Levi subgroup and $z$ in the anti-contraction group. Let $pi$ be the mapping from the cartesian product of the contraction group, Levi subgroup and anti-contraction group to $Omega$ which maps $(x,y,z)$ to $xyz$. We show: $Omega$ is open in $G$ and $pi$ is {e}tale for suitable immersed Lie subgroup structures on the three subgroups just mentioned. Moreover, we study group-theoretic properties of contraction groups and anti-contraction groups.
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.
These are the lecture notes of a 2-hour mini-course on Lie groups over local fields presented at the Workshop on Totally Disconnected Groups, Graphs and Geometry at the Heinrich-Fabri-Institut Blaubeuren in May 2007. The goal of the notes is to provide an introduction to p-adic Lie groups and Lie groups over fields of formal Laurent series, with an emphasis on relations to the structure theory of totally disconnected, locally compact groups. In particular, they contain a discussion of the scale, tidy subgroups and contraction groups for automorphisms of Lie groups over local fields. Special attention is paid to the case of Lie groups over local fields of positive characteristic.
We prove that the outer automorphism group $Out(G)$ is residually finite when the group $G$ is virtually compact special (in the sense of Haglund and Wise) or when $G$ is isomorphic to the fundamental group of some compact $3$-manifold. To prove these results we characterize commensurating endomorphisms of acylindrically hyperbolic groups. An endomorphism $phi$ of a group $G$ is said to be commensurating, if for every $g in G$ some non-zero power of $phi(g)$ is conjugate to a non-zero power of $g$. Given an acylindrically hyperbolic group $G$, we show that any commensurating endomorphism of $G$ is inner modulo a small perturbation. This generalizes a theorem of Minasyan and Osin, which provided a similar statement in the case when $G$ is relatively hyperbolic. We then use this result to study pointwise inner and normal endomorphisms of acylindrically hyperbolic groups.
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.
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