1. The question
Finite quotients provide a way to study an infinite group through finite calculations. If two elements are different, one hopes that some finite image of the group will still distinguish them.
Residual finiteness says that this hope always succeeds. Hyperbolicity imposes a very different kind of control: the group’s large-scale geometry has uniformly thin triangles, resembling negative curvature. Does that geometric discipline force enough finite quotients to detect every element?
Family 252 claims a negative answer. Its manuscript constructs a torsion-free word-hyperbolic group with a nonidentity element killed by every homomorphism to a finite group.
The same element is then invisible to every finite-dimensional linear representation over every commutative field. The source proves the existence of such an element among a fixed finite family, but does not identify which member of that family it is. That distinction is part of the stated result.
2. Background and essential definitions
Seeing an infinite group through finite images
A homomorphism preserves multiplication. When is finite, it gives a finite image of . Its kernel consists of elements that become the identity in that image.
The group is residually finite if every has some finite image in which . Equivalently, the finite residual
contains only the identity.
For example, the integers are residually finite: a nonzero integer remains nonzero modulo some positive integer. No single modulus distinguishes every pair, but the whole family of finite quotients does.
Failure of residual finiteness means that at least one nonidentity element disappears in every finite image. It does not mean that all finite images are trivial. Other elements may remain visible.
Hyperbolicity and torsion
Choose a finite generating set for , and form its Cayley graph. Vertices are group elements; edges multiply by generators; distance counts the fewest generator steps.
The group is word-hyperbolic if some constant makes every shortest-path triangle -thin: every point on one side is within distance of the other two sides. Changing the finite generating set changes constants but not the property.
Trees are the simplest examples. Products of two infinite trees behave differently: they contain flat planes, so the required uniform thinness fails. Thus nonpositive curvature is a broader notion than word-hyperbolicity.
An element is torsion if some positive power equals the identity. A torsion-free group has no such elements except the identity. The new claim retains torsion-freeness, so the failure of finite detection is not blamed on finite-order elements.
Linear representations
A finite-dimensional linear representation over a commutative field is a homomorphism
The group is linear over if one such representation is injective. A classical theorem of Malcev says that finitely generated linear groups over commutative fields are residually finite, including in positive characteristic.
Consequently, a finitely generated group with a nontrivial finite residual cannot be faithfully represented by finite matrices over any such field. The manuscript records the stronger common-kernel formulation: every element of lies in the kernel of every one of these representations.
3. Development history and previous partial results
The residual-finiteness question is part of the program surrounding Gromov’s theory of hyperbolic groups. Hyperbolicity gives powerful control over geometry and algorithms, but it does not directly produce finite quotients.
Ilya Kapovich and Wise proved equivalences between several class-wide statements. All hyperbolic groups are residually finite if and only if every nontrivial hyperbolic group has a nontrivial finite quotient. They also related the same universal assertion to every hyperbolic group being virtually torsion-free, meaning torsion-free after passage to a finite-index subgroup.
These equivalences do not say that each individual failure has all the corresponding properties. For example, a particular group may fail residual finiteness while still having nontrivial finite quotients.
Agol, Groves and Manning showed that a positive answer for all hyperbolic groups would imply separability of every quasiconvex subgroup of every hyperbolic group. Quasiconvexity means geodesics joining subgroup elements stay uniformly near the subgroup. Separability asks that elements outside a subgroup can be distinguished from it in finite quotients. Their argument needs residual finiteness of certain hyperbolic quotient groups, explaining the importance of the class-wide hypothesis.
Additional geometric structure does give positive results. Haglund and Wise developed special cube complexes, whose groups have strong linear and residual properties. Agol proved that a hyperbolic group acting properly and cocompactly on a CAT(0) cube complex is virtually special. Such a group is residually finite.
Earlier examples showed that nonpositive curvature by itself was insufficient. Wise constructed compact nonpositively curved square complexes with non-residually-finite fundamental groups. Burger and Mozes constructed finitely presented simple groups acting on products of trees. Those products contain flat planes, so these were not counterexamples to the hyperbolic-group question.
Nonlinearity was also already known among hyperbolic groups. Michael Kapovich constructed examples with severe restrictions on finite-dimensional representations, and Tholozan and Tsouvalas constructed residually finite nonlinear hyperbolic groups. The latter result makes the logical distinction especially clear: lack of a faithful linear representation does not alone establish failure of residual finiteness.
The proposed advance is to combine torsion-free hyperbolicity with a genuinely nontrivial finite residual in one group.
4. What the new manuscripts claim
The manuscript dated 23 September 2026 asserts that there exists a torsion-free word-hyperbolic group that is not residually finite.
Its group is the fundamental group of a finite Euclidean triangle complex. Such a complex is assembled from finitely many vertices, edges and Euclidean triangles, with compatible edge lengths. In particular, the resulting fundamental group has a finite presentation.
The construction supplies a fixed finite family of nontrivial elements represented by rectangular edge loops. The paper shows that one member of this family belongs to . The family and the complex are fixed independently of any finite quotient under consideration.
The proof is existential about the selected member. It establishes that one fixed nonidentity element is killed by every finite quotient; it does not choose a potentially different element for each quotient and stop there.
As a classical consequence, for every commutative field , every finite dimension , and every representation ,
Thus one common nonidentity element is killed simultaneously by all these representations, across all fields and dimensions. The group is not linear over any commutative field.
Using the Kapovich–Wise equivalences, the paper also records existence of some nontrivial hyperbolic group with no nontrivial finite quotient and of some hyperbolic group that is not virtually torsion-free. These are further existence consequences about the class. They are not additional descriptions of the torsion-free group explicitly constructed here.
5. Why the advance matters
Importance: 5/5, provisional. The claimed example resolves a central question about whether coarse negative curvature forces finite-quotient approximation. It separates a powerful geometric condition from a fundamental algebraic finiteness property.
The torsion-free conclusion strengthens the separation. The finite residual survives even when finite-order phenomena are excluded, and the group still has the finite description supplied by a finite triangle complex.
The common-kernel statement also gives a strong limitation on matrix models. Every finite-dimensional linear representation over every commutative field misses the same nonidentity element. This is more informative than simply saying that each candidate faithful representation fails somewhere.
The result would also clarify the role of cubulation and specialness in positive theorems. Their extra geometry cannot be discarded and replaced by hyperbolicity alone.
6. The main idea
Make geometry and finite algebra pull in opposite directions
The construction must satisfy two demands simultaneously. Local geometry must force hyperbolicity and torsion-freeness, and must certify that the chosen rectangular loops represent nonidentity elements. Algebraic relations must then prevent all those loops from surviving together in any finite quotient.
The source begins with affine lines in a vector space over a large finite prime field. Each line has a marked point, which is removed from its incidence list. A point and a line label are adjacent when the point lies on the line away from its mark.
Random sampling followed by deletion removes short cycles from the incidence graph while retaining many incidences outside a small exceptional set. A graph’s girth is the length of its shortest cycle, so this produces large girth together with substantial degree.
The remaining incidences are arranged into blocks with Cartesian-product coordinates. Each line label contributes an array of edges between two vertices. Additional edges through a third vertex complete suitable triples to triangles. Shifts in the array coordinates are chosen so that the relevant vertex links inherit the large-girth control.
Angular links certify nontrivial loops
The link of a vertex records the directions of edges leaving that vertex and the angles between them inside incident triangles. Traveling around a loop in the link measures the total angle encountered around the vertex.
The manuscript proves a criterion requiring every nonempty cyclically reduced closed link walk to have angular length at least
for some fixed . It claims that the resulting finite triangle complex has torsion-free hyperbolic fundamental group.
The criterion also says that a closed edge path whose turns all have link distance at least cannot be nullhomotopic. The specially chosen rectangular loops satisfy this local condition, so they are nontrivial in the original group.
The geometric argument uses reduced planar fillings and angular Gauss–Bonnet estimates. The strict angle excess controls interiors of filling diagrams, leading to thinness estimates and then to thin triangles in a Cayley graph. This geometry supplies the nontriviality that the finite-quotient contradiction will need.
Finite separation would make one matrix both large-rank and small-rank
Suppose every rectangular word survived in some finite quotient. Since there are only finitely many such words, take the product of their detecting quotient maps. This gives one finite image in which all of them survive simultaneously.
Represent that finite image by permutation matrices acting on its elements. The edge arrays become block matrices. Survival of the rectangular words forces substantial real rank in these arrays.
On the other hand, the triangle relations make sums over each Cartesian block factor through comparatively small spaces. Factorization bounds matrix rank from above.
The argument must compare these constraints over the finite prime field fixed when the complex was built. It cannot select a new characteristic after learning the size of the finite quotient. The source uses bounds on integer minors, including Hadamard’s determinant inequality and divisibility, to preserve a fixed fraction of the rank uniformly in that quotient size.
Finally, product Lagrange interpolation at the marked points gives a tensor identity. One side has large rank contributed by the edge arrays. The other has small rank because of the local factorizations and the small exceptional set. The initial parameters make these bounds incompatible.
Therefore simultaneous finite separation is impossible. If no individual rectangular word lay in the finite residual, the product construction would have supplied exactly that simultaneous separation. Hence one fixed nontrivial word lies in every finite-quotient kernel.
7. Scope, limitations, and open questions
The theorem produces an example, not a classification of hyperbolic groups with nontrivial finite residual. The positive results for virtually special and other residually finite classes remain separate statements.
The source does not identify which member of the specified finite family is invisible in every finite quotient. Nor does it assert that this particular group has no nontrivial finite quotients at all.
The linear corollary concerns finite-dimensional representations over commutative fields. It does not claim that every representation is trivial, and it makes no assertion about skew fields or infinite-dimensional representations.
The finite-field parameters and the complex are fixed before any finite quotient is tested. This uniformity is essential: a family of examples adapted to larger and larger quotients would not establish a single non-residually-finite group.
There is also no conflict with the sphere-boundary recognition claim in family 246, which has a special boundary hypothesis and gives additional structure in that restricted class. The present theorem concerns hyperbolic groups without that sphere-boundary assumption.
Further questions include describing the constructed finite residual more explicitly and understanding which additional geometric conditions ensure finite-quotient detection. The claimed counterexample establishes that hyperbolicity and torsion-freeness alone do not suffice.
8. Sources and further reading
Gromov, Hyperbolic groups (1987), provides the general framework. Ilya Kapovich and Wise, The equivalence of some residual properties of word-hyperbolic groups, Journal of Algebra 223 (2000), 562–583, gives the class-wide equivalences. Agol, Groves and Manning, Residual finiteness, QCERF and fillings of hyperbolic groups, Geometry & Topology 13 (2009), 1043–1073, explains the separability implications.
Haglund and Wise, Special cube complexes (2008), and Agol’s The virtual Haken conjecture give important positive cases. Wise’s Complete square complexes (2007) and Burger–Mozes’ Finitely presented simple groups and products of trees (1997) concern the broader nonpositively curved setting.
Malcev’s On isomorphic matrix representations of infinite groups (1940) supplies the linear residual-finiteness theorem. Tholozan and Tsouvalas, Residually finite non-linear hyperbolic groups, Commentarii Mathematici Helvetici 100 (2025), 1–9, explains why nonlinearity alone is a different conclusion.
Source manuscripts
- M1. OpenAI, A torsion-free hyperbolic group that is not residually finite (September 23, 2026).
Sources consulted on 22 October 2026 at repository revision 1581d4c. Download the manuscript citations.