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Cowling’s conjecture: the almost connected case

A mix of human-written and AI-generated textHuman understanding: some partsmath.FA — Functional Analysismath.GR — Group Theorymath.OA — Operator Algebras

Contributed by Cédric ARHANCET ↗

Version 1 / Oct 29, 2026 / CC BY-SA 4.0

Abstract

We prove that a locally compact group GG with amenable group G/G0G/G^0 of components has the Haagerup property if and only if it is weakly amenable with constant one. This proves Cowling's conjecture for connected Lie groups and almost connected locally compact groups. We also characterize weak amenability for arbitrary connected Lie groups and prove the product formula for the Cowling--Haagerup constant in terms of the simple factors of a Levi decomposition. Consequently, the constant is invariant under local isomorphism and decreases under quotients of connected locally compact groups. For groups with amenable group of components, it is unchanged by every closed central quotient. We deduce applications to lattices and their central quotients. The proof combines Ozawa's invariant-state obstruction with averaging over central characters and the tensor identity for induced representations.

Provenance statement

The author acknowledges the use of AI tools for language polishing, exploratory mathematical discussions and feedback during the development and preparation of this manuscript.

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Version history

  1. v1Submitted by Cédric ARHANCETInitial depositCurrentOct 28, 2026