arXiv math.GT digest — 29 May 2026 (16 papers)

Today's listing for the Friday 29 May 2026 session brings 16 papers to math.GT: 5 new submissions, 2 cross-lists from group theory, and 9 replacements. Themes range from discrete lattice-knot geometry and trisection theory to surface-group representations in PU(2,1), combinatorial descriptions of closed 3-manifolds, and minimal-area homotopy theory.

arXiv:2605.28891 · Ulysse Remfort-Aurat (I2M)

Abstract. Let be the fundamental group of a closed orientable surface of genus . The outer automorphism group naturally acts on the character variety for any Lie group $G$. We consider the set of simple-stable representations which are modelled on Minsky's primitive-stable representations. We prove that the set of conjugacy classes of simple-stable representations of in is a domain of discontinuity for this action, strictly larger than the set of conjugacy classes of convex cocompact representations.

Key result. The simple-stable locus in is a domain of discontinuity for the -action, and it strictly contains the convex cocompact representations.

Techniques. The work adapts Minsky's primitive-stability framework from to the complex-hyperbolic setting. Simple curves in replace primitive elements; the key analytical tool is controlling the dynamics of simple geodesics under the PU(2,1) action.

Proof idea. One verifies that the simple-stability condition is open and -invariant, then shows proper discontinuity of the -action by demonstrating that only finitely many outer automorphisms can send a simple-stable representation into any fixed compact set in the character variety. The strict inclusion of convex cocompacts uses explicit non-convex-cocompact examples that remain simple-stable.

arXiv:2605.29160 · Makoto Ozawa

Abstract. We develop a framework for discrete p-density and compression-radius profiles of lattice knots. For lattice polygons representing a fixed knot type, we define scale-free density quantities by dividing lattice length by chord-length spread functionals, and we define corresponding compression radii using a raw lattice thickness convention. These profiles are studied both on length-filtered lattice representative sets and on seed-generated finite search spaces arising from local move systems such as BFACF moves. The framework is intended as a computable counterpart of continuous scale-free geometric invariants of knot types. It separates density, compression, and ropelength-like quantities, and records how these quantities behave under length filtration and finite move-graph exploration. We emphasize throughout the distinction between exhaustive lattice-knot invariants and seed-generated computational profiles. As a case study, we compute raw p-density and compression-radius values along an explicit BFACF merge certificate connecting a 30-edge simple cubic figure-eight seed to its reflected mirror seed at length bound N=32. The extracted path has 21 states and 20 BFACF moves and passes through a 32-edge connecting state. Along this path, density and compression-radius values are not monotone: the connecting state has larger raw density and slightly larger -compression radius than the seeds, while an intermediate 30-edge state has smaller compression radius but larger density. This illustrates that density minimization, compression minimization, and ropelength-like minimization define distinct finite-state optimization problems.

Key result. A discrete, scale-free framework for p-density and compression radii of lattice knots is developed; the figure-eight case study shows these three quantities are not simultaneously minimized along a BFACF path, establishing they define genuinely distinct optimization problems.

Techniques. Lattice knot combinatorics (BFACF moves), chord-length spread functionals for scale-free normalization, and computational exploration of move-graph paths. Supplementary code and data are deposited at Zenodo (doi:10.5281/zenodo.20419442).

Proof idea. The framework is largely constructive: define the invariants, implement them, then exhibit the figure-eight BFACF path explicitly. The non-monotonicity is verified by direct computation on the 21-state, 20-move path, not by a general theorem, which is precisely the point — the paper shows that no simple monotonicity principle holds.

arXiv:2605.29252 · Natsuya Takahashi

Abstract. We classify knot traces with trisection genus at most 2. We give infinitely many knots whose traces have trisection genus 3, and infinitely many knots whose traces have trisection genus 4. We also show that there exist infinite families of knots whose traces have arbitrarily large trisection genus. In addition, we determine or give sharp bounds for the trisection genus of the traces of several well-known knots, such as the figure-eight knot, the $(p, pq+1)$-torus knots, and the $(-2, 3, 2n-1)$-pretzel knots.

Key result. Complete classification of knot traces with trisection genus ; infinite families with genus 3, 4, and arbitrarily large; explicit sharp values for the figure-eight, $(p, pq+1)$-torus, and $(-2,3,2n-1)$-pretzel traces.

Techniques. Trisection diagrams and Heegaard-splitting-style decompositions of 4-manifolds, combined with knot-theoretic surgery descriptions of trace exteriors.

Proof idea. The classification for genus proceeds by enumerating possible trisection diagrams of the relevant complexity and matching them to trace surgery descriptions. For the lower-bound results (arbitrarily large genus), one passes to an algebraic obstruction — likely derived from intersection forms or handle complexity — that grows with the chosen knot family.

arXiv:2605.29443 · Stavros Garoufalidis, Rinat Kashaev, Sakie Suzuki

Abstract. It is well known that every compact oriented 3-manifold admits an ideal triangulation, and that any two such triangulations with at least two ideal tetrahedra are related by a sequence of Pachner 2-3 moves. Motivated by constructions in quantum topology, we give a combinatorial description of closed 3-manifolds in terms of ordered ideal triangulations and ordered Pachner 2-3 and 0-2 moves.

Key result. Closed oriented 3-manifolds admit a combinatorial description via ordered ideal triangulations, with moves being ordered versions of the Pachner 2-3 and 0-2 moves. This provides a combinatorial foundation tailored to quantum-topological invariant constructions.

Techniques. Ordered ideal triangulations (where tetrahedra carry a linear ordering on vertices), ordered Pachner moves, and the classical result relating any two ideal triangulations of the same manifold via 2-3 moves.

Proof idea. The standard result shows unordered ideal triangulations are connected by 2-3 moves. The authors add an ordering structure and show that ordered 2-3 and 0-2 moves are sufficient to connect any two ordered ideal triangulations, giving a presentation suitable for building quantum invariants that depend on the ordering.

arXiv:2605.29474 · Lia Buchbinder, Yunjia Kou, Bala Krishnamoorthy, Kevin R. Vixie

Abstract. We study the existence of area-minimizing homotopies between homotopic curves in the plane. While the classical Plateau problem establishes the existence of least-area surfaces spanning a single Jordan curve, the corresponding existence theory for homotopies between curves is more subtle and is not directly covered by the same framework. Existing results in the plane are mainly based on combinatorial and algebraic methods, such as decomposing curves into self-overlapping subcurves. These methods are highly effective in the planar setting, but they are often tied to special classes of curves and rely strongly on the local structure of the self-intersections, frequently assuming transverse crossings. In contrast, our approach is geometric and variational, and does not depend on the local structure of the self-intersections. In this paper, we develop a variational existence theory for minimum-area homotopies of immersed planar curves. Our approach adapts classical minimal surface methods by lifting an immersed planar curve with self-intersections into higher co-dimension, where it becomes embedded. For such a lifted curve, we apply Douglas's solution of the Plateau problem to obtain an area-minimizing disk. For closed curves of class , we prove uniform convergence of the Douglas minimizers and show that the limiting map minimizes area among all spanning maps of the original planar curve. We then extend the construction to closed Lipschitz curves using approximation and Sobolev compactness arguments. Since the limiting minimizing disk lies in the original plane, it directly produces a null homotopy whose swept area is minimal among all admissible homotopies of the original curve. In this way, the construction connects Plateau theory with minimal homotopy area minimization.

Key result. For any closed (resp. Lipschitz) immersed planar curve, a minimum-area null homotopy exists; the result holds without any assumption on the local structure of self-intersections.

Techniques. Lifting to higher codimension to embed the self-intersecting curve; Douglas's Plateau problem solution for area-minimizing disks; uniform convergence of Douglas minimizers ( case); Sobolev compactness and approximation arguments (Lipschitz case).

Proof idea. Lift the planar immersed curve to an embedded curve in higher-dimensional space. Apply Douglas's theorem to get an area-minimizing spanning disk there. Show (via uniform convergence) that the minimizers converge as the lift height goes to zero, and that the limiting map projects to a null homotopy in the plane whose swept area achieves the minimum over all admissible homotopies.

arXiv:2605.28963 (cross-list from math.GR) · Ilaria Castellano, Bianca Marchionna, Brita Nucinkis, Yuri Santos Rego

Abstract. We propose the systematic study of presentations that can be generalised over a continuous open group monomorphism. Presentations with this property can turn well-known presentations such as those for orientable surface groups, Artin groups, and some Thompson groups, into topological groups with a prescribed open subgroup. Later we focus on right-angled Artin groups (RAAGs) and introduce a notion of topological RAAGs. Our approach differs from lattice envelopes and produces examples of locally compact (LC) groups that contain RAAGs as discrete subgroups, but generally not as lattices. We investigate some geometric aspects of topological RAAGs, with a special emphasis on compactness properties of LC ones. This includes a study of universal Salvetti-type complexes which may be of independent interest. These complexes share some properties with buildings. Although in some cases they are CAT(0) cube complexes and provide models for classifying spaces, in other cases they are not even uniquely geodesic. For a large class of examples we establish high connectivity properties for these complexes. This yields novel examples of LC groups with prescribed compactness properties or rational cohomological dimension. We note that the Bestvina-Brady machinery does not automatically generalise to this setting; nevertheless, we extend the Bieri-Stallings construction to obtain totally disconnected locally compact (TDLC) groups of type but not . Along the way we record counterparts of cohomological results, such as a Mayer-Vietoris sequence and Künneth formula in discrete (co)homology for TDLC groups, which have not appeared elsewhere in the literature. Despite our non-discrete LC focus we obtain, as by-product, new examples of discrete groups with controlled finiteness properties including, for every , a Thompson-like Bieri-Stallings group of type but not .

Key result. A theory of topological RAAGs (locally compact groups containing RAAGs as discrete, generally non-lattice subgroups) is developed; universal Salvetti-type complexes provide models for classifying spaces in many cases; new TDLC groups of type are produced for all $n$.

Techniques. Generalisable presentations over continuous open monomorphisms; CAT(0) cube complex geometry; Bestvina-Brady Morse theory (with limitations noted); Bieri-Stallings construction; discrete cohomology for TDLC groups (Mayer-Vietoris, Künneth).

Proof idea. The authors first axiomatise when a group presentation can be "lifted" along a continuous open monomorphism to yield a topological group with a prescribed open subgroup. Applying this to RAAG presentations yields the LC-RAAG family. Finiteness properties are then read off from connectivity of the associated Salvetti-type complexes, extending the classical Bestvina-Brady argument where it applies.

arXiv:2605.29422 (cross-list from math.GR) · Takatoshi Hama

Abstract. We show that the affine cactus group is a CAT(0) group for all degrees. Furthermore, we show that the affine cactus group of degree three is a hyperbolic group.

Key result. Every affine cactus group is CAT(0); additionally is hyperbolic.

Techniques. CAT(0) cubical or polyhedral complex constructions; Gromov-hyperbolicity criteria (likely thin-triangle or fellow-traveller).

Proof idea. Construct a CAT(0) space on which the affine cactus group acts geometrically, then verify the CAT(0) curvature condition. For , demonstrate that the CAT(0) space is hyperbolic (e.g. by exhibiting a linear isoperimetric inequality or verifying the Rips condition).

arXiv:2501.10558 (replaced) · Adrien Rodau

Abstract. We introduce a new topological invariant of complex line arrangements in the complex projective plane, derived from the interaction between their complement and the boundary of a regular neighbourhood. The motivation is to identify Zariski pairs which have the same combinatorics but different embeddings. Building on ideas developed by B. Guerville-Ballé and W. Cadiegan-Schlieper, we consider the inclusion map of the boundary manifold to the exterior and its effect on homology classes. A careful study of the graph Waldhausen structure of the boundary manifold allows to identify specific generators of the homology. Their potential images are encoded in a group, the graph stabiliser, with a nice combinatorial presentation. The invariant related to the inclusion map is an element of this group. Using a computer implementation in Sage, we compute the invariant for some examples and exhibit new Zariski pairs.

Key result. A new topological invariant for complex line arrangements distinguishing Zariski pairs, constructed from the induced map on homology of the boundary manifold inclusion; new Zariski pairs are exhibited via Sage computation.

Techniques. Graph Waldhausen decomposition of boundary manifolds; homology of the complement and boundary; the graph stabiliser group; computer-assisted invariant computation in Sage.

Proof idea. Analyse the long exact sequence of the pair (complement, boundary) and track which homology classes of the boundary manifold map non-trivially into the complement. The graph-theoretic description of the boundary manifold gives explicit generators; the image of each generator under inclusion is an element of the graph stabiliser, producing a computable invariant that distinguishes arrangements with identical combinatorial data.

arXiv:2504.04407 (replaced, to appear in Algebraic and Geometric Topology) · Wei Liao, Baohua Xie

Abstract. We prove that a family of complex hyperbolic ultra-parallel -triangle group representations, where , is discrete and faithful if and only if the isometry is non-elliptic for some positive integer $n$. Additionally, we investigate the special case where and provide a substantial improvement upon the main result by Monaghan, Parker, and Pratoussevitch.

Key result. Full discreteness criterion for complex hyperbolic ultra-parallel triangle groups (): discrete and faithful iff is non-elliptic for some ; improved result for .

Techniques. Complex hyperbolic geometry; isometry classification (elliptic/parabolic/loxodromic) in ; Poincaré polyhedron theorem or analogues for discreteness.

Proof idea. For discrete and faithful representations, the non-ellipticity of specific word isometries is necessary (they would otherwise generate a dense subgroup). For the converse, construct a fundamental domain using the Poincaré polyhedron theorem and verify all cycle conditions, with non-ellipticity of the specified words ensuring the face-pairings are compatible.

arXiv:2509.03396 (replaced) · Advika Rajapakse

Abstract. For an arbitrary link , Sarkar-Scaduto-Stoffregen construct a family of spatial refinements of even and odd Khovanov homology. We give a computation of on these spaces, determining their stable homotopy types for all knots $K$ up to 11 crossings. We also prove that the Steenrod squares , defined by Schütz do arise as Steenrod squares on these spaces.

Key result. is computed on the Sarkar-Scaduto-Stoffregen spatial refinements for all knots up to 11 crossings; the Schütz operations and are identified as genuine Steenrod squares. (Revision: typos corrected, Schütz formula identification corrected, Conway mutation information added.)

Techniques. Khovanov homotopy types; Steenrod operations on CW-spectra; Sarkar-Scaduto-Stoffregen spatial refinements; comparison with Schütz's algebraic construction.

Proof idea. From the explicit cell structure of the spatial refinements, compute via the standard definition on a CW-spectrum (attaching map analysis). The identification with Schütz's , follows by comparing the algebraic formulas with the geometric computation.

arXiv:2603.01079 (replaced) · Ilya Alekseev, Ivan Nasonov, Gaiane Panina

Abstract. Let be an oriented sphere bundle supporting an affine transverse foliation. We give an upper bound for the Euler number of the bundle. We also give a new and elementary proof of the following fact: if the fundamental group of is amenable, then the Euler number of the bundle vanishes.

Key result. An upper bound for the Euler number of a sphere bundle admitting an affine transverse foliation; new elementary proof that amenability of forces the Euler number to vanish.

Techniques. Affine transverse foliations; Euler class / Euler number of sphere bundles; bounded cohomology arguments (amenability); foliated bundle theory.

Proof idea. The upper bound on the Euler number comes from the constraints imposed by the existence of a flat transverse connection (the affine structure). The amenable-base result is reproved elementarily, presumably avoiding the bounded-cohomology machinery of earlier proofs, by exploiting the averaging properties of amenable groups directly at the foliation level.

arXiv:2604.13194 (replaced) · Ayodeji Lindblad

Abstract. For $X$ any complete intersection of even complex dimension or any connected sum thereof (or, more generally, any space among certain broad classes of smooth manifolds), we concretely construct diffeomorphisms $a, c$ of punctured $X$ rel boundary whose commutator $[a,c]$ represents the smooth mapping class (rel boundary) of the boundary Dehn twist. This shows that boundary Dehn twists on 4-manifolds known to be nontrivial in the smooth mapping class group rel boundary by work of Baraglia-Konno, Kronheimer-Mrowka, J. Lin, and Tilton become trivial after abelianization, generalizing work of Y. Lin, who applied an argument based on the global Torelli theorem and an obstruction of Baraglia-Konno to prove that the abelianized boundary Dehn twist on the punctured $K3$ surface is trivial.

Key result. For a broad class of 4-manifolds (complete intersections of even complex dimension, connected sums thereof, and more), the boundary Dehn twist is a commutator in the smooth mapping class group rel boundary, hence vanishes in the abelianization. New results and refined exposition were added in this revision.

Techniques. Smooth 4-manifold topology; mapping class groups rel boundary; Baraglia-Konno and Kronheimer-Mrowka gauge-theory results; global Torelli theorem (for context/history).

Proof idea. Construct explicit diffeomorphisms $a$ and $c$ of the punctured manifold rel boundary such that $[a,c]$ is smoothly isotopic to the boundary Dehn twist. The construction is concrete — not abstract existence — which is the technical contribution. The abelianization triviality then follows formally from $[a,c] = 1$ in any abelian quotient.

arXiv:2605.24610 (replaced) · Roberto De Leo

Abstract. We present a method to build free immersions in critical dimension on $m$-tori for $m = 2, 3, 4, 5$ by using a factorization trick inspired by tori immersions in critical dimension. As an application, we show that the set of smooth free maps from a closed surface $M$ to is nonempty. In particular, every closed surface embeds freely in .

Key result. Every closed surface admits a free embedding into ; free immersions in critical dimension on $m$-tori ($m = 2,3,4,5$) are constructed explicitly.

Techniques. Free maps (immersions whose first-order partial derivatives at every point span the ambient tangent space in the strongest sense); factorization tricks; critical-dimension immersion theory.

Proof idea. For tori, use a factorization of the embedding map that leverages the product structure to construct first-order full rank at each point. Transfer the argument to closed surfaces by using the torus as a model, patching together free immersion pieces. The embedding conclusion comes from the fact that, in the critical codimension, immersions can often be perturbed to embeddings.

arXiv:2503.22470 (replaced, to appear in Commentarii Mathematici Helvetici) · Philippe Eyssidieux, Louis Funar

Abstract. We provide new constraints for algebro-geometric subgroups of mapping class groups, namely images of fundamental groups of curves under complex algebraic maps to the moduli space of smooth curves. Specifically, we prove that the restriction of an infinite, finite rank unitary representation of the mapping class group of a closed surface to an algebro-geometric subgroup should be infinite, when the genus is at least 3. In particular the restriction of most Reshetikhin-Turaev representations of the mapping class group to such subgroups is infinite. To this purpose we use deep work of Gibney, Keel and Morrison to constrain the Shafarevich morphism associated to a linear representation of the fundamental group of the compactifications of the moduli stack of smooth curves studied in our previous work. As an application we prove that universal covers of most of these compactifications are Stein manifolds.

Key result. For genus : any infinite finite-rank unitary representation of the mapping class group restricts to an infinite representation on any algebro-geometric subgroup; most Reshetikhin-Turaev representations therefore have infinite image on such subgroups. Universal covers of most compactifications of moduli stacks of smooth curves are Stein manifolds.

Techniques. Algebro-geometric subgroups of mapping class groups; Shafarevich morphism; Gibney-Keel-Morrison results on the geometry of ; Reshetikhin-Turaev TQFT representations; Stein manifold theory.

Proof idea. Translate the question about unitary representations into a question about the Shafarevich morphism of the relevant compactifications of . Use Gibney-Keel-Morrison to show this morphism is non-trivial, which forces the restricted representation to be infinite. The Stein manifold conclusion follows from the structure of the universal cover deduced from these constraints.

arXiv:2605.01103 (replaced, cross-listed from quant-ph / math-ph / math.SG) · Maurice de Gosson

Abstract. We introduce a geometric formulation of quantum indeterminacy from which the standard uncertainty inequalities emerge as necessary consequences. Our approach is based on convex geometry in phase space and on methods from symplectic topology, and does not rely on statistical descriptors such as variances or covariances. Instead, we associate to empirical position and momentum data convex bodies whose mutual relations encode the fundamental constraints of quantum mechanics. The central tools are h-polar duality and symplectic capacities, which provide intrinsic, coordinate-free bounds on admissible phase-space configurations. Within this framework, the Robertson-Schrödinger inequalities arise naturally as manifestations of deeper geometric and topological principles. This perspective suggests that quantum indeterminacy is not primarily a statistical phenomenon, but rather a structural property of phase space governed by symplectic covariance.

Key result. The Robertson-Schrödinger uncertainty inequalities are derived as consequences of a geometric/symplectic-topological framework, without invoking variances or covariances; quantum indeterminacy is recast as a structural property of symplectic phase space.

Techniques. Convex bodies in phase space; h-polar duality; symplectic capacities; symplectic topology.

Proof idea. Associate convex bodies to joint position-momentum data. Use h-polar duality (analogous to Legendre-Fenchel duality but adapted to symplectic geometry) and symplectic capacity bounds to derive constraints on admissible configurations. Show these constraints, when translated back into probabilistic language, exactly recover the Robertson-Schrödinger inequalities.

arXiv:2605.25265 (replaced) · Ilya Kapovich

Abstract. By the classic results of Fricke and Klein, for every word $w$ in the free group $F(a,b)$ there exists a unique integer trace polynomial such that for all . We study quantitative aspects of trace polynomials. We prove an exact formula for the leading homogeneous part of for every nontrivial cyclically reduced word . In particular, , and the sharp general bounds hold. We obtain explicit exponential upper bounds for the and norms of and exhibit examples with exponential coefficient growth at rate , where is the golden ratio. For random freely reduced, cyclically reduced, and positive words of length $n$, the support size of grows at least quadratically in $n$ and the total bit-size grows at least as . As a consequence, any algorithm computing in totally expanded form has worst-case and generic-case time complexity bounded below by ; a deterministic algorithm achieving time and space is also given.

Key result. An exact formula for ; sharp bounds ; golden-ratio exponential coefficient growth; quadratic support-size growth and cubic bit-size growth for random words; lower bound (and upper bound) for expanded computation. (Revision strengthens support-size to quadratic and bit-size to cubic growth, with complexity consequences.)

Techniques. Fricke-Klein trace identities; combinatorics of cyclic words in $F(a,b)$ (letter-pair counting); probabilistic arguments for random word models; polynomial algebra over .

Proof idea. The degree formula follows from tracking how sub-word patterns ($ab$, , etc.) contribute cancellations in the leading term via the Fricke-Klein recursion. Coefficient size bounds come from bounding the number of monomials and their integer coefficients using the recursion. The complexity lower bound is a direct consequence of the established bit-size growth: any correct algorithm must write down an output of size , so it takes at least that long.