What question did this study set out to answer?

To determine the conditions under which the action of algebraic groups is topologically amenable.

June 14, 2026Open Access

Amenable actions of real and p -adic algebraic groups

Key Points

To determine the conditions under which the action of algebraic groups is topologically amenable.
Discussed the relationship between group actions and point stabilizers.
Utilized results from Borel–Serre regarding locally compact groups and spaces.
Examined smooth actions to establish criteria for topological amenability.
Established that the action of G on V is topologically amenable if all point stabilizers are solvable-by-compact.
Combined findings showing amenability relates to the topology of the group acting on variety.
Provided a structural insight into the groups and their actions in the context of algebraic geometry.

Abstract

Let K be a locally compact field of characteristic 0. Let G be a linear algebraic group defined over K , acting algebraically on an algebraic variety V . We prove that the action of G ( K ) (the group of K -rational points of G ) on V ( K ) is topologically amenable, if and only if all point stabilizers in G ( K ) are solvable-by-compact. This follows by combining a result by Borel–Serre 5with the following fact: let G be a second countable locally compact group acting continuously on a second countable locally compact space Y . If the action G ↷ Y is smooth (i.e. the Borel structure on G ∖ Y is countably separated), then topological amenability of G ↷ Y is equivalent to amenability of all point stabilizers in G .

Read Full Paperexternally

Perguntar à IA

Bookmark

View Full Paper