Random generation in semisimple algebraic groups over local fields
نویسنده
چکیده
Let G be a semisimple algebraic group over a local field K of characteristic p. If the universal covering map Gsc → G is inseparable then no open subgroup of G(K) is finitely generated. Otherwise, for any compact open subgroup G with probability 1 two randomly chosen elements generate an open subgroup of G(K). Our main tool is a recent theorem by R. Pink characterizing compact groups linear over a local field.
منابع مشابه
Tits p-indexes of semisimple algebraic groups
The first author has recently shown that semisimple algebraic groups are classified up to motivic equivalence by the local versions of the classical Tits indexes over field extensions, known as Tits p-indexes. We provide in this article the complete description of the values of the Tits p-indexes over fields. From this exhaustive study, we also deduce criteria for motivic equivalence of semisim...
متن کاملCentral extensions of p-adic algebraic groups by finite p-groups
Some problems on algebraic groups over global fields like the congruence subgroup problem involve the determination of topological central extensions of the adelic group which, in turn, leads naturally to the study of topological central extensions of p-adic Lie groups by finite groups like the group of roots of unity in the p-adic field. Moreover, central extensions of semisimple p-adic Lie gr...
متن کاملFiniteness theorems for algebraic groups over function fields
We prove the finiteness of class numbers and Tate-Shafarevich sets for all affine group schemes of finite type over global function fields, as well as the finiteness of Tamagawa numbers and Godement’s compactness criterion (and a local analogue) for all such groups that are smooth and connected. This builds on the known cases of solvable and semisimple groups via systematic use of the recently ...
متن کاملInvariant Measures for Algebraic Actions, Zariski Dense Subgroups and Kazhdan’s Property (t )
Let k be any locally compact non-discrete field. We show that finite invariant measures for k-algebraic actions are obtained only via actions of compact groups. This extends both Borel’s density and fixed point theorems over local fields (for semisimple/solvable groups, resp.). We then prove that for k-algebraic actions, finitely additive finite invariant measures are obtained only via actions ...
متن کاملThe plancherel formula for sl(2) over a local field.
More than two decades ago, in his classical paper on the irreducible unitary representations of the Lorentz group, V. Bargmann initiated the concrete study of Fourier analysis on real Lie groups and obtained the analogue of the classical Fourier expansion theorem in the case of the Lorentz group. Since then the general theory for real semisimple Lie groups has been extensively developed, chiefl...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2002