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arxiv: 2408.13350 · v3 · pith:LKICNIB4new · submitted 2024-08-23 · 🧮 math.GR · math.KT· math.OA

Conditional representation stability, classification of *-homomorphisms, and relative eta invariants

Rufus Willett This is my paper

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classification 🧮 math.GR math.KTmath.OA
keywords quasi-representationsrepresentation approximationC*-algebrasK-theoryeta invariantsfundamental groupsBaumslag-Solitar groupsthree-manifolds
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The pith

Quasi-representations of certain low-dimensional groups can be approximated by honest representations when topological obstructions vanish.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper explores whether unitary quasi-representations of groups can be approximated by actual representations once known topological obstructions disappear. It proves this is possible in a weak sense for fundamental groups of closed surfaces, certain Baumslag-Solitar groups, free-by-cyclic groups, and many fundamental groups of three-manifolds. The approach relies on K-theoretic techniques originating from assembly maps and the Elliott program, including a new stable uniqueness theorem applicable to non-exact C*-algebras and an analysis of finite-coefficient K-theory maps using relative eta invariants. Although the proofs use this machinery, the main results are stated in elementary terms without requiring K-theory knowledge. This partially generalizes earlier results for the free abelian group of rank two and the Klein bottle group.

Core claim

For the listed low-dimensional groups, if the known obstructions to approximation vanish, then quasi-representations can be approximated by honest representations at least in a weak sense. The proof proceeds by establishing a stable uniqueness theorem for non-exact C*-algebras and relating K-theory maps with finite coefficients to relative eta invariants.

What carries the argument

Stable uniqueness theorem for non-exact C*-algebras together with the analysis of maps on K-theory with finite coefficients in terms of relative eta invariants.

Load-bearing premise

The new stable uniqueness theorem works for non-exact C*-algebras and the K-theory maps with finite coefficients can be analyzed using relative eta invariants.

What would settle it

A counterexample consisting of a specific group from the list and a quasi-representation with vanishing obstructions that cannot be approximated by any honest representation would falsify the result.

read the original abstract

A quasi-representation of a group is a map from the group into a matrix algebra (or similar object) that approximately satisfies the relations needed to be a representation. Work of many people starting with Kazhdan and Voiculescu, and recently advanced by Dadarlat, Eilers-Shulman-S\o{}rensen and others, has shown that there are topological obstructions to approximating unitary quasi-representations of groups by honest representations, where `approximation' is understood to be with respect to the operator norm. The purpose of this paper is to explore whether approximation is possible if the known obstructions vanish, partially generalizing work of Gong-Lin and Eilers-Loring-Pedersen for the free abelian group of rank two, and the Klein bottle group. We show that this is possible, at least in a weak sense, for some `low-dimensional' groups including fundamental groups of closed surfaces, certain Baumslag-Solitar groups, free-by-cyclic groups, and many fundamental groups of three manifolds. The techniques used in the paper are $K$-theoretic: they have their origin in Baum-Connes-Kasparov type assembly maps, and in the Elliott program to classify $C^*$-algebras; Kasparov's bivariant KK-theory is a crucial tool. The key new technical ingredients are: a stable uniqueness theorem in the sense of Dadarlat-Eilers and Lin that works for non-exact $C^*$-algebras; and an analysis of maps on $K$-theory with finite coefficients in terms of the relative eta invariants of Atiyah-Patodi-Singer. Despite the proofs going through $K$-theoretic machinery, the main theorems can be stated in elementary terms that do not need any $K$-theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper studies approximation of unitary quasi-representations of groups by honest representations (in the operator norm) when known topological obstructions vanish. It proves that such approximation is possible in a weak sense for fundamental groups of closed surfaces, certain Baumslag-Solitar groups, free-by-cyclic groups, and many fundamental groups of 3-manifolds. The proofs rely on a new stable uniqueness theorem valid for non-exact C*-algebras (in the sense of Dadarlat-Eilers and Lin) together with an analysis of maps on K-theory with finite coefficients via relative eta invariants of Atiyah-Patodi-Singer; the main theorems are stated in elementary terms without K-theory.

Significance. If the new stable uniqueness theorem and the eta-invariant analysis hold, the work provides a partial generalization of the Gong-Lin and Eilers-Loring-Pedersen results for Z^2 and the Klein bottle group. It connects Baum-Connes-Kasparov assembly maps and the Elliott classification program to concrete approximation questions for low-dimensional groups, and the elementary formulation of the main theorems is a strength for broader accessibility.

minor comments (2)
  1. The abstract states that the main theorems can be formulated without K-theory, but the introduction should include an explicit elementary statement of at least one main theorem (e.g., for surface groups) to make this claim immediately verifiable by readers.
  2. The precise class of 'many fundamental groups of three manifolds' to which the results apply is not delimited in the abstract; a short paragraph in §1 listing the exact hypotheses (e.g., which 3-manifold groups satisfy the required K-theoretic conditions) would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its connections to the Elliott program and Baum-Connes assembly, and the recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper proves its central results via newly established ingredients (a stable uniqueness theorem valid for non-exact C*-algebras and an analysis of finite-coefficient K-theory maps via relative eta invariants) that are derived within the manuscript itself from Kasparov KK-theory and Atiyah-Patodi-Singer invariants. These are applied to the listed low-dimensional groups after the obstructions vanish. The derivation relies on external established literature (Baum-Connes-Kasparov assembly maps, Dadarlat-Eilers, etc.) rather than self-citations or redefinitions of its own inputs; the main theorems are stated elementarily without K-theory. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard K-theory axioms and the new technical results proved inside the paper; no free parameters or invented entities are introduced in the abstract description.

axioms (2)
  • standard math Properties of Kasparov's bivariant KK-theory
    Described as a crucial tool for the K-theoretic techniques.
  • standard math Baum-Connes-Kasparov type assembly maps
    Origin of the techniques used in the paper.

pith-pipeline@v0.9.0 · 5859 in / 1318 out tokens · 31553 ms · 2026-05-25T08:17:59.765428+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A secondary pairing between K-theory and K-homology, relative eta invariants, and zeta maps

    math.KT 2026-05 unverdicted novelty 6.0

    Introduces a secondary pairing between subgroups of K-homology and K-theory valued in Q/Z that detects classes missed by the primary pairing and relates it to relative eta invariants, the Thomsen exact sequence, and z...

  2. The Local Lifting Property, Property FD, and stability of approximate representations

    math.GR 2026-03 unverdicted novelty 6.0

    3-manifold groups, limit groups, and selected one-relator and right-angled Artin groups possess the local lifting property and property FD, implying flexible stability of their approximate representations.

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