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arxiv: 2606.19606 · v1 · pith:OMU2HPVFnew · submitted 2026-06-17 · 🧮 math.GR

Outer automorphism groups and the Atiyah Conjecture

Sam P. Fisher , Andrew Ng This is my paper

Pith reviewed 2026-06-26 18:32 UTC · model grok-4.3

classification 🧮 math.GR
keywords outer automorphism groupAtiyah conjecturevon Neumann dimensionright-angled Artin grouppro-p completionzero divisor conjecturesurface group
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The pith

The von Neumann dimension function of Out(G) takes values in a discrete subgroup of the rationals for surface groups, free groups and right-angled Artin groups.

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

The paper shows that for G the fundamental group of a compact surface, a finitely generated free group, or a finitely generated right-angled Artin group, the von Neumann dimension function on Out(G) lands inside a discrete subgroup of Q. This follows from proving the Strong Atiyah Conjecture for a torsion-free subgroup of finite index in Out(G). The authors also show that for any field K a suitable such subgroup H has its group algebra K[H] embedding into a division ring, so the Zero Divisor Conjecture holds. The proofs proceed by establishing the same statements first for an open subgroup of Out of the pro-p completion of G. A reader cares because the result sharply restricts the possible L2-dimensions that can arise from these outer automorphism groups.

Core claim

For G the fundamental group of a compact surface, a finitely generated free group or a finitely generated right-angled Artin group, the von Neumann dimension function of Out(G) is valued in a discrete subgroup of Q. This is obtained by establishing the Strong Atiyah Conjecture for a torsion-free subgroup of Out(G) of finite index. For every field K there likewise exists a torsion-free finite-index subgroup H of Out(G) such that K[H] embeds into a division ring. These statements are proved by first establishing the analogous claims for a suitable open subgroup of Out of the pro-p completion of G.

What carries the argument

The Strong Atiyah Conjecture for a torsion-free finite-index subgroup of Out(G), transferred from the pro-p completion of G via an open subgroup of its outer automorphism group.

If this is right

  • The Strong Atiyah Conjecture holds for the indicated torsion-free finite-index subgroups of Out(G).
  • The Zero Divisor Conjecture holds for K[H] over any field K.
  • An automorphism of a free nilpotent group is inner if and only if it induces an inner automorphism on the pro-p completion.
  • The von Neumann dimension function on these Out(G) is supported on a discrete subgroup of Q.

Where Pith is reading between the lines

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

  • The same discreteness might hold for outer automorphism groups of other classes of groups if the pro-p reduction technique extends.
  • L2-Betti numbers of Out(G) would then also lie in that same discrete subgroup of Q.
  • The appendix result on free nilpotent groups suggests a general pattern relating inner automorphisms before and after pro-p completion.

Load-bearing premise

The algebraic and dimension properties verified on the pro-p completion and its outer automorphism group lift to the original discrete group G and to Out(G).

What would settle it

An explicit element in the group von Neumann algebra of Out(G) whose dimension lies outside every discrete subgroup of Q would falsify the claim.

read the original abstract

Let $G$ be the fundamental group of a compact surface, a finitely generated free group, or more generally a finitely generated right-angled Artin group. We prove that the von Neumann dimension function of $\mathrm{Out}(G)$ is valued in a discrete subgroup of $\mathbb Q$. This is accomplished by establishing the Strong Atiyah Conjecture for a torsion-free subgroup of $\mathrm{Out}(G)$ of finite index. We also prove that for every field $\mathbb K$, there exists a torsion-free subgroup $H \leqslant \mathrm{Out}(G)$ of finite index such that $\mathbb K[H]$ embeds into a division ring, and hence satisfies the Zero Divisor Conjecture. These results are obtained by establishing analogous ones for a suitable open subgroup of $\mathrm{Out}(\mathbf G)$ and its completed group algebra, where $\mathbf G$ denotes the pro-$p$ completion of $G$. In an appendix, the first author shows that an automorphism of a free nilpotent group is inner if and only if it induces an inner automorphism of its pro-$p$ completion.

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 / 0 minor

Summary. The paper claims that for G the fundamental group of a compact surface, a finitely generated free group, or a finitely generated right-angled Artin group, the von Neumann dimension function of Out(G) takes values in a discrete subgroup of Q. This is achieved by establishing the Strong Atiyah Conjecture for a torsion-free finite-index subgroup of Out(G). It further claims that for any field K there exists such a subgroup H with K[H] embedding into a division ring (hence satisfying the Zero Divisor Conjecture). Both results are obtained by proving analogous statements for a suitable open subgroup of Out(G-hat) and its completed group algebra, where G-hat is the pro-p completion of G. An appendix shows that an automorphism of a free nilpotent group is inner if and only if it induces an inner automorphism on its pro-p completion.

Significance. If the central claims hold, the work supplies new families of groups (outer automorphism groups of surface groups, free groups, and RAAGs) for which the Strong Atiyah Conjecture is verified, yielding discreteness of L2-dimensions and progress on the zero-divisor conjecture. The reduction to pro-p completions offers a potentially reusable technique for transferring algebraic properties from profinite to discrete settings.

major comments (2)
  1. [abstract (reduction paragraph)] The load-bearing reduction step (abstract, paragraph beginning 'These results are obtained by establishing analogous ones...'): it is not shown how discreteness of the von Neumann dimension function on the discrete group Out(G) follows from the corresponding property on the completed group algebra of an open subgroup of Out(G-hat). In particular, the correspondence between L2-dimensions defined via the group von Neumann algebra of Out(G) and the dimension function on the completed algebra must be established explicitly so that no denominators outside a discrete subgroup of Q are introduced when passing from the open subgroup of Out(G-hat) back to the finite-index torsion-free subgroup of the discrete Out(G).
  2. [appendix] The appendix treats only the narrower case of free nilpotent groups; it is unclear whether the inner-automorphism correspondence established there is used to control the finite-index subgroup or the embedding property in the main theorems, or whether it is merely illustrative.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the exposition of our reduction arguments. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [abstract (reduction paragraph)] The load-bearing reduction step (abstract, paragraph beginning 'These results are obtained by establishing analogous ones...'): it is not shown how discreteness of the von Neumann dimension function on the discrete group Out(G) follows from the corresponding property on the completed group algebra of an open subgroup of Out(G-hat). In particular, the correspondence between L2-dimensions defined via the group von Neumann algebra of Out(G) and the dimension function on the completed algebra must be established explicitly so that no denominators outside a discrete subgroup of Q are introduced when passing from the open subgroup of Out(G-hat) back to the finite-index torsion-free subgroup of the discrete Out(G).

    Authors: We agree that the explicit correspondence is not detailed in the current manuscript. The reduction relies on the fact that the open subgroup of Out(G-hat) maps onto a finite-index subgroup of Out(G) with controlled kernel, combined with the compatibility of the von Neumann dimension with the completion map, but this chain is only sketched. In the revised version we will add a dedicated paragraph (or short subsection) in the introduction or Section 2 that spells out the dimension correspondence, verifies that the denominators remain inside the same discrete subgroup of Q, and cites the relevant functoriality properties of L2-dimension under finite-index inclusions and profinite completions. revision: yes

  2. Referee: [appendix] The appendix treats only the narrower case of free nilpotent groups; it is unclear whether the inner-automorphism correspondence established there is used to control the finite-index subgroup or the embedding property in the main theorems, or whether it is merely illustrative.

    Authors: The appendix result is used in the proofs: for free groups and RAAGs the relevant automorphisms factor through nilpotent quotients, and the inner-automorphism correspondence on the pro-p completion is invoked to guarantee that the chosen open subgroup of Out(G-hat) descends to a torsion-free finite-index subgroup of Out(G) while preserving the embedding property into a division ring. We will insert a short clarifying sentence (with a forward reference) at the end of the introduction and again in the relevant proof sections to make this dependence explicit rather than leaving it implicit. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on independent pro-p reductions and external group theory

full rationale

The paper establishes the Strong Atiyah Conjecture for a torsion-free finite-index subgroup of Out(G) by proving analogous statements for an open subgroup of Out of the pro-p completion G-hat and its completed algebra, then transfers the discreteness of von Neumann dimensions back to the discrete case. No step reduces a claimed prediction or result to a fitted parameter, self-defined quantity, or load-bearing self-citation chain by the paper's own equations. The appendix result on nilpotent groups is internal to the manuscript and does not create a self-referential loop. The central claims rest on standard pro-p completion techniques and group algebra embeddings that are not shown to be tautological with the target discreteness statement.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms or invented entities are stated.

pith-pipeline@v0.9.1-grok · 5719 in / 1187 out tokens · 23979 ms · 2026-06-26T18:32:16.457982+00:00 · methodology

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