Pseudo-Anosov subgroups in surface bundles over tori are convex cocompact.
problem Understanding the structure of pseudo-Anosov subgroups in surface bundles over tori.
method Using the Birman exact sequence to show convex cocompactness.
result Finitely generated, purely pseudo-Anosov subgroups are convex cocompact in surface bundles over tori.
We show that sufficiently irreducible Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.
We show that sufficiently irreducible totally non-symplectic Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.
Paper explains dynamics of homeomorphisms to mapping tori geometry.
problem Understanding dynamics of end-periodic homeomorphisms.
method Illustration-driven overview of recent results.
result Analogue of Brock's theorem for infinite-type surfaces.
The invariant measured foliations of a pseudo-Anosov homeomorphism induce a natural (singular) Sol structure on mapping tori of surfaces with pseudo-Anosov monodromy. We show that when the pseudo-Anosov φ:S→S has orientable foliations and does not have 1 as an eigenvalue of the induced cohomology action on…
New triangulations encode flows with vanishing polynomial.
problem Constructing veering triangulations with vanishing taut polynomial.
method Using connections between veering triangulations and pseudo-Anosov flows.
result Created arbitrarily large veering triangulations with vanishing taut polynomial.
Study bounds topological entropy of maps on surfaces with punctures based on mapping torus homology.
problem Relating topological entropy of pseudo-Anosov maps to homology of mapping tori.
method Analyzing the topological entropy of pseudo-Anosov maps on surfaces with punctures and relating it to the rank of the first homology of their mapping tori.
result Entropy of a pseudo-Anosov map is bounded by a formula involving the genus, number of punctures, and homology rank.
Lower bound on volumes of special mapping tori.
problem Calculating the minimum volume of compactified mapping tori.
method Using strongly irreducible end-periodic homeomorphisms and properties of pants graphs.
result Volume of compactified mapping tori is comparable to the translation length of the homeomorphism on pants graphs.
Authors create non-isometric 3-orbifolds with identical topology and volume.
problem Finding non-isometric hyperbolic 3-orbifolds with the same topological type and volume.
method Constructing pairs of non-isometric hyperbolic 3-orbifolds with the same topological type and volume.
result Demonstrated the existence of non-isometric hyperbolic 3-orbifolds with the same topological type and volume.
Quantum Teichmüller invariants studied for 3-manifolds.
problem Quantum invariants of 3-manifolds.
method Characterization through intertwiners and topological quantum field theories.
result Invariants of mapping tori defined by traces of intertwiners.
Study on 3-manifolds admitting pseudo-Anosov maps on subsurfaces.
problem Which 3-manifolds admit pseudo-Anosov maps on incompressible subsurfaces?
method Determine self-homeomorphisms of 3-manifolds that restrict to pseudo-Anosov maps on subsurfaces.
result Self-homeomorphisms of irreducible 3-manifolds are isotopic to partially pseudo-Anosov homeomorphisms.
The paper proves left-orderability for certain Dehn fillings of pseudo-Anosov mapping tori.
problem Left-orderability of fundamental groups in Dehn fillings of pseudo-Anosov mapping tori.
method Two approaches: one using R-covered foliations and the other using one-sided branching. result All such Dehn fillings have left-orderable fundamental groups.
This paper connects veering triangulations to pseudo-Anosov flows on 3-manifolds.
problem Understanding the dynamics of pseudo-Anosov flows on 3-manifolds.
method Building a dictionary between veering triangulations and pseudo-Anosov flows, using canonical circular orders and link spaces.
result A bijection between veering triangulations and pseudo-Anosov flows on 3-manifolds is established.
We show how to construct an ideal triangulation of a mapping torus of a pseudo-Anosov map punctured along the singular fibers. This gives rise to a new conjugacy invariant of mapping classes, and a new proof of a theorem of Farb-Leininger-Margalit. The approach in this paper is based on ideas of Hamenstadt.
We will discuss theoretical and experimental results concerning comparison of entropy of pseudo-Anosov maps and volume of their mapping tori. Recent study of Weil-Petersson geometry of the Teichmüller space tells us that they admit linear inequalities for both sides under some bounded geometry condition. We construct a…
Constructs graph manifolds with many Anosov flows.
problem Finding graph manifolds supporting multiple Anosov flows.
method Cutting geodesic flows, pulling back to finite covers, and gluing compatible pairs of flows.
result Constructs graph manifolds with at least n Anosov flows for any n.
New compactification of Teichmüller space via renormalized volume.
problem Compactify Teichmüller space with new distance.
method Horocompactification with renormalized volume.
result Translation length of pseudo-Anosov mapping classes equals hyperbolic volume of mapping tori.
In this paper, we investigate the ergodic and rigidity properties of weakly hyperbolic group actions. Motivated by classical theorems describing Anosov diffeomorphisms, we obtain two main results: First, all C^2 volume preserving weakly hyperbolic actions on closed manifolds are ergodic. This result generalizes Anosov'…
The main result of this paper is a universal finiteness theorem for the set of all small dilatation pseudo-Anosov homeomorphisms, ranging over all surfaces. More precisely, we consider pseudo-Anosovs F:S to S with |chi(S)| log(lambda(F)) bounded above by some constant, and we prove that, after puncturing the surfaces a…
We consider the pseudo-Anosov elements of the mapping class group of a surface of genus g that fix a rank k subgroup of the first homology of the surface. We show that the smallest entropy among these is comparable to (k+1)/g. This interpolates between results of Penner and of Farb and the second and third authors, who…
We first prove rigidity results for pseudo-Anosov flows in prototypes of toroidal 3-manifolds: we show that a pseudo-Anosov flow in a Seifert fibered manifold is up to finite covers topologically equivalent to a geodesic flow and we show that a pseudo-Anosov flow in a solv manifold is topologically equivalent to a susp…
We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism). This result is the main part in the proof of local smooth rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov…
Thanks to a recent result by Jean-Marc Schlenker, we establish an explicit linear inequality between the normalized entropies of pseudo-Anosov automorphisms and the hyperbolic volumes of their mapping tori. As its corollaries, we give an improved lower bound for values of entropies of pseudo-Anosovs on a surface with f…
Study partially hyperbolic dynamics on 3-manifolds with quasi-isometric center.
problem Characterize dynamics on 3-manifolds with specific center properties.
method Analyzes partially hyperbolic diffeomorphisms with quasi-isometric center under non-wandering conditions.
result Volume-preserving diffeomorphisms are ergodic without su-tori, confirming a conjecture. The study connects translation length to manifold structure, proving bounds and identifying finite types.
problem Understanding the structure of 3-manifolds via pseudo-Anosov mapping classes and their translation lengths.
method Proving bounds on translation lengths and constructing specific 3-manifolds from mapping tori.
result Finite set of 3-manifolds can be derived from pseudo-Anosov mapping classes with bounded translation length.
The study of pseudo-Anosovs through mapping torus geometry.
problem Understanding pseudo-Anosov diffeomorphisms via mapping torus geometry.
method Analyzing simplicial complexes and mapping tori to relate dynamics and geometry.
result Relating pseudo-Anosov actions to fixed points in isotopy classes.
Study mapping class groups on invariant incompressible surfaces in reducible 3-manifolds.
problem Understanding the effect of mapping class groups on incompressible surfaces in reducible 3-manifolds.
method Analyzing the mapping class group of reducible 3-manifolds and their incompressible surfaces.
result Characterize reducible 3-manifolds that admit Anosov tori.
We advocate the use of cluster algebras and their y-variables in the study of hyperbolic 3-manifolds. We study hyperbolic structures on the mapping tori of pseudo-Anosov mapping classes of punctured surfaces, and show that cluster y-variables naturally give the solutions of the edge-gluing conditions of ideal tetrahedr…
Non-coherence proven for certain groups with specific mapping tori.
problem Proving non-coherence for groups with specific mapping tori.
method Analyzing the structure of groups as extensions and properties of mapping tori.
result Groups with certain mapping tori are not coherent.
New 4D homeomorphism shows surprising similarities to 2D.
problem Finding a 4D pseudo-Anosov homeomorphism.
method Building a specific example of a 5-manifold fibered over the circle.
result Compact 4-manifold with unique immersed and embedded surfaces.
Study of Witten-Reshetikhin-Turaev invariants for mapping tori.
problem Asymptotic expansion of Witten-Reshetikhin-Turaev invariants of mapping tori.
method Geometric quantization of moduli spaces of flat connections, Picard-Lefschetz theory for Laplace integrals.
result Full asymptotic expansion provided for pseudo-Anosov mapping classes on a punctured torus.
Study of flows on 7D manifolds with holomorphic properties.
problem Characterizing transversely holomorphic partially hyperbolic flows.
method Analyzing flows with biholomorphic holonomy pseudo-group, proving properties under integrable subcenter distribution.
result Flow projects to a transversely holomorphic Anosov flow on a 5D manifold.
Study integral simplicial volume of cyclic covers of torus bundles.
problem Asymptotic behavior of integral simplicial volume of cyclic covers.
method Integral filling volume invariant for monodromy, bounds established.
result Established both lower and upper bounds for the limit of integral simplicial volume.
For each surface S of genus g>2 we construct pairs of conjugate pseudo-Anosov maps, φ1 and φ2, and two non-equivalent covers pi:S~⟶S, i=1,2, so that the lift of φ1 to S~ with respect to p1 coincides with that of φ2 with respect to p2. The…
Geometric methods show cosets of mapping class groups for 3-manifolds.
problem Understanding quantum representations of surface mapping class groups.
method Geometric methods and pseudo-Anosov monodromies.
result Cosets of abelian and free subgroups found in mapping class groups.
Thanks to a theorem of Brock on comparison of Weil-Petersson translation distances and hyperbolic volumes of mapping tori for pseudo-Anosovs, we prove that the entropy of a surface automorphism in general has linear bounds in terms of Gromov norm of its mapping torus from below and in bounded geometry case from above. …
The paper calculates intertwiners for a torus and proves a conjecture about their limits.
problem Relating quantum invariants to hyperbolic geometry using intertwiners.
method Explicit calculation of intertwiners for a closed torus and periodic diffeomorphisms.
result The limit superior of the trace of intertwiners is zero for certain diffeomorphisms.
Recently, Ian Agol introduced a class of "veering" ideal triangulations for mapping tori of pseudo-Anosov homeomorphisms of surfaces punctured along the singular points. These triangulations have very special combinatorial properties, and Agol asked if these are "geometric", i.e. realised in the complete hyperbolic met…
This research studies end-periodic mapping tori and their hyperbolic structures.
problem Understanding the geometry and dynamics of end-periodic mapping tori.
method Analyzes invariant laminations and hyperbolic structures of mapping tori.
result Establishes a relationship between the geodesic length of boundary components and the infimum of geodesic lengths in hyperbolic structures.
New method constructs actions on R capturing foliations, proving left-orderability.
problem Left-orderability of 3-manifold fundamental groups.
method Constructs actions of π1(M) on R, capturing foliations.
result Proves left-orderability of fundamental groups of 3-manifolds.
The study shows how to create co-orientable taut foliations in specific Dehn fillings.
problem Creating co-orientable taut foliations in Dehn fillings of pseudo-Anosov mapping tori.
method Analyzing the stable foliation and using Dehn filling with specific multislopes.
result Dehn fillings of pseudo-Anosov mapping tori with co-orientation-reversing monodromy admit co-orientable taut foliations.
Study minimal translation lengths on curve complexes, providing bounds and constructing examples.
problem Understanding minimal translation lengths on curve complexes and their relation to Teichmüller spaces.
method Analysed the curve complex analog of Teichmüller spaces, providing lower and upper bounds and constructing specific examples.
result Lower bound on minimal asymptotic translation length on curve complexes interpolates known results on Teichmüller spaces.
Study Anosov representations of reducible suspensions of hyperbolic groups.
problem Characterize dynamical properties of reducible suspensions of Anosov representations.
method Analyzing linear representations of non-elementary hyperbolic groups, focusing on weak unipotent actions on subspaces.
result Characterize when reducible suspensions are discrete and faithful, quasi-isometrically embedded, and Anosov.
The paper explores Anosov flows on torus bundles and provides new insights.
problem Investigating Anosov flows on affine torus bundles and their properties.
method Surgery type construction of Anosov flows and dichotomy result for fiberwise Anosov flows.
result New insights into Anosov flows, including non-transitive flows in odd dimensions.
Constructs Anosov flows in hyperbolic 3-manifolds, disproving a conjecture.
problem Proving existence of infinitely many distinct Anosov flows in certain 4-manifolds.
method Using Cannon-Thurston maps and pseudo-Anosov quasigeodesic flows.
result Infinitely many distinct Anosov flows in some 4-manifolds.
Let Mφ be a surface bundle over a circle with monodromy φ:S→S. We study deformations of certain reducible representations of π1(Mφ) into SL(n,C), obtained by composing a reducible representation into SL(2,C) with the irreducible representation $\text{SL}(2,\mathb…
Develops theory of Anosov representations for Fuchsian groups, showing stability and analytical properties.
problem Understanding geometrically finite Fuchsian groups and their representations.
method Theory of Anosov representations, type-preserving deformations, limit maps, relative Anosov and dominated representations.
result Cusped Hitchin representations are Borel Anosov, stable under deformations, and limit maps vary analytically.
Bi-contact surgery operations can be applied to Anosov flows.
problem Characterizing Anosov flows and their properties.
method Metric and contact geometric characterizations, Liouville geometry, Reeb dynamics.
result Bi-contact surgery operations can be applied to Anosov flows.