Classifying quasi-representations of groups via their characters

Abstract

For every countable discrete group, we construct a character for finite-dimensional unitary quasi-representations, with values in the inverse limit of the equivariant K-theory of its Rips complexes. The character is a complete invariant for quasi-representations up to stable equivalence for amenable groups and, more generally, for groups satisfying the Haagerup property and QR-quasidiagonality. For Haagerup groups satisfying quantitative QR-quasidiagonality, we also obtain a quantiative version of the theorems. These conclusions apply when the maximal group C^*-algebra is quasidiagonal and has the lifting property, including amenable groups, free groups, and closed orientable surface groups. For surface groups, rank and winding number give complete invariants for the classification. Finally, composing with the equivariant Chern character gives a delocalized cohomological obstruction to stable equivalence of a quasi-representation with genuine representations.

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BibTeX RIS

BibTeX

@misc{weinberger2026classifying,
  title = {{Classifying quasi-representations of groups via their characters}},
  author = {Shmuel Weinberger and Jianchao Wu and Guoliang Yu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17196},
  eprint = {2610.17196},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17196},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - Classifying quasi-representations of groups via their characters
AU  - Shmuel Weinberger
AU  - Jianchao Wu
AU  - Guoliang Yu
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.17196
UR  - https://arxiv.org/abs/2610.17196
AB  - For every countable discrete group, we construct a character for finite-dimensional unitary quasi-representations, with values in the inverse limit of the equivariant K-theory of its Rips complexes. The character is a complete invariant for quasi-representations up to stable equivalence for amenable groups and, more generally, for groups satisfying the Haagerup property and QR-quasidiagonality. For Haagerup groups satisfying quantitative QR-quasidiagonality, we also obtain a quantiative version of the theorems. These conclusions apply when the maximal group C^*-algebra is quasidiagonal and has the lifting property, including amenable groups, free groups, and closed orientable surface groups. For surface groups, rank and winding number give complete invariants for the classification. Finally, composing with the equivariant Chern character gives a delocalized cohomological obstruction to stable equivalence of a quasi-representation with genuine representations.
ER  -

Abstract

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