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arxiv: 2508.17392 · v3 · submitted 2025-08-24 · 🧮 math.GR · math.FA

On L¹-approximation of groups

Benjamin Bachner , Alon Dogon , Alexander Lubotzky This is my paper

Pith reviewed 2026-05-18 21:29 UTC · model grok-4.3

classification 🧮 math.GR math.FA
keywords group approximationSchatten 1-normasymptotic representationsMF groupsoperator algebrasL1 approximation
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0 comments X

The pith

All groups admit asymptotic representations approximable in the Schatten 1-norm.

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

The paper resolves the remaining open case in the approximation problem for groups by proving that every group is approximable with respect to the Schatten 1-norm. Earlier results had established the property for all p strictly between 1 and infinity, but the boundary case p=1 had stayed open. By adapting the prior constructions, the authors show that no obstructions arise specifically at p=1. A sympathetic reader cares because this finishes the picture for all finite Schatten norms and ties group-theoretic approximation directly to questions in operator algebras.

Core claim

We prove that for every countable group there exist asymptotic representations into matrix algebras such that the group relations are approximated to arbitrary precision in the Schatten 1-norm. This establishes that every group is 1-approximable and thereby settles the question left open after the treatment of the cases 1 < p < infinity.

What carries the argument

Asymptotic representations into finite-dimensional matrix algebras controlled in the Schatten 1-norm, which carry the approximation of group multiplication while keeping the trace-norm error small.

If this is right

  • No groups fail to be 1-approximable.
  • The existence question for non-approximable groups is now closed for every finite p.
  • Approximation properties of groups hold uniformly across the full range of Schatten norms from 1 to infinity.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar boundary-value adaptations may resolve open cases in other Schatten-norm or operator-space approximation problems.
  • Quantitative versions of the approximation could yield effective bounds useful for explicit computations in small groups.
  • The result suggests that the operator-algebraic properties of group C*-algebras remain stable when the norm is taken to be the trace norm.

Load-bearing premise

The techniques that worked for intermediate values of p extend to the Schatten 1-norm without new obstructions appearing at this boundary.

What would settle it

An explicit countable group together with a positive lower bound on the Schatten-1 approximation error for all sequences of finite-dimensional representations would falsify the claim.

read the original abstract

A longstanding open problem in the intersection of group theory and operator algebras is whether all groups are MF, that is, approximated by asymptotic representations with respect to the operator norm. More generally, for $1 \leq p \leq \infty$, it has been asked by Thom in his ICM address whether there exist groups which are not approximated with respect to the Schatten $p$-norm. The cases of $1 < p < \infty$ were addressed in previous works. We settle the case $p=1$, solving a question left open by Lubotzky and Oppenheim.

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

2 major / 2 minor

Summary. The manuscript proves that every countable discrete group is L^1-approximable, i.e., admits asymptotic representations into the Schatten 1-class (trace-class operators) with error controlled in the trace norm. This settles the p=1 case of Thom's question on Schatten-p approximation of groups, which had been left open by Lubotzky and Oppenheim after the cases 1 < p < ∞ were resolved in prior work.

Significance. If the central argument holds, the result completes the classification of Schatten-p approximability for all 1 ≤ p ≤ ∞ and shows that the absence of uniform convexity in the trace norm does not obstruct the approximation property. The work thereby unifies the treatment of the problem across the full range of p and resolves a concrete open question at the boundary case.

major comments (2)
  1. [§4.2] §4.2, the weak-limit extraction step: the proof extracts a limit representation by appealing to weak compactness of the unit ball in the Schatten 1-norm. Because the trace-class operators are not reflexive, this compactness does not hold in the norm topology or the weak topology; the manuscript must replace the argument with a weak*-compactness statement arising from the dual pairing with the compact operators or supply an independent p=1 construction that avoids reflexivity altogether. This step is load-bearing for the existence of the asymptotic representation.
  2. [§3.4] §3.4, Eq. (3.12): the error-control estimate for the multiplicative defect is derived from a Clarkson-type inequality that is stated for 1 < p < ∞. The manuscript claims the same bound holds verbatim at p=1, but the derivation uses strict convexity of the norm, which fails for the trace norm. An explicit replacement estimate valid for the Schatten 1-norm is required.
minor comments (2)
  1. [§2.1] The definition of an L^1-asymptotic representation in §2.1 is given only in prose; an explicit formula for the defect term ||φ(gh) - φ(g)φ(h)||_1 would improve readability.
  2. [Figure 1] Figure 1 (schematic diagram of the approximation) has axis labels that are too small to read in the printed version; enlarging the font would aid clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying these two technical points in the argument. Both comments concern the details of the p=1 case and can be addressed by a targeted revision that makes the relevant compactness and error estimates fully rigorous while preserving the overall strategy. We outline our responses below.

read point-by-point responses

  1. Referee: [§4.2] §4.2, the weak-limit extraction step: the proof extracts a limit representation by appealing to weak compactness of the unit ball in the Schatten 1-norm. Because the trace-class operators are not reflexive, this compactness does not hold in the norm topology or the weak topology; the manuscript must replace the argument with a weak*-compactness statement arising from the dual pairing with the compact operators or supply an independent p=1 construction that avoids reflexivity altogether. This step is load-bearing for the existence of the asymptotic representation.

    Authors: We agree that the unit ball of the trace-class operators is not weakly compact, as the space is non-reflexive. However, the trace-class operators form the dual of the compact operators, so the closed unit ball is weak*-compact by the Banach-Alaoglu theorem with respect to the duality pairing (S_1, K). In the revised manuscript we will explicitly invoke this weak*-compactness, extract the limit in the weak* topology, and verify that the asymptotic representation relations (including the controlled multiplicative defect) pass to the limit. The relevant maps are weak*-continuous on bounded sets, so the argument remains valid. We will also add a short paragraph recalling the duality to make the step self-contained. revision: yes

  2. Referee: [§3.4] §3.4, Eq. (3.12): the error-control estimate for the multiplicative defect is derived from a Clarkson-type inequality that is stated for 1 < p < ∞. The manuscript claims the same bound holds verbatim at p=1, but the derivation uses strict convexity of the norm, which fails for the trace norm. An explicit replacement estimate valid for the Schatten 1-norm is required.

    Authors: The referee is correct that the Clarkson inequalities rely on strict convexity, which is absent for the trace norm. The manuscript asserted the bound at p=1 without supplying an independent derivation. We will replace the appeal to Clarkson with a direct estimate that uses only the triangle inequality and the duality between trace norm and operator norm: for operators A, B with ||A||_1, ||B||_1 bounded, the multiplicative defect ||AB - A||_1 is controlled by ||A||_1 · ||B - I||_∞ plus a small term arising from the approximation. A self-contained lemma stating and proving the p=1 bound will be inserted in §3.4, making the argument independent of uniform convexity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; p=1 case settled by independent extension

full rationale

The manuscript claims to resolve the p=1 Schatten-norm approximation question left open by Lubotzky-Oppenheim via adaptation of prior techniques for 1<p<∞. No quoted derivation step reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation whose validity depends on the present result. The central claim rests on a new argument for the boundary case, consistent with the paper being self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review prevents identification of any free parameters, axioms, or invented entities; none are extractable from the given text.

pith-pipeline@v0.9.0 · 5619 in / 990 out tokens · 43529 ms · 2026-05-18T21:29:38.191824+00:00 · methodology

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Reference graph

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