1 · The essentials
Can a mathematical system always be understood through its finite versions? The supplied manuscript answers this negatively for a particular class of groups—systems of reversible operations—called word-hyperbolic groups. It constructs one in which no nonidentity operation returns to the identity simply by being repeated a positive number of times. Yet finite versions cannot distinguish all of its elements from the identity. The repository summary adds that one fixed nonidentity element is invisible in every representation by finite-sized matrices over any commutative field. Consequently, those matrix representations cannot faithfully describe the whole group either. The main construction has a Lean formalization, although the supplied text does not establish that this formalization also covers the additional matrix-representation claim.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine trying to identify an object using a collection of simplified photographs. Here, every finite photograph misses a distinction that really exists; the repository summary further says that one fixed distinction remains invisible to every allowed matrix-based photograph.
Why it matters
The manuscript answers the residual-finiteness question for hyperbolic groups negatively by constructing a counterexample. Its example is also torsion-free, so the failure occurs even under that additional restriction. The repository summary describes a further limitation: no faithful finite-dimensional linear representation exists over any commutative field.
2 · Going further
The precise main question concerns residual finiteness: for each nonidentity element of a group, is there a homomorphism to a finite group that sends it to a nonidentity element? The manuscript constructs a torsion-free word-hyperbolic group for which the answer is no. Torsion-free means that for a positive integer implies . Word-hyperbolicity is the geometric condition on the group specified in the title and abstract.
The repository summary states a stronger representation-theoretic property of the constructed group : there is a fixed element such that every finite-dimensional representation satisfies , for every commutative field . Thus no such representation is injective, and is not linear over any commutative field. The important quantifier is that the same element is invisible in every one of these representations.
The supplied abstract describes the approach only as a construction; it gives no construction details or intermediate argument to explain further. Only one manuscript is supplied, so no relationship with companion manuscripts can be documented. That main manuscript is marked Lean-formalized, while the stronger statement about linear representations appears in the repository summary; the supplied material does not specify whether that additional statement is included in the formalization.
Useful notions
- Groups and the identity element
- Homomorphisms between groups
- Finite groups
- Finite-order elements
- Matrix representations
- Geometric conditions on groups
What was known before / what is new
Before
The abstract identifies residual finiteness for hyperbolic groups as the question being addressed. The supplied material gives no earlier partial results, historical attribution, or prior construction to compare with this one.
New
The main manuscript constructs a torsion-free word-hyperbolic group that is not residually finite, and this main result has a Lean formalization. The repository summary additionally claims that a single nonidentity element is killed by every finite-dimensional linear representation over every commutative field. This implies that the constructed group is not linear over any such field.
Limits and assumptions
This is an existence result for a constructed torsion-free word-hyperbolic group, not a claim that every group in that class fails to be residually finite or linear. The representation statement concerns finite-dimensional representations over commutative fields. The main result is marked Lean-formalized, but the supplied text does not detail the formalization's scope or confirm coverage of the stronger representation statement. It provides neither the construction nor the argument's intermediate steps, and it identifies no remaining open questions.
Mini glossary
- Torsion-free —
- A group is torsion-free if only its identity element can become the identity after being repeated a positive number of times.
- Residually finite —
- A group is residually finite if every nonidentity element remains nonidentity under some homomorphism to a finite group.
- Finite-dimensional linear representation —
- A finite-dimensional linear representation is a homomorphism from a group to a group of invertible matrices of a fixed finite size over a field.
- Faithful representation —
- A representation is faithful if it is injective, so distinct group elements have distinct images.
- Linear over a field —
- A group is linear over a field if it has a faithful finite-dimensional linear representation over that field.
Keywords : Hyperbolic groups · Residual finiteness · Linear representations
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 torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question negatively. One fixed nonidentity element is killed by every finite-dimensional linear representation over every commutative field, so the group is not linear over any such field.
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
A torsion-free hyperbolic group that is not residually finite
23/09/2026We construct a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question for hyperbolic groups negatively.