Proves purely pseudo-Anosov groups are convex cocompact.
problem Understanding the properties of pseudo-Anosov groups in mapping class groups.
method Proves convex cocompactness using group theory.
result Groups as described are convex cocompact in mapping class groups.
Proves certain subgroups of genus 2 handlebody group are convex cocompact.
problem Characterizing subgroups of genus 2 handlebody group.
method Proving convex cocompactness of purely pseudo-Anosov subgroups.
result Finitely generated, purely pseudo-Anosov subgroups are convex cocompact.
Generic pseudo-Anosov mapping classes in mapping class groups.
problem Understanding the prevalence of pseudo-Anosov mapping classes.
method Proving genericity with respect to specific notions of genericity.
result Pseudo-Anosov mapping classes are generic in mapping class groups.
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.
Loxodromic elements are pseudo-Anosov on specific graphs.
problem Characterizing loxodromic elements in specific groups.
method Analyzing subgroups acting on multiarc and curve graphs, and the handlebody group on disk graphs.
result Loxodromic elements are pseudo-Anosov on witness graphs.
The study shows pseudo-Anosovs are common in mapping class groups.
problem Counting pseudo-Anosovs in mapping class groups.
method Using weakly contracting isometries and Morse elements.
result Pseudo-Anosovs are generic in mapping class groups.
New findings on generating mapping class groups using pseudo-Anosov elements.
problem Generating mapping class groups using specific types of elements.
method Proving the generation of mapping class groups by pseudo-Anosov elements and conjugate reducible but not periodic elements.
result The mapping class group can be generated by two conjugate pseudo-Anosov elements with arbitrarily large dilatations for surfaces of genus greater than or equal to nine.
Study non-transitive pseudo-Anosov flows using group actions.
problem Characterize pseudo-Anosov flows in 3-manifolds.
method Extend pseudo-Anosov action to non-transitive flows, use group actions on orbit spaces and boundary at infinity.
result Pseudo-Anosov flows in 3-manifolds are determined by their group actions on boundary at infinity.
Exponential proportion of pseudo-Anosovs in mapping class groups.
problem Proportion of non-pseudo-Anosov mapping classes in a ball of radius R.
method Using word metric and finite generating sets, we show that the proportion decreases exponentially.
result Proportion of non-pseudo-Anosov mapping classes decreases exponentially.
Minimal pseudo-Anosov dilatations in hyperelliptic handlebody groups are asymptotically 1/g.
problem Understanding the asymptotic behavior of minimal pseudo-Anosov dilatations in hyperelliptic handlebody groups.
method Analyzing the hyperelliptic handlebody group and comparing it to the mapping class group.
result The logarithm of the minimal dilatation is comparable to 1/g for hyperelliptic handlebody groups of genus g.
Study shows a significant portion of mapping class group elements are pseudo-Anosov.
problem Determining the proportion of pseudo-Anosov elements in mapping class groups.
method Examined the Cayley graph of mapping class groups with respect to various generating sets, analyzing the proportion of pseudo-Anosov elements as the radius of the ball increases.
result The proportion of pseudo-Anosov elements is bounded away from zero, contradicting the well-known conjecture.
New formula connects braid dilatation with fixed points.
problem Understanding the relationship between braid dilatation and fixed points.
method Using Jiang and Zheng's braid group representations.
result Formula for dilatation via fixed points.
Study shows certain subgroups of fibered 3-manifolds are convex cocompact.
problem Understanding subgroups of fibered 3-manifolds in mapping class groups.
method Used the Birman exact sequence to show convex cocompactness.
result Finitely generated pseudo-Anosov subgroups are convex cocompact.
New subgroup behavior in genus-2 mapping class group identified.
problem Understanding subgroups in genus-2 mapping class group.
method Analyzing purely pseudo-Anosov subgroups as convex cocompact.
result Finitely-generated, purely pseudo-Anosov subgroups are convex cocompact.
Study of pseudo-Anosov actions on SU(2)-character variety for genus 2 surfaces.
problem Characterizing pseudo-Anosov actions on SU(2)-character variety. method Analysis of mapping class group subgroups and their pseudo-Anosov elements.
result Existence of a subgroup containing infinitely many pseudo-Anosov elements with invariant rational functions.
Random walks on mapping classes lead to pseudo-Anosov maps in the principal stratum.
problem Understanding the behavior of random walks on mapping class groups.
method Analyzing random walks with specific subgroup properties and pseudo-Anosov maps.
result Almost every infinite sample path of random walks contains pseudo-Anosov maps with invariant geodesics in the principal stratum.
The paper finds pseudo-Anosov-like maps on an infinite ladder surface.
problem Exploring dynamics on infinite surfaces.
method Lifts Penner-type pseudo-Anosov maps from a closed surface to an infinite ladder surface.
result Existence and properties of pseudo-Anosov-like maps on the infinite ladder surface.
The study finds that many mapping class groups have normal generators.
problem Finding normal generators in mapping class groups.
method Provided a criterion for normal closure and applied it to show normal generators for specific classes of mapping classes.
result Many nontrivial periodic mapping classes and pseudo-Anosov mapping classes are normal generators.
The abstract discusses various open problems in mapping class groups.
problem Open problems in mapping class groups of surfaces.
method Discussion of open problems.
result Various open problems in mapping class groups.
We show that, for any (symmetric) finite generating set of the Torelli group of a closed surface, the probability that a random word is not pseudo-Anosov decays exponentially in terms of the length of the word.
New growth rate for pseudo-Anosov conjugacy classes in Teichmüller space.
problem Understanding growth rates of conjugacy classes in Teichmüller space.
method Analyzing pseudo-Anosov mapping classes and their conjugacy classes in Teichmüller space.
result The number of lattice points of pseudo-Anosov conjugacy classes intersecting a closed ball of radius R is coarsely asymptotic to \(e^{\frac{h}{2}R}\).
Study reveals uniform difference in stretch factors between genus two handlebody group and outer automorphism group.
problem Analyzing the relationship between stretch factors in genus two handlebody group and outer automorphism group.
method Examined natural homomorphism from genus g handlebody group to outer automorphism group of free groups, focusing on pseudo-Anosov mapping classes and their stretch factors.
result Minimum stretch factor in genus two handlebody group is less than ten times the stretch factor of fully irreducible outer automorphism.
The study finds all trace field degrees for Torelli group mappings.
problem Identifying all possible trace field degrees for Torelli group mappings.
method Using Thurston-Veech construction of pseudo-Anosov maps, and providing examples of stretch factors with specific algebraic degrees.
result All integers 1≤d≤3g−3 are trace field degrees for g≥2. The paper proves the maximum algebraic degree of pseudo-Anosov stretch factors and finds the possible degrees of trace fields.
problem Justify Thurston's remark about the maximum algebraic degree of pseudo-Anosov stretch factors.
method Novel asymptotic irreducibility criterion for polynomials and a construction algorithm.
result Determine the set of possible algebraic degrees of pseudo-Anosov stretch factors on almost all finite type surfaces.
Metric WPD for pseudo-Anosov maps shows many unbounded quasi-morphisms.
problem Understanding quasi-morphisms on pseudo-Anosov maps.
method Metric weak proper discontinuity (WPD) for pseudo-Anosov maps.
result Existence of many unbounded quasi-morphisms on homeomorphisms with large fixed sets.
Quadratic-time algorithm computes stretch factors and foliations for pseudo-Anosov mapping classes.
problem Computing stretch factors and foliations for pseudo-Anosov mapping classes efficiently.
method Quadratic-time algorithm using input word and length as complexity measure.
result First algorithm to compute stretch factors and foliations in sub-exponential time.
We compute the growth series and the growth functions of reducible and pseudo-Anosov elements of the pure mapping class group of the sphere with four holes with respect to a certain generating set. We prove that the ratio of the number of pseudo-Anosov elements to that of all elements in a ball with center at the ident…
The study examines translation lengths of pseudo-Anosov maps on curve graphs.
problem Estimating translation lengths of pseudo-Anosov maps on curve graphs.
method Analyzing geodesic axes and powers of Dehn twists.
result Determining minimal translation lengths and optimizing map ratios.
We establish a criterion for certain mapping classes of a surface homeomorphisms to be pseudo-Anosov in terms of the geometry of hyperbolic 3-manifolds and Gromov-hyperbolic surface group extensions. Specifically, any element of the fundamental group of a surface S gives rise to a mapping class on the punctured surface…
We exhibit a pseudo-Anosov homeomorphism of a surface S which acts trivially on the first homology group of S and whose flux is non zero
Study Veech groups in fibered 3-manifolds, proving no parabolics for fibers.
problem Characterizing Veech groups in fibered 3-manifolds.
method Analyzing pseudo-Anosov monodromies and foliations, proving properties of Veech groups.
result Veech groups in fibers typically contain no parabolic elements.
Given any generating set of any pseudo-Anosov-containing subgroup of the mapping class group of a surface, we construct a pseudo-Anosov with word length bounded by a constant depending only on the surface. More generally, in any subgroup G we find an element f with the property that the minimal subsurface supporting a …
We prove that generic elements of braid groups are pseudo-Anosov, in the following sense: in the Cayley graph of the braid group with n ≥ 3 strands, with respect to Garside's generating set, we prove that the proportion of pseudo-Anosov braids in the ball of radius l tends to 1 exponentially quickly as l tends to i…
The symplectic representation of mapping classes is not surjective for certain types of mapping classes.
problem The surjectivity of the symplectic representation of mapping classes, particularly pseudo-Anosov ones, is not always preserved.
method Explicit construction of symplectic matrices with a bi-Perron leading eigenvalue that cannot be represented by orientable pseudo-Anosov mapping classes.
result The symplectic representation of orientable pseudo-Anosov mapping classes is not surjective.
Answering a question of Farb-Leininger-Margalit, we give explicit lower bounds for the dilatations of pseudo-Anosov mapping classes lying in the kth term of the Johnson filtration of the mapping class group.
Let Gamma_k be the lower central series of a surface group Gamma of a compact surface S with one boundary component. A simple question to ponder is whether a mapping class of S can be determined to be pseudo-Anosov given only the data of its action on Gamma/Gamma_k for some k. In this paper, to each mapping class f whi…
The study examines pseudo-Anosov maps on curve complexes and their asymptotic translation lengths.
problem Analyzing the behavior of pseudo-Anosov maps on curve complexes.
method Using sequences of fibers and monodromies in the fibered cone, the asymptotic translation length of pseudo-Anosov maps is studied.
result The asymptotic translation length of pseudo-Anosov maps on curve complexes behaves asymptotically like 1/∣χ(Rn)∣2. We show that the kernel of Burau(4)⊗Zp, the reduced Burau representation with coefficients in Zp of the 4-braid group B4, consists only of pseudo-Anosov braids.
Geodesics in mapping class group show unexpected behaviors.
problem Understanding geodesics and quasi-geodesics in mapping class groups.
method Constructing explicit examples and analyzing projections.
result Geodesics in mapping class group can have unexpected properties, e.g., not being quasi-geodesics.
New examples of surface bundles found over surfaces.
problem Finding compact atoroidal surface bundles.
method Type-preserving homomorphism from knot complement to mapping class group.
result Infinitely many commensurability classes of surface subgroups.
New insights into the geometry of flows on 3-manifolds.
problem Understanding the geometry of flows on 3-manifolds.
method Analyzing the action of pseudo-Anosov flows on Gromov-hyperbolic spaces.
result Genericity of non-periodic elements in the fundamental group.
We prove that, in the l-ball of the Cayley graph of the braid group with n⩾3 strands, the proportion of rigid pseudo-Anosov braids is bounded below independently of l by a positive value.
Simplified approach to pseudo-Anosov flows on 3-manifolds.
problem Complexity in understanding pseudo-Anosov flows on 3-manifolds.
method Streamlined framework called Anosov-like group actions.
result Unified and simplified presentation of pseudo-Anosov flows.
Galois conjugates of pseudo-Anosov stretch factors are dense in the complex plane.
problem Understanding the distribution of Galois conjugates of pseudo-Anosov stretch factors.
method Using Penner's construction of pseudo-Anosov mapping classes, we show density in the complex plane.
result Galois conjugates of stretch factors arising from Penner's construction are dense in the complex plane.
The paper shows contracting elements are exponentially generic in various groups.
problem Establishing genericity of contracting elements in different groups.
method Statistically convex-cocompact actions and properties of specific elements in groups.
result Exponential genericity of contracting elements in various groups.
We prove that the set of non-pseudo-Anosov elements in the Torelli group is exponentially small.
This paper contains two main results. The first is the existence of an equivariant Weil-Petersson geodesic in Teichmueller space for any choice of pseudo-Anosov mapping class. As a consequence one obtains a classification of the elements of the mapping class group as Weil-Petersson isometries which is parallel to the T…
Study Goeritz groups of link decompositions, focusing on their asymptotic behavior.
problem Understanding the asymptotic behavior of Goeritz groups for link decompositions.
method Defined Goeritz groups for link decompositions, analyzed their properties, and discussed their asymptotic behavior.
result Discussed the asymptotic behavior of minimal pseudo-Anosov entropies and related it to Goeritz groups of Heegaard splittings.