Isometric Group Actions on Banach Spaces and Representations Vanishing at Infinity

By Yves Cornulier, Romain Tessera, Alain Valette. 2006

53 citations recorded by the source corpus.

Our main result is that the simple Lie group G = Sp(n, 1) acts metrically properly isometrically on Lp(G) if p > 4n + 2. To prove this, we introduce Property $ \left( {\text{BP}_\text{0}^\text{V} } \right) $, with V being a Banach space: a locally compact group G has Property $ \left( {\text{BP}_\text{0}^\text{V} } \right) $ if every affine isometric action of G on V, such that the linear part is a C0-representation of G, either has a fixed point or is metrically proper. We prove that solvable groups, connected Lie groups, and linear algebraic groups over a local field of characteristic zero, have Property $ \left( {\text{BP}_\text{0}^\text{V} } \right) $. As a consequence, for unitary representations, we characterize those groups in the latter classes for which the first cohomology with respect to the left regular representation on L2(G) is nonzero; and we characterize uniform lattices in those groups for which the first L2-Betti number is nonzero.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1840cf78f29244b14ae55825d7e25f42fb2168b4

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In Algebra and Number Theory

  1. Andreas Thom
  2. J. Peterson
  3. Adrian Ioana
  4. Cyril Houdayer
  5. S. Vaes
  6. Sorin Popa
  7. Thomas Sinclair
  8. I. Chifan

In Mathematical Physics

  1. Andreas Thom
  2. J. Peterson
  3. Adrian Ioana
  4. Ben Hayes
  5. Cyril Houdayer
  6. S. Vaes
  7. Sorin Popa
  8. Thomas Sinclair

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