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On actions on the real line of some finitely generated groups

By Nicolás Matte Bon

Appears in collection : Big Mapping Class Groups and Diffeomorphism Groups / Gros groupes modulaires et groupes de difféomorphismes

It is well-known that a finitely generated group acts faithfully on the real line if and only if it is left-orderable. When this is the case, it is natural to study the possible representations of G into the group of homeomorphisms the real line. Several questions can be asked: how many dynamically distinct representations does G admit, and is there a 'nice' invariant that distinguishes such representations under (semi-)conjugacy? Which such representations can be conjugated into the group of diffeomorphisms (of a given regularity C^r)? Which such representations are rigid under perturbations, or when can two representations be deformed into one another by a continuous path? An object of great help in addressing such questions is the Deroin space of the group G, which is a compact space endowed with a flow which encodes all possible actions of G on the line (whose construction is based on work of Deroin-Kleptsyn-Navas-Parwani). Which I will survey some results that address these questions for various groups, including finitely generated solvable groups and a class of groups which includes Thompson's group F. The talk is based on joint works with J. Brum, C. Rivas and M. Triestino.

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Citation data

  • DOI 10.24350/CIRM.V.19968403
  • Cite this video Matte Bon Nicolás (10/13/22). On actions on the real line of some finitely generated groups. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19968403
  • URL https://dx.doi.org/10.24350/CIRM.V.19968403

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