1 · The essentials
These manuscripts study expressions that combine elements of a group with numerical coefficients. They focus on special objects that stay unchanged when multiplied by themselves. A trace extracts information from these objects and sorts it into classes of group elements related by a change of viewpoint. The main manuscript shows that, even when certain infinite sums are allowed, this information can appear in only finitely many classes. Every such class must consist of elements that return to the identity after finitely many repetitions. The statement applies to every discrete group. A companion manuscript, marked as Lean-formalized, gives a related result for finite sums and a consequence excluding all but two self-reproducing elements under additional assumptions.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine a sorting machine that places a summary of a collection into labeled drawers. The main result says that only finitely many drawers can receive anything, and each occupied drawer must have a particular allowed label. This illustrates the restriction on the summary, not a restriction on every item in the original collection.
Why it matters
The main manuscript gives a uniform answer to the ℓ¹-Bass conjecture without restricting which discrete group is used. The companion connects a related trace restriction to a concrete algebraic conclusion: under its hypotheses, an element of a group ring that squares to itself must be either 0 or 1. Its Lean formalization provides a verification component for this family, with the supplied formalization label attached specifically to the companion.
2 · Going further
For a discrete group , the main manuscript considers the complex group algebra , whose coefficients are absolutely summable. Its precise statement concerns every idempotent matrix over this algebra, meaning . The Hattori–Stallings trace records information by conjugacy class. The manuscript shows that this trace is supported on finitely many conjugacy classes, all consisting of finite-order elements: outside those classes, its coefficients vanish.
The companion treats the complex group ring and the Hattori–Stallings trace of a finitely generated projective module. Its stated Bass trace result says that this trace is supported on conjugacy classes of finite-order elements, again for every discrete group. Thus the two manuscripts address related trace restrictions in different settings: absolutely summable expressions in the main manuscript and finite group-ring expressions in the companion.
The companion then gives a consequence for a torsion-free group and every commutative unital domain of characteristic zero: the only idempotents in are and . This is the characteristic-zero Kaplansky idempotent statement described in its abstract. The repository summary also attributes the integral Bass trace conjecture to the algebraic companion, although the supplied companion abstract states the complex group-ring version.
The abstracts do not describe the argument establishing either trace restriction, so no proof mechanism can be extracted from the supplied text. The logical connection they do provide is that the companion's trace result yields its idempotent conclusion. The companion is explicitly marked as Lean-formalized; the supplied material does not specify which individual claims its formalization covers or identify a separate formalization of the statement.
Useful notions
- Groups and the identity element
- Finite-order elements and conjugacy classes
- Group rings and absolutely summable coefficients
- Idempotent elements and matrices
- Traces and projective modules
What was known before / what is new
Before
The source identifies the questions as the ℓ¹-Bass conjecture, the Bass trace conjecture, and Kaplansky's idempotent conjecture. It gives no earlier partial results, historical development, or account of previous methods, so it does not support a more detailed comparison with prior work.
New
The main manuscript shows the ℓ¹-Bass statement for every discrete group, including the requirement that the trace occupy only finitely many finite-order conjugacy classes. The Lean-labeled companion states the complex group-ring Bass result for every discrete group and the characteristic-zero Kaplansky consequence for every torsion-free group over the specified coefficient domains. The repository summary additionally reports an integral Bass trace result.
Limits and assumptions
The main statement is about idempotent matrices over the complex algebra and the support of their Hattori–Stallings traces, not the support of the matrices themselves. The companion's idempotent conclusion requires a torsion-free group and a commutative unital coefficient domain of characteristic zero. A Lean formalization is explicitly reported for the companion, but the supplied text does not delimit its coverage claim by claim or establish formal verification of the separate ℓ¹ manuscript. No remaining open cases are listed; conclusions outside the stated coefficient and group hypotheses are not supplied.
Mini glossary
- Idempotent —
- An element or square matrix that satisfies .
- Conjugacy class —
- The collection of group elements of the form for a fixed element as ranges over the group.
- Finite-order element —
- A group element some positive power of which is the identity.
- Support —
- The positions, here conjugacy classes, where the coefficients of the trace are nonzero.
- Torsion-free group —
- A group whose only finite-order element is the identity.
- Hattori–Stallings trace —
- The trace used in these statements to associate conjugacy-class information to idempotent matrices or finitely generated projective modules.
Keywords : ℓ¹-Bass conjecture · Hattori–Stallings trace · Kaplansky idempotent conjecture
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)
Proves the ℓ1-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over are supported on finitely many finite-order conjugacy classes. The algebraic companion proves the integral Bass trace conjecture and Kaplansky's idempotent conjecture for torsion-free groups over every commutative unital characteristic-zero domain.
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.
Manuscripts in this family
The ℓ¹-Bass Conjecture for Discrete Groups
05/10/2026We prove the ℓ1-Bass conjecture for every discrete group. The Hattori–Stallings trace of every idempotent matrix over the complex ℓ1 group algebra is supported on finitely many conjugacy classes of finite-order elements.
The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture
24/09/2026We prove the complex group-ring Bass trace conjecture for every discrete group: the Hattori–Stallings trace of a finitely generated projective module over its complex group ring is supported on conjugacy classes of finite-order elements. As a consequence, for every torsion-free group G and every commutative unital domain R of characteristic zero, the only idempotents in are 0 and 1. This proves Kaplansky's idempotent conjecture in characteristic zero.