Zero divisors without finite-order group elements

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Family 196 · Algebra
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-21

1. The question

Ordinary nonzero real numbers never multiply to zero. Matrices can: two nonzero matrices may have a zero product. Which behavior should one expect from a group algebra, where formal sums of group elements are multiplied using the group law?

Finite-order group elements create an immediate source of zero divisors. Kaplansky’s zero-divisor conjecture asks whether that is the only obstruction: if a group is torsion-free, must its group algebra over every field have no nonzero zero divisors?

Family 196 claims a negative answer. It constructs a torsion-free group and two nonzero finite sums over the field with two elements whose product vanishes. The algebraic cancellation is driven by a simple parity observation. The difficult part is making that cancellation coexist with torsion-freeness and ensuring that the factors themselves survive the imposed relations.

2. Background and essential definitions

Formal sums with a group multiplication rule

For a field $K$ and a group $G$, the group algebra $K[G]$ consists of finite sums

$$\alpha=\sum_{g\in G}a_g g,$$

where only finitely many coefficients $a_g\in K$ are nonzero. Addition combines coefficients, and multiplication distributes using the multiplication in $G$. Thus the coefficient of $u$ in a product is the sum of $a_gb_h$ over all pairs with $gh=u$.

A zero divisor is a nonzero element that multiplies another nonzero element to zero. If $g$ has finite order $m>1$, then

$$(1-g)(1+g+\cdots+g^{m-1})=1-g^m=0.$$

Both factors are nonzero because the indicated group elements are distinct. A group is torsion-free when its identity is the only element of finite order. Removing finite order removes this elementary construction, but the conjecture asks whether more elaborate cancellations remain possible.

The field $\mathbb F_2$ has elements zero and one, with $1+1=0$. A finite sum over this field vanishes precisely when every group element occurs an even number of times. This turns the desired product cancellation into a parity problem.

Unique product is stronger than torsion-freeness

A group has the unique-product property if every two finite nonempty subsets $A,B$ have some product with exactly one representation $ab$. Applying this to the supports of two nonzero group-algebra elements supplies a coefficient that cannot cancel, so their product is nonzero.

Failure of unique product is weaker than having zero divisors. Two or more representations may have odd total multiplicity; over $\mathbb F_2$, three copies still sum to one. The construction needs an even number of occurrences for every product, not merely the absence of unique representations.

A finitely presented group is specified by finitely many generators and relations. A classifying space for a group is a connected space with that fundamental group and contractible universal cover. A finite two-dimensional classifying space gives strong finiteness and topological control; in particular, its fundamental group cannot contain nontrivial finite-order elements.

3. Development history and previous partial results

The question appears in Higman’s 1940 work on units in group rings and in Kaplansky’s 1956 conference problems, published in 1957. Its simplicity conceals the challenge of controlling all finite cancellation patterns at once.

Higman proved positive results for groups whose nontrivial finitely generated subgroups map onto the integers, the local-indicability setting. Kropholler, Linnell, and Moody established the absence of zero divisors for torsion-free elementary amenable groups, even with division-ring coefficients. More recent division-ring embedding results of Fisher and Sánchez-Peralta cover torsion-free virtually compact special groups and substantial classes of three-manifold groups.

These theorems show that torsion-freeness combines with additional structure to give strong algebraic consequences. They do not apply to every torsion-free group.

Rips and Segev constructed torsion-free groups without unique product. Their work showed that one natural route to proving the conjecture could not be universal. Later graphical small-cancellation treatments clarified such constructions. But non-unique-product examples alone did not produce the parity cancellation required here.

The neighboring unit conjecture asks whether every invertible group-algebra element is a scalar multiple of a group element. Gardam disproved it over $\mathbb F_2$ and later over $\mathbb C$. A nontrivial unit is not a zero divisor, so those results address a different conjecture.

Mineyev developed product structures and geometric criteria aimed at constructing units or zero divisors. Garg and Mineyev studied their finite-support combinatorics computationally. Shin established an obstruction to a particular orientable no-fold, triple-girth route. The present manuscript describes its own arrangement-exclusion mechanism rather than invoking those criteria.

Its topological strategy has antecedents in Gromov’s random graphical presentations and subsequent work of Ollivier, Gruber, and Bereznyuk. The new claim combines deliberately prescribed parity with probabilistic control of the relations, rather than taking a generic random presentation with no coefficient constraints.

4. What the new manuscripts claim

The manuscript, dated 23 September 2026, claims that there is a finitely presented torsion-free group $G$ and elements $\alpha,\beta\in\mathbb F_2[G]$ such that

$$\alpha\ne0,\qquad \beta\ne0,\qquad \alpha\beta=0.$$

Moreover, $G$ has a finite two-dimensional classifying space. This supplies a counterexample within a class of groups with a finite topological model, rather than relying on infinitely many defining relations.

The construction is probabilistic and existential. It proves that suitable finite graph matchings exist, but does not list matchings from which a concrete presentation and explicit expanded zero-divisor factors can be immediately read off. Finitely presented does not mean that the paper displays a short numerical instance.

The coefficient field in the claimed example is $\mathbb F_2$. This is enough to disprove a conjecture quantified over all fields. The theorem does not assert the same cancellation over characteristic-zero fields or over every characteristic.

5. Why the advance matters

Provisional importance: 5/5. If correct, the result disproves one of the central structural conjectures about torsion-free group algebras. It shows that eliminating finite-order elements does not eliminate every mechanism for zero multiplication.

The example would also distinguish several neighboring properties sharply: torsion-freeness, unique product, absence of zero divisors, and restrictions on units are related but not interchangeable. Earlier counterexamples to stronger properties did not automatically settle this one.

The construction’s design is useful conceptually. A local parity condition guarantees algebraic cancellation regardless of later random choices. Probability is then used to impose global geometric safeguards. This separates the part that must happen from the bad topological configurations that must be excluded.

6. The main idea

Two labelled graphs supply the factors

Take two finite graphs whose oriented edges carry group-generator labels; reversing an edge replaces its label by the inverse. At each vertex, no outgoing label is repeated. Map both graphs into a rose, a graph with one vertex and one loop for each generator.

Attach cones over the graph components, thereby imposing relations that kill the labels of closed graph paths. In the resulting group, choose a root in one component of each graph. A path from the first root to a vertex $x$ has a well-defined group label $g_x$, because alternative paths differ by a closed path. Define $h_y$ in the other component similarly.

Using the full vertex sets $A'$ and $B'$ of the chosen components, set

$$\alpha=\sum_{x\in A'}g_x, \qquad \beta=\sum_{y\in B'}h_y^{-1}.$$

These sums can contain repeated group labels. The construction does not require all their non-root labels to be distinct.

Odd degrees force even cancellation

Let $S_x$ and $S_y$ be the outgoing signed-label sets. The graphs are prescribed so that every cross-intersection $S_x\cap S_y$ has odd size.

Now form a graph on pairs $(x,y)$. Connect pairs by taking one step with the same label $t$ in both original graphs. The represented product remains unchanged:

$$(g_xt)(h_yt)^{-1}=g_xh_y^{-1}.$$

Every vertex in this pair graph has odd degree. The degree-sum formula says that a finite graph has an even number of odd-degree vertices, so every connected component has an even number of vertices. All those vertices contribute the same group element to $\alpha\beta$. Their contributions cancel in $\mathbb F_2$.

This proves the desired zero product once the graphs and their quotient group satisfy the remaining safeguards.

Preserve nonzero factors and forbid torsion

The first safeguard is root separation: a path from a chosen root to any other vertex in its component must have nonidentity label. Then the root contributes the identity exactly once to each factor, so neither factor vanishes. Distinguishing every pair of non-root vertices is unnecessary.

The second safeguard is asphericity of the coned complex, meaning that its higher homotopy groups vanish. The manuscript proves this directly through the exclusion of reduced spherical arrangements. It does not claim that these graphs automatically satisfy a familiar small-cancellation or nonpositive-curvature criterion.

The outgoing labels are arranged using lines in a finite projective plane, together with three extra letters. Distinct projective lines meet once, and the extra-letter intersections preserve odd parity. Random matchings then join compatible label occurrences into edges.

Those vertex types also give exponential decay for a squared-word counting expression, controlling coincidences between labelled paths. The argument must track repeated traversal of the same edges and distinguish genuinely different lifts from coincident paths. It cannot treat all path occurrences as independent random events.

Reduce arbitrarily large diagrams to bounded tests

A bad sphere could involve arbitrarily many paths, so a naive union bound over a fixed diagram size would not suffice. The manuscript uses planar separation, based on the Lipton–Tarjan separator theorem, to extract a bounded configuration from any sufficiently complicated spherical arrangement.

Long boundaries are split and closed at controlled cost, and planar counting limits the number of comparison intervals. The tolerated proportion of unmatched path positions is fixed before the size bounds. The same finite class of forbidden patterns can then detect arbitrarily large bad arrangements.

The probabilistic estimates exclude those patterns for some choice of finite matchings. Cone-picture arguments turn that exclusion into both root separation and asphericity, completing the claimed construction.

7. Scope, limitations, and open questions

The theorem concerns existence over $\mathbb F_2$, not an explicit small presentation or a minimum support size for the zero divisors. Finding practical concrete instances is a different problem from the probabilistic existence statement.

The two factors need not have all distinct support labels before coefficient collection. What matters is that the identity coefficient survives in each and every product coefficient cancels. This distinction avoids assuming a stronger graph-embedding statement than the construction uses.

The claimed group need not belong to the classes covered by existing positive theorems, and the result does not negate those theorems. It also makes no general characteristic-zero zero-divisor claim.

Finally, this family should be distinguished from failure of direct finiteness, discussed in family 197. A zero product, a nontrivial unit, and a one-sided inverse without a two-sided inverse are different algebraic phenomena, even when related geometric methods appear in their study.

8. Sources and further reading

Higman’s group-ring work and Kaplansky’s Problems in the theory of rings (1957) give the historical problem. Kropholler, Linnell, and Moody’s positive theorem and Fisher–Sánchez-Peralta’s division-ring embeddings illustrate important cases where zero divisors are excluded.

Rips and Segev’s non-unique-product construction and Gardam’s unit-conjecture counterexample concern distinct neighboring questions. The Lipton–Tarjan separator theorem supplies the planar reduction used in the new argument.

Brown’s Cohomology of Groups and Hatcher’s Algebraic Topology explain classifying spaces and the topological language behind the torsion-free safeguard.

Source manuscripts

• M1. OpenAI, A Torsion-Free Group Algebra with Zero Divisors (September 23, 2026).

Sources consulted on 21 October 2026 at repository revision 1581d4c. Download the manuscript citations.