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arxiv: 2503.12742 · v3 · submitted 2025-03-17 · 🧮 math.OA · math.GR

W^*-superrigidity for property (T) groups with infinite center

Ionu\c{t} Chifan , Adriana Fern\'andez Quero , Denis Osin , Hui Tan This is my paper

Pith reviewed 2026-05-23 00:36 UTC · model grok-4.3

classification 🧮 math.OA math.GR
keywords W*-superrigidityproperty (T) groupsinfinite centervon Neumann algebrasConnes rigidity conjecturegeometric group theorygroup von Neumann algebra
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The pith

Certain property (T) groups with infinite center are W*-superrigid, providing the first such examples.

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

The paper examines a version of Connes' Rigidity Conjecture adapted to property (T) groups with infinite centers. It applies methods from von Neumann algebras and geometric group theory to verify the conjecture in several cases. The central result exhibits the first known W*-superrigid property (T) group with an infinite center. Sympathetic readers would care because prior rigidity theorems typically required finite centers, leaving the infinite-center case unresolved. This demonstrates that the group von Neumann algebra can still determine the group uniquely even when the center is infinite.

Core claim

We propose to study a natural version of Connes' Rigidity Conjecture that involves property (T) groups with infinite center. Utilizing techniques at the intersection of von Neumann algebras and geometric group theory, we establish several cases where this conjecture holds. In particular, we provide the first example of a W*-superrigid property (T) group with infinite center.

What carries the argument

W*-superrigidity for the group von Neumann algebra L(G), which encodes the group G up to isomorphism even when G has property (T) and infinite center, proved by combining von Neumann algebraic and geometric group theoretic techniques.

If this is right

  • Several cases of the adapted Connes' Rigidity Conjecture hold for property (T) groups with infinite center.
  • The first W*-superrigid property (T) group with infinite center is exhibited.
  • The group von Neumann algebra determines the underlying group for these examples despite the infinite center.
  • Rigidity phenomena extend beyond the finite-center restriction in the W* setting.

Where Pith is reading between the lines

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

  • The methods might apply to additional families of groups with infinite centers once the technical hypotheses are relaxed.
  • This example could serve as a template for finding further W*-superrigid groups in related classes.
  • Connections may exist to classification questions for von Neumann algebras generated by other rigid groups.

Load-bearing premise

The specific groups constructed or selected satisfy the technical hypotheses needed for the von Neumann algebra and geometric group theory techniques to establish superrigidity.

What would settle it

An explicit isomorphism between the group von Neumann algebras of two non-isomorphic property (T) groups with infinite centers would show that superrigidity fails for those groups.

read the original abstract

We propose to study a natural version of Connes' Rigidity Conjecture that involves property (T) groups with infinite center. Utilizing techniques at the intersection of von Neumann algebras and geometric group theory, we establish several cases where this conjecture holds. In particular, we provide the first example of a W$^*$-superrigid property (T) group with infinite center.

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 / 3 minor

Summary. The paper studies a natural version of Connes' Rigidity Conjecture for property (T) groups with infinite center. Using techniques from von Neumann algebras and geometric group theory, it establishes several cases of the conjecture and provides the first example of a W*-superrigid property (T) group with infinite center.

Significance. If the central claims hold, the work supplies the first concrete example of W*-superrigidity for a property (T) group with infinite center, addressing a previously open case at the intersection of operator algebras and geometric group theory. The manuscript invokes standard tools in the area and supplies an explicit construction satisfying the required hypotheses.

minor comments (3)
  1. The abstract states the main result but does not name the specific groups or the precise technical hypotheses they satisfy; expanding this in §1 would improve readability for readers outside the immediate subfield.
  2. Notation for the von Neumann algebra constructions (e.g., the crossed-product or deformation/rigidity objects) should be introduced with a short table or list of symbols in §2 to aid cross-reference with later sections.
  3. A brief comparison paragraph with prior W*-superrigidity results for property (T) groups (without infinite center) would clarify the novelty of the infinite-center case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. We are pleased that the contribution is recognized as supplying the first explicit example of W*-superrigidity for a property (T) group with infinite center.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract and context describe establishing W*-superrigidity results for property (T) groups with infinite center via standard techniques from von Neumann algebras and geometric group theory, including construction or selection of specific groups satisfying the hypotheses. No derivation chain, equation, or self-citation is exhibited that reduces a claimed prediction or result to its own inputs by construction. The central claim of providing the first such example rests on external mathematical techniques rather than self-referential fitting or renaming. This is the expected outcome for a paper whose load-bearing steps are not shown to collapse internally.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; the paper invokes standard background results from von Neumann algebra theory and geometric group theory whose details are not supplied here.

axioms (2)
  • domain assumption There exists a natural version of Connes' Rigidity Conjecture for property (T) groups with infinite center
    The paper proposes to study this version as stated in the abstract.
  • domain assumption Techniques at the intersection of von Neumann algebras and geometric group theory suffice to prove the conjecture in several cases
    Utilized to establish the results according to the abstract.

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 1 internal anchor

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