A torsion-free group algebra that is not directly finite

1 · The essentials

Can two objects undo each other in one order but fail to do so in the reverse order? These manuscripts construct examples where the answer is yes, in a system that combines the arithmetic of a field with the multiplication rules of a group. The key relation is $ab = 1$ but $ba \neq = 1$ : order changes whether the two objects cancel. The main manuscript shows this even for a group in which no nonidentity element returns to the identity through repeated multiplication. Its coefficients come from the field with just two elements. Companion constructions also give update rules for configurations that never merge distinct inputs yet leave some possible outputs unreachable. Lean formalizations accompany the characteristic-two, determinant, and odd-characteristic results, but the supplied descriptions do not explicitly identify the torsion-free strengthening as covered by those formalizations.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine two operations on an arrangement of objects: doing one and then the other restores the original arrangement, while reversing their order does not. This captures the one-way cancellation at issue, but not the specific arithmetic and group structure required by the manuscripts.

Why it matters

The torsion-free example shown in the main manuscript contradicts Kaplansky's direct-finiteness conjecture even under the additional restriction of having no torsion. The companions connect this failure to counterexamples to Gottschalk's surjunctivity conjecture and to the unrestricted group-ring Determinant Conjecture. Together, they concern limits on cancellation, configuration updates, and determinants in settings built from groups.

2 · Going further

A group algebra $K [G]$ consists of finite formal sums of group elements with coefficients in a field $K$ , with multiplication extending that of the group. Direct finiteness is the requirement that $ab = 1$ imply $ba = 1$ . The main manuscript shows a counterexample over $F_{2}$ with $G$ finitely presented and torsion-free; it also states that $G$ admits a finite two-dimensional classifying complex. Thus the failure does not require a nonidentity group element of finite order.

The construction strategy disclosed by the abstracts is to supply a group, a coefficient field, and finite sums witnessing the failure of direct finiteness. No further construction or argument details are supplied. The Lean-formalized characteristic-two companion constructs a finite field $K$ of characteristic two and a finitely presented nonsofic group $G$ , together with $a, b \in K [G]$ satisfying $ab = 1$ and $ba \neq = 1$ . The same elements define a cellular automaton on the full configuration space $K^{G}$ that is injective but not surjective. Its abstract does not specify that this group is torsion-free or that its field is exactly $F_{2}$ .

The odd-characteristic companion gives a related Lean-formalized counterexample for one specified odd prime $p$ , using a field of order $p^{4}$ and a finitely generated group containing torsion. It again supplies finite sums with $ab = 1$ but $ba \neq = 1$ , and an injective, nonsurjective cellular automaton on $K^{G}$ . This extends the family beyond characteristic two, but the stated result is not a claim for every odd prime.

The Lean-formalized determinant companion concerns a different obstruction: a square matrix over $Z [G]$ , for a finitely generated group $G$ , that is invertible over $Q [G]$ but has Fuglede–Kadison determinant strictly between zero and one. The logarithmic integral defining that determinant is finite. The abstract presents this as a counterexample to the unrestricted group-ring Determinant Conjecture, without giving a construction-level link to the other examples.

Useful notions

Groups and their identity elements
Fields and characteristic
Finite sums in a group algebra
Left and right inverses
Injective and surjective maps

What was known before / what is new

Before

The source identifies Kaplansky's direct-finiteness conjecture as the conjecture challenged by the group-algebra constructions. It also names Gottschalk's surjunctivity conjecture and the unrestricted group-ring Determinant Conjecture as targets of companion counterexamples. It supplies no chronology or account of earlier partial results.

New

The main manuscript shows a torsion-free, finitely presented counterexample over $F_{2}$ , with a finite two-dimensional classifying complex. The Lean-formalized characteristic-two companion supplies explicit finite-sum witnesses and an injective, nonsurjective cellular automaton. Another Lean-formalized companion gives such witnesses in one odd characteristic, with torsion present. The determinant companion supplies a Lean-formalized counterexample with rational invertibility and determinant strictly between zero and one.

Limits and assumptions

The torsion-free result is stated over the two-element field, whereas the characteristic-two formalized companion specifies a finite field of characteristic two without identifying it as the two-element field. The main direct-finiteness failure has a Lean-formalized companion, but the supplied scope does not explicitly cover the torsion-free strengthening or the finite two-dimensional classifying complex. The odd-characteristic result concerns one specified odd prime and a group containing torsion, not every odd characteristic or a torsion-free example there. The determinant example concerns the unrestricted conjecture and has a finite defining logarithmic integral. The abstracts do not describe the internal proof steps or identify further open cases.

Mini glossary

Group algebra: An algebra of finite formal sums of group elements with field coefficients, whose multiplication extends the group's multiplication.
Direct finiteness: The property that whenever $ab = 1$ , the reverse product also satisfies $ba = 1$ .
Torsion-free group: A group in which no nonidentity element has a positive power equal to the identity.
Finitely presented group: A group describable using finitely many generators and finitely many defining relations.
Full configuration space: The space $K^{G}$ of all assignments of a value in $K$ to every element of $G$ .
Injective but not surjective: A map is injective but not surjective when distinct inputs have distinct outputs, yet some elements of its target are never outputs.

Keywords : Direct finiteness · Group algebras · Counterexamples

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽₂ is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective cellular automata on all configurations, refuting Gottschalk's surjunctivity conjecture. Another counterexample is an integral group-ring matrix, invertible over the rational group ring, with Fuglede–Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

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lean/docs/197.md

Manuscripts in this family

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A Torsion-Free Group Algebra That Is Not Directly Finite

04/10/2026

To be verified

We construct a finitely presented torsion-free counterexample to Kaplansky's direct finiteness conjecture over the field of two elements. The group admits a finite two-dimensional classifying complex.

PDF GitHub folder

A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two

23/09/2026

Verified by Lean

We disprove Kaplansky's direct-finiteness conjecture by constructing a finite field K of characteristic two, a finitely presented group G, and finite sums $a, b \in K [G]$ with $ab = 1$ but $ba \neq = 1$ . The group G is nonsofic. The same elements define a cellular automaton on K^G that is injective but not surjective, disproving Gottschalk's surjunctivity conjecture.

PDF GitHub folder Lean formalization

A Counterexample to the Group-Ring Determinant Conjecture

23/09/2026

Verified by Lean

We disprove the unrestricted group-ring Determinant Conjecture. We construct a finitely generated group G and a square matrix over $Z [G]$ that is invertible over $Q [G]$ and has Fuglede–Kadison determinant strictly between zero and one. The logarithmic integral defining the determinant is finite.

PDF GitHub folder Lean formalization

A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic

26/09/2026

Verified by Lean

We construct a counterexample to Kaplansky's direct-finiteness conjecture in odd characteristic. For one specified odd prime p, we obtain a field K of order p⁴, a finitely generated group G containing torsion, and finite sums $a, b \in K [G]$ with $ab = 1$ but $ba \neq = 1$ . The same elements define a cellular automaton on K^G that is injective but not surjective.

PDF GitHub folder Lean formalization