Survey on invariant quasimorphisms and their relation to stable commutator length.
problem Understanding the relationship between invariant quasimorphisms and stable commutator length.
method Review of existing methods and examples in invariant quasimorphisms and their relation to stable commutator length.
result The existence of non-extendable invariant quasimorphisms is closely related to the behavior of stable mixed commutator length.
Quasimorphisms and pseudo-Anosov flows
problem connections between quasimorphisms and pseudo-Anosov flows
method showing connections between quasimorphisms and pseudo-Anosov flows
result upper bounds on exponential growth rate of closed orbits in pseudo-Anosov flows
Let G^ be a group and G its normal subgroup. In this paper, we study G^-invariant quasimorphisms on G which appear in symplectic geometry and low dimensional topology. As its application, we prove the non-existence of a section of the flux homomorphism on closed surfaces of higher genus. We also prove…
New quasimorphisms constructed for manifolds with negative curvature.
problem Building quasimorphisms on manifolds with negative curvature.
method Generalizing Barge and Ghys construction to Lie groups.
result Counterexamples to Kapovich and Fujiwara theorem for Lie groups.
Study cup products on CAT(0) cube complexes, proving quasimorphisms' vanishing results.
problem Understanding cup products in higher bounded cohomology groups.
method Extending vanishing results from Brooks quasimorphisms to median quasimorphisms on CAT(0) cube complexes.
result Vanishing results for cup products of median quasimorphisms on CAT(0) cube complexes.
New proof shows lower bound for commutator length in RAAGs.
problem Finding lower bounds for commutator length in RAAGs.
method Using letter-quasimorphisms to create negatively curved angle structures.
result Establishes a sharp lower bound of 1/2 for stable commutator length in RAAGs.
We study the construction of quasimorphisms on groups acting on trees introduced by Monod and Shalom, that we call median quasimorphisms, and in particular we fully characterise actions on trees that give rise to non-trivial median quasimorphisms. Roughly speaking, either the action is highly transitive on geodesics, i…
Develops thermodynamic formalism for quasimorphisms on negatively curved spaces.
problem Analyzing quasimorphisms on negatively curved spaces.
method Thermodynamic formalism framework, Banach isomorphism, weak Livšic cohomology.
result Establishes Central Limit Theorem and invariance principle for unbounded quasimorphisms.
This paper extends quasimorphism results to nonorientable surfaces.
problem Understanding quasimorphisms on nonorientable surface diffeomorphism groups.
method Constructing infinitely many quasimorphisms on the identity component of nonorientable surface diffeomorphism groups.
result The space of nontrivial quasimorphisms on the identity component of the diffeomorphism group of a closed nonorientable surface is infinite-dimensional.
Paper proves non-extendability of quasimorphism and non-vanishing of Reznikov's class.
problem Non-extendability of Shelukhin's quasimorphism and non-vanishing of Reznikov's class.
method Analyzing specific symplectic manifolds like blow-ups of tori and products of surfaces.
result Non-extendability of Shelukhin's quasimorphism and non-vanishing of Reznikov's class for certain symplectic manifolds.
The paper extends quasimorphisms on subgroups to larger groups.
problem Extending quasimorphisms from subgroups to larger groups.
method Provides a general sufficient condition for extendability of quasimorphisms on subgroups.
result New results for quasimorphisms on normal subgroups, including bi-Lipschitz equivalence of stable commutator length and group-theoretic Dehn filling.
New quasimorphisms show stable commutator lengths are not equivalent.
problem Equivalence of stable commutator lengths in groups.
method Invariant quasimorphisms for groups acting on the circle.
result Stable commutator lengths are not bi-Lipschitzly equivalent.
Proof of boundedness of quasimorphisms for certain Lie groups.
problem Bounding quasimorphisms on lattices in Lie groups.
method Thermodynamic formalism applied to bounded cohomology.
result Every π-quasimorphism on irreducible uniform lattices is bounded.
Study mapping class group action on de Rham quasimorphisms, finding no fixed points.
problem Action of mapping class group on de Rham quasimorphisms.
method Examined the action of mapping class group on de Rham classes in bounded cohomology of a hyperbolic surface.
result No fixed points in the action of mapping class group on de Rham quasimorphisms.
The paper studies quasimorphisms on density-preserving diffeomorphisms of the Möbius band.
problem Exploring quasimorphisms on groups of diffeomorphisms of non-orientable manifolds.
method Investigates the group of density-preserving diffeomorphisms on the Möbius band and shows the existence of unbounded quasimorphisms.
result The group of density-preserving diffeomorphisms on the Möbius band admits countably many unbounded quasimorphisms.
We extend the definition of Weinstein's Action homomorphism to Hamiltonian actions with equivariant moment maps of (possibly infinite-dimensional) Lie groups on symplectic manifolds, and show that under conditions including a uniform bound on the symplectic areas of geodesic triangles the resulting homomorphism extends…
We prove that the group of compactly supported symplectomorphisms of the standard symplectic ball admits a continuum of linearly independent real-valued homogeneous quasimorphisms. In addition these quasimorphisms are Lipschitz in the Hofer metric and have the following property: the value of each such quasimorphism on…
The paper studies quasimorphisms and distortion in homeomorphism groups of manifolds.
problem Identifying quasimorphisms that extend continuously to homeomorphism groups.
method Analyzing Gambaudo-Ghys and Polterovich quasimorphisms, extending them continuously, and presenting applications.
result Conditions for undistorted homeomorphisms and unbounded bi-invariant metrics.
Constructs a bilinear form from a quasimorphism on symplectic manifold groups.
problem Understanding symplectic group properties through quasimorphisms and bilinear forms.
method Develops machinery to construct a real-valued bilinear form from a quasimorphism on the commutator subgroup of symplectic group.
result The constructed bilinear form b controls extendability of quasimorphisms and triviality of characteristic classes. Unified proof of four Bavard dualities and new results on quasimorphisms.
problem Comparing stable commutator length and quasimorphisms on groups.
method Expository account and new strengthening of Bavard duality, providing complete proofs.
result Generalized mixed Bavard duality, recovering all previous dualities.
Let S be a compact oriented surface. We construct homogeneous quasimorphisms on Diff(S,area), on Diff0(S,area) and on Ham(S) generalizing the constructions of Gambaudo-Ghys and Polterovich. We prove that there are infinitely many linearly independent homogeneous quasimorphisms on Diff(S,area), on $Diff_0(…
Bavard proved a duality theorem between commutator length and quasimorphisms. Burago, Ivanov and Polterovich introduced the notion of a conjugation-invariant norm which is a generalization of commutator length. Entov and Polterovich proved that Oh-Schwarz spectral invariants are subset-controlled quasimorphisms which a…
Study of quasimorphisms and bounded cohomology in braided Thompson groups.
problem Investigate quasimorphisms and bounded cohomology in braided versions of Thompson groups.
method Analyze quasimorphisms and bounded cohomology of various braided Thompson groups.
result Found infinite-dimensional spaces of quasimorphisms in some braided Thompson groups and trivial second bounded cohomology in others.
The study examines spaces of non-extendable quasimorphisms for group pairs.
problem Analyzing the space of non-extendable quasimorphisms for group pairs.
method Established a five-term exact sequence of cohomology relative to bounded subcomplex.
result Proved stable commutator length equivalent to stable mixed commutator length for certain pairs.
We prove that the group of area-preserving diffeomorphisms of the 2-sphere admits a non-trivial homogeneous quasimorphism to the real numbers with the following property. Its value on any diffeomorphism supported in a sufficiently small open subset of the sphere equals to the Calabi invariant of the diffeomorphism. Thi…
Study groups admitting unbounded quasimorphisms to R with coarsely-connected quasikernel.
problem Understanding PD3 groups and their properties. method Coarse generalization of Shapiro's lemma, homological isoperimetric inequalities, and Margolis's coarse homological algebra.
result Groups admitting unbounded quasimorphisms to R with coarsely-connected quasikernel are either torus or Klein-bottle bundles over S^1, or quasiisometric to Riemannian manifolds.
We prove the vanishing of the cup product of the bounded cohomology classes associated to any two Brooks quasimorphisms on the free group. This is a consequence of the vanishing of the square of a universal class for tree automorphism groups.
Establishes a duality theorem connecting quasimorphisms and commutator lengths in group theory.
problem Connecting quasimorphisms and commutator lengths in group theory.
method Geometric interpretation and algebraic proof of (G,N)-commutator lengths. result Bi-Lipschitz equivalence of scl on [G,N] under certain conditions. We construct new families of quasimorphisms on many groups acting on CAT(0) cube complexes. These quasimorphisms have a uniformly bounded defect of 12, and they "see" all elements that act hyperbolically on the cube complex. We deduce that all such elements have stable commutator length at least 1/24. The group actions…
Uniform undistortion in cyclic subgroups of certain groups.
problem Understanding undistorted subgroups in group actions.
method Using quasimorphisms and hierarchically hyperbolic groups.
result Sharp examples of undistorted subgroups in hierarchically hyperbolic groups.
Classifies 3-manifold groups with equivariant hierarchically hyperbolic structures.
problem Classifying 3-manifold groups with equivariant hierarchically hyperbolic structures.
method Construction of suitable quasimorphisms on Seifert pieces to construct actions on quasi-lines.
result 3-manifold groups admit equivariant hierarchically hyperbolic structures.
Uniform spectral gap found for stable commutator length in hyperbolic 2-orbifolds.
problem Understanding stable commutator length in 3-manifolds.
method Explicit quasimorphisms for generic case, hyperbolic geometry for exceptional case.
result Explicit uniform gap of 1/36 for all orbifolds except a sphere with three cone points.
The study shows that several properties are not profinite invariants.
problem Determining which properties are profinite invariants.
method Combining Rips constructions and iterated group-theoretic Dehn filling on hyperbolic virtually special groups.
result Several properties (stable commutator length, quasimorphisms, property NL, property FW∞, property FA, and non-abelian free subgroups) are not profinite invariants. We give a group theoretic characterization of geodesics with superlinear divergence in the Cayley graph of a right-angled Artin group A(G) with connected defining graph G. We use this to determine when two points in an asymptotic cone of A(G) are separated by a cut-point. As an application, we show that if G does not d…
We prove that manifolds with complicated enough fundamental group admit measure-preserving homeomorphisms which have positive stable fragmentation norm with respect to balls of bounded measure.
Let Fn be the free group on n generators and Γg the surface group of genus g. We consider two particular generating sets: the set of all primitive elements in Fn and the set of all simple loops in Γg. We give a complete characterization of distorted and undistorted elements in the corresponding Aut-in…
New coefficient detects irrational rotation behavior on infinite-type surfaces.
problem Detecting irrational rotation behavior on surfaces of infinite type.
method Introducing a new quasimorphism, the Dehn twist coefficient, and proving its properties.
result The Dehn twist coefficient can have image all of R for some infinite-type surfaces.
Duality result connects bounded cohomology to relative Gromov seminorm.
problem Understanding finer invariants in geometric group theory.
method Algebraic interpretation via Hopf formula and relative Gromov seminorm.
result Bavard duality for relative Gromov seminorm.
These notes are the English version of the paper "Hyperbolicité du graphe des rayons et quasi-morphismes sur un gros groupe modulaire". The mapping class group Gamma of the complement of a Cantor set in the plane arises naturally in dynamics. We show that the ray graph, which is the analog of the complex of curves for …
Let F be the fundamental group of S, where S is a compact, connected, oriented surface with negative Euler characteristic and nonempty boundary. (1) The projective class of the chain \partial S in B_1(F) intersects the interior of a codimension one face of the unit ball in the stable commutator length pseudo-norm. (2) …
New result on symplectomorphisms on surfaces, showing vanishing cup product of fluxes.
problem Understanding commuting symplectomorphisms on surfaces.
method Refinement of non-extendability result for Py's Calabi quasimorphism.
result Vanishing cup product of fluxes for commuting symplectomorphisms.
Characterizes fractional Dehn twist coefficient and proves slice-Bennequin inequality.
problem Understanding the fractional Dehn twist coefficient and its relation to smooth slice genus.
method Characterization of FDTC and establishing the slice-Bennequin inequality.
result Affine linear lower bound for smooth slice genus in terms of FDTC.
This paper has two parts, on Baumslag-Solitar groups and on general G-trees. In the first part we establish bounds for stable commutator length (scl) in Baumslag-Solitar groups. For a certain class of elements, we further show that scl is computable and takes rational values. We also determine exactly which of these el…
We show that a finite collection of stable subgroups of a finitely generated group has finite height, finite width and bounded packing. We then use knowledge about intersections of conjugates to characterize finite families of quasimorphisms on hyperbolically embedded subgroups that can be to simultaneously extended to…
The paper examines boundedness of diffeomorphism groups of manifold pairs, focusing on the circle case.
problem Boundedness of conjugation invariant norms on diffeomorphism groups of manifold pairs.
method Analyzes the relation among norms, uses quasimorphisms and rotation angles to derive bounds.
result The diffeomorphism group D is uniformly weakly simple and bounded when dimMeq2,4. We define families of invariants for elements of the mapping class group of S, a compact orientable surface. Fix any characteristic subgroup H of pi_1(S) and restrict to J(H), any subgroup of mapping classes that induce the identity modulo H. To any unitary representation, r of pi_1(S)/H we associate a higher-order rho…
We prove that the autonomous norm on the group of Hamiltonian diffeomorphisms of the two-dimensional torus is unbounded. We provide explicit examples of Hamiltonian diffeomorphisms with arbitrarily large autonomous norm. For the proofs we construct quasimorphisms on Ham(T2) and some of them are Calabi.
Paper introduces 'zippers' for constructing universal circles.
problem Constructing universal circles for hyperbolic 3-manifolds.
method Introduces zippers to directly construct universal circles.
result New and direct way to construct universal circles.