Group Theory
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Paper Nonsingular systems of equations over arbitrary groups
Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie's conjecture. The coefficient group…
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Paper An infinite finitely presented residually finite 2-group and a finitely presented nil algebra
Constructs an infinite finitely presented residually finite group whose elements all have finite 2-power order, answering the finitely presented Burnside problem negatively even in this class. The construction also…
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Paper Amenability, unitarizability, and strong Ulam stability
Resolves Dixmier's problem for all discrete groups: amenability is equivalent to every uniformly bounded Hilbert-space representation being similar to a unitary representation. For countable discrete groups, amenability…
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Paper A group without fixed price
Constructs a finitely generated group with two essentially free probability-measure-preserving actions of different costs, answering the general fixed-price problem negatively. Its Bernoulli action has cost bounded away…
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Paper Gersten’s conjecture and virtual compact specialness of one-relator groups
Proves Gersten's conjecture: every finitely generated one-relator group containing no Baumslag–Solitar subgroup \mathrm{BS}(m,n), with m,n\ne0, is word-hyperbolic. It also proves that every word-hyperbolic…
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Paper A hyperbolic group without a geometric CAT(0) action
Answers negatively whether every word-hyperbolic group is a CAT(0) group. Constructs one with a finite classifying space but no proper cocompact isometric action on any proper complete $`\mathop{\mathrm{CAT}}\nolimits…
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Paper Quasi-isometric recognition of virtually polycyclic groups
Proves that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic, resolving the Eskin–Fisher–Whyte lattice-recognition conjecture. Equivalently, a…
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Paper Thompson's group F is nonamenable
Proves that Thompson's group F, the group of dyadic piecewise linear homeomorphisms of the interval, is nonamenable, resolving its longstanding amenability problem. () Manuscript September 23, 2026 · · Lean…
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Paper A torsion-free hyperbolic group that is neither residually finite nor linear over any field
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…
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Paper An infinite finitely presented simple amenable group
Constructs an infinite finitely presented simple amenable group, answering the longstanding question of whether these properties can occur simultaneously. () Manuscript September 23, 2026 · · Lean formalization…
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Paper Classifying spaces and geometric obstructions for Artin groups
The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin K(\pi,1) conjecture. Arbitrary intersections of its parabolic subgroups are parabolic, proving the Parabolic Intersection…
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Paper Boone–Higman embeddings with higher finiteness
A finitely generated group has decidable word problem exactly when it embeds in a finitely presented simple group, proving the Boone–Higman conjecture. The target can have type F∞: a classifying space with finitely many…
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Paper Cannon's conjecture
Every word-hyperbolic group with boundary homeomorphic to S2 admits a proper cocompact isometric action on hyperbolic three-space with finite kernel, proving Cannon's conjecture. Every torsion-free such group is…
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Paper A finitely generated Eilenberg–Ganea counterexample
Constructs a finitely generated residually finite group with integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture. It has no two-dimensional classifying space,…
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