arXiv:2608.16581v1

Ergodic-transformation centralizers and essentially non-compact graphing symmetry

Alexandru Chirvasitu

math.DSmath.COmath.GR20B2722F5028A6037A2528A0522C0505C6305C76

Paper metadata

Original submission: August 17, 2026
Claimed MSC: 20B27, 22F50, 28A60, 37A25, 28A05, 22C05, 05C63, 05C76
arXiv categories: math.DS, math.CO, math.GR
Paper license: arXiv non-exclusive distribution license ↗
Comments: 6 pages + references

Abstract

We prove that for every ergodic transformation $T$ on an infinite standard probability space both the automorphism group (i.e. centralizer) $\mathrm{Aut}(T)$ and its reversing automorphism group are realizable as symmetry groups of graphings. This is an analogue of Sabidussi's realization of arbitrary graph-automorphism groups, and provides numerous examples of graphing automorphism groups carrying no compatible compact topology, answering a question of Lovasz'. Another consequence of discussion and ensuing constructions is the existence of large mutually locally-globally equivalent graphing families with highly variable symmetry.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

The reversing centralizer and the ordinary centralizer of every ergodic transformation on an infinite standard probability space are correctly realized as automorphism groups of ergodic graphings; the stated local-global-equivalence and non-compactness consequences also follow.

Theorem 1.7Correct

Automorphisms of cyclic graphings and their finite convex combinations

Pages 4–5 · Theorem 1.7 · arXiv:2608.16581v1

For an aperiodic ergodic transformation $T$ , each component of $C_T$ is a bi-infinite path. A graphing automorphism has a constant orientation sign on each orbit, and ergodicity makes that sign almost everywhere constant, giving exactly $\operatorname{Aut}^{\pm}(T)$ . In a finite weighted disjoint union, the copies are the atoms of the ergodic decomposition, so an automorphism may permute precisely the equal-weight copies; this yields the stated wreath product. All these graphings are hyperfinite and have the same rooted local law, so the cited hyperfinite equivalence theorem gives local-global equivalence.

Corollary 1.9Correct

Reversing groups are graphing symmetry groups and need not admit compact topologies

Page 5 · Corollary 1.9 · arXiv:2608.16581v1

The single-copy case of Theorem 1.7 realizes $\operatorname{Aut}^{\pm}(T)$ while remaining local-global equivalent to every irrational cyclic graphing. Choosing a mixing transformation with centralizer generated by its powers makes the reversing group countably infinite, with the centralizer of index at most two. An infinite compact Hausdorff group has at least continuum cardinality, so this group cannot carry a compatible compact Hausdorff group topology.

Theorem 1.10Correct

Degree-three oriented gadgets realize the ordinary centralizer

Pages 5–6 · Theorem 1.10 and gadget (1-3) · arXiv:2608.16581v1

Replacing each unoriented orbit edge by the displayed finite asymmetric gadget makes every vertex type and the direction from $x$ to $Tx$ recoverable from finite graph neighborhoods. Hence every graphing automorphism restricts to a measure-preserving transformation commuting with $T$ , and every such centralizer element extends uniquely over the corresponding gadget copies. The copies and connecting maps are standard Borel probability spaces and measure-preserving isomorphisms, so the mass-transport identity holds. Connected gadgets preserve ergodicity and have maximum degree three.

02Proofs4 reported findingsCorrect

The orbit-orientation argument, ergodic-decomposition wreath product, local-global equivalence input, compact-group specialization, and asymmetric-gadget construction are correct and complete.

Equation (1-1)Correct and complete

The graphing automorphism group is the reversing-centralizer wreath product

Pages 4–5 · proof of Theorem 1.7(1) · arXiv:2608.16581v1

An automorphism of a bi-infinite path either preserves or reverses its orientation, with the sign constant along the path. The measurable sign set is invariant under $T$ and therefore null or conull. This identifies $\operatorname{Aut}(C_T)$ with $\operatorname{Aut}^{\pm}(T)$ . Ergodic components of a finite convex combination can only be permuted among equal weights, giving the claimed semidirect wreath-product structure.

Local-global equivalence inputCorrect and complete

Hyperfinite graphings with the common path law are local-global equivalent

Page 5 · proof of Theorem 1.7(1) · arXiv:2608.16581v1

Because the underlying ergodic transformations act on infinite standard probability spaces, they are aperiodic almost everywhere, so every rooted local neighborhood is the corresponding neighborhood in the bi-infinite path. The graphings are hyperfinite, and the cited local-global convergence theorem identifies hyperfinite graphings with the same local statistics.

Hatami–Lovász–Szegedy, Limits of local-global convergent graph sequences ↗

Equation (1-2)Correct and complete

Compact monothetic rotations have the stated extended centralizer

Pages 4–5 · proof of Theorem 1.7(2) · arXiv:2608.16581v1

A compact group with a dense cyclic subgroup is abelian. If a measurable automorphism $\phi$ commutes with translation by the topological generator $a$ , then $s^{-1}\phi(s)$ is invariant under that ergodic translation and is therefore almost everywhere constant; thus $\phi$ is a translation. Composing a reversing automorphism with inversion reduces it to this centralizer, yielding $G\rtimes\mathbb Z/2$ .

Gadget (1-3)Correct and complete

The replacement graphing is measurable, balanced, ergodic, and orientation-rigid

Page 6 · proof of Theorem 1.10 · arXiv:2608.16581v1

Each internal gadget position is represented by an equal-weight copy of the original standard probability space, and every gadget edge is the graph of a measure-preserving Borel isomorphism between two copies. Summing these paired edge graphs verifies the mass-transport identity. Degree patterns and distances to the asymmetric leaves distinguish all positions and both ends of the gadget, forcing orientation preservation. Finally, every invariant measurable union in the replacement projects to a $T$ -invariant set in the original graphing, so ergodicity is retained.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.16581v1
Authors listed: Alexandru Chirvasitu
Audit date: August 18, 2026

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Ergodic-transformation centralizers and essentially non-compact graphing symmetry

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Automorphisms of cyclic graphings and their finite convex combinations

Reversing groups are graphing symmetry groups and need not admit compact topologies

Degree-three oriented gadgets realize the ordinary centralizer

The graphing automorphism group is the reversing-centralizer wreath product

Hyperfinite graphings with the common path law are local-global equivalent

Compact monothetic rotations have the stated extended centralizer

The replacement graphing is measurable, balanced, ergodic, and orientation-rigid

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