Group Theory
Finite groups, topological groups, representation theory, cohomology, classification and structure
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Every continuous reparameterization is matched to a group metric; word metrics yield flows with all orbits of integer length.
· “A geometric correspondence for reparameterizations of geodesic flows”
Higher-dimensional Brin-Thompson groups control torsion yet embed the rationals in continuum many distinct ways.
With clique number at most n, about 1.414^n abelian subgroups always suffice, and extraspecial 2-groups show this is optimal.
· “Sharp Asymptotics for Abelian Covers of Groups with Bounded Noncommutativity”
Short proof: any non-precompact group has a uniformly continuous function whose orbit is an ℓ1 basis.
Kleinian groups are quasi-isometric exactly when Bowditch boundaries agree on carpets and cut-point orders.
Every graph that looks planar from far away has a finite combinatorial recipe from planar graphs.
For n≥3, PSL_n(Z)'s reduced C*-algebra has no exotic invariant subalgebras — rigidity matches the von Neumann case.
Phase/CZ circuits and mirrors span the whole code-preserving Clifford group; two-local stacks go full on 78 codes.
· “Beyond transversality: structure of Clifford circuits for CSS codes”
A Sylow-subgroup argument settles the open problem and narrows the next case, p=97, to a rigid form.
An explicit brace has a solvable additive group but a multiplicative group with an insoluble PSL2(7) quotient.
· “A Counterexample to Byott's Conjecture for Finite Skew Braces”
For finitely presented groups the Gödel numbers of true modal sentences match those of true arithmetic up to computable mapping.
Modal group theory shows propositional validities under embeddings are exactly S4.2 and interprets true arithmetic.
Density-1/2 sets satisfy |A Δ (ax+b)| = o(p) for all |a|,|b| = o(log p), solving Green's open problem 90.
· “Almost Affine Invariance Over Prime Fields: Green Problem 90”
The group is amenable exactly when F is, yet its minimal line actions lack a Borel transversal for conjugacy.
The generating sets also describe the commutator subgroup and recover Johnson's abelianization of the Torelli group.
Even counting up to relabelling, degrees n=p^m yield at least 2^{(1/p-o(1)) n log n} classes.
· “Asymptotic enumeration of minimally transitive permutation groups”
Even with infinitely many cusps, first-return data and a renormalized product keep orbit information computable.
· “Zeta renormalization and pressure at infinity for an infinitely cusped tree lattice”
This paper proves that a single comonad on sets recovers the full isomorphism type of a group.
The equivalence is witnessed by an explicit metric, and a dichotomy splits the subgroups into countable or Fσδ-complete.
· “Complexity and Polishability of characterized subgroups on the unit circle”
2-step nilpotent Lie groups and their lattices fill loops with area n^2 log n, ending the power-only pattern.
New proof settles the 25-year-old Isaacs–Navarro–Wolf conjecture for finite solvable groups
For modules over free group algebras, the average number of homomorphisms to a random n-dimensional module is eventually rational.
· “Average Numbers of Homomorphisms to Random Modules over Free Group Algebras”
Orbits on pairs of j-elements define a Coxeter graph; twisting along it always preserves regularity.
· “Coset geometries acting on their elements: The j-diagonals twisting”
Homology exponents depend only on degree and group exponent; solvable and nilpotent groups get explicit e-power bounds.
High-genus finite-cover epimorphisms from a 1989 construction are proven equivalent to the standard splitting of S^3.
· “Equations in Products of Free Groups and 3-Manifold Groups II: Olshanskii Epimorphisms”
A growth exponent near 0.1523 settles the hyperfiniteness question for amenable actions of that growth.
A long cycle spanning three quarters of the letters is enough, and two classes also cover the symmetric group.
· “On Conjugacy Classes of Derangements in Symmetric and Alternating Groups”
Every non-adjacent Nielsen generator is an iterated commutator, so two short relation lists define the whole group.
· “Presentations of Lower-Triangular Subgroups of operatorname{Aut}(F_n)”
Gorenstein homological dimension equals Hirsch length in virtually soluble groups, even with torsion.
A categorical construction over free presentations recovers the group-ring quotients, with a matching Lie-ring version.
· “Dimension quotients as boundary limits: the general case”
Representations, Hopf algebras, and analytic coordinates are interchangeable toolkits for pro-p groups.
This settles which finite groups are forcible: abelian groups whose p-parts are cyclic times elementary abelian.
Only coordinate shifts and one flip survive; the injective monoid endomorphisms reduce to the identity.
Almost-polynomial maps of G reduce to almost-homomorphisms of Pol_s(G); polynomial soficity adds no new groups.
· “Polynomial permutation stability, soficity, and universal polynomial groups”
All 37 irreducible-character cases of the order-four permanental-dominance conjecture are proved.
· “The General Subgroup Permanental-Dominance Conjecture in Order Four”
The equality links element-order sums to Burnside rings, refining the crossed Burnside rank and a Cartan determinant.
· “Deflation map and the sum of inverses of the element orders in finite groups”
PD4-pairs assemble from balls and algebraic pieces exactly when a group-theoretic cohomology vanishes
Four near-flip measures are either below a bound just above 1/2 or hit a short list ending at 1 — the gap is forced by group-index…
· “A probabilistic approach to the Yang--Baxter equation and skew braces”
The paper identifies the minimal set of representations needed to compute the gap exactly for every choice of rates.
· “Random walks on wreath products and spectral gaps for coloured interchange processes”
The proof also forces vanishing first ℓ²-Betti number and anti-treeability.
A small 8-generator construction answers an open 2-complex question and also refutes the Lie-ring analogue.
For every Grigorchuk group, all three 3-generator Cayley graphs have connective constant at least the golden mean.
A faithful complex representation of H×K needs δ(H)+δ(K) dimensions; δ now tabulated for all groups of order p^6.
All orientably regular maps with this linear symmetry group admit a reflection, extending a 1974 result.
· “A note on the reflexibility of regular maps with automorphism group PGL(2,q)”
New local-density test splits the property factor by factor; a second result hits every density n/m.
· “Characterization of Weak EKR Groups and Intersection Densities with Prescribed Point Stabilizers”
Every finite simple group of Lie type over fields of size at most 8 contains a conjugacy class whose square is the whole group, completing…
· “Thompson's Conjecture for Finite Simple Groups of Lie Type over Small Fields”
When the degree is prime to p, the p-part of the conductor equals the torus character's, except at level one.
A construction builds graphings whose symmetries match any ergodic map's centralizer, even when no compact topology fits.
· “Ergodic-transformation centralizers and essentially non-compact graphing symmetry”
It proves the conjectured character-square sum bound for all primes, with equality tied to a normal complement.
Every lower-central quotient is free abelian, with an explicit basis of weighted commutators.
· “The Lower Central Series of Right Lower-Triangular Nielsen Automorphism Groups”
The count decides how many type III1 factors share a given continuous core.
· “Non-uniqueness of continuous trace-scaling flows on certain full factors”
The order of a simple group is forced small when few p-elements meet a capped centralizer exponent.
A complete structural test for when a quotient of an F-inverse monoid preserves the maximum-element operation.
The new divisibility implies the exponent conjecture for every group-theoretical semisimple Hopf algebra over any field.
· “Exponents of factorized groups and Kashina's conjecture for group-theoretical Hopf algebras”
A cut-and-paste topological proof shows every Schur multiplier generator is toral, linking rationality and bordism.
· “Topological perspectives on the vanishing of some Bogomolov multipliers”
The paper settles exactly which infinite simple locally finite groups can satisfy a mixed identity.
Iterated Galois groups die out unless the map factors through an algebraic curve, settling a problem open since 1985.
· “The inverse Galois problem of iterated Galois groups and their fixed-point proportion”
One small defect matrix tells how many eigenvalues can be irrational and how large the splitting field can grow.
· “Splitting fields and spectral invariants of character degree graphs in solvable groups”
The residual property behind virtual-fibering proofs now holds for pure braid groups.
If a hyperbolic graph splits as a tree of hyperfinite bags, its boundary action is hyperfinite too.
· “Hyperfiniteness of boundary actions via tree decompositions”
A new topological definition separates the Tits building from all exotic alternatives for groups of Lie type of rank 2.
For two-generated monoids, determinant independence from the cocycle makes the trivial character check enough.
· “Homogeneous Weights on Semigroup Algebras of Finite Commutative Semigroups”
The same divisor-indexed vectors cover element orders, class sizes, and character degrees.
· “Arithmetic invariants for finite simple and related groups”
Two are impossible, and explicit triples are given for dimension five.
· “On minimal involutive generic sets of Extended Special Linear group ESL_n(mathbb{F}_p)”
Superstability adds weak noetherianity; ideal generating sizes give the full stability spectrum.
· “On the abstract elementary class of acts with pure embeddings”
Making every non-power deck 2-transitive, the proof yields Alt or Sym and settles two conjectures.
· “Cyclic Shuffle Groups: Universal Two-Transitivity and Complete Classification”
A generating function with closed local factors gives exact counts for every n, and sp(4) is verified by computer.
· “Commuting pairs in symplectic Lie algebras over finite fields of characteristic two”
Every other component is an isolated vertex or an explicit two-vertex pair in PGL_r(q) or PGU_r(q).
· “Components of the Divisibility Graph of Finite Groups of Lie Type in Defining Characteristic 2”
The result pins down when the Lefschetz map for endotrivial complexes is surjective: exactly over F2.
Explicit generating pairs stay distinct until a trivial coordinate is added.
· “Nielsen classes in outer automorphism groups of free groups”
One trichotomy classifies PL circle groups and blocks hyperbolic manifold actions in all dimensions at least 3.
· “Hyperbolicity and obstructions to PL actions of the circle”
For every d≥2 and every generating set, the diagonal operator on the lamplighter group fails to metrize the state space.
· “Word-Length Spectral Triples of (mathbb{Z}/2mathbb{Z})wrmathbb{F}_(d) Are Not Metric”
Finite many Morita classes for each quaternion group, completing Donovan's conjecture for tame blocks.
A family of d-generated subgroups with coprime indices forces a finite group to need at most d+1 generators.
· “Generation of finite groups from subgroups of coprime index”
For virtual knot groups, circularly orderable is the same as left-orderable — a dichotomy proved through one relative quotient.
The paper swaps the intricate original construction for an iterative one, keeping the same transfer and normality properties.
The rank is the only part that cannot be compressed; the transverse directions shrink by half.
· “Kakeya sets and dimension compression in compact Lie groups”
Counterexamples on any probability space settle a 2010 question of Pestov.
The proof turns the conjecture into an orbit statement: faithful modules contain a vector with a sufficiently large orbit.
These coset geometries are flag-transitive and firm; the q=2 unitriangular case fills a missing family.
