pseudo-Anosov Research Articles

Weil–Petersson translation length and manifolds with many fibered fillings

We prove that any mapping torus of a pseudo-Anosov mapping class with bounded normalized Weil–Petersson translation length contains a finite set of transverse and level closed curves with the property that drilling out this set of curves results in one of a finite number of cusped hyperbolic 3–manifolds. Moreover, the set of resulting manifolds depends only on the bound for normalized translation length. This gives a Weil–Petersson analog of a theorem of Farb–Leininger–Margalit [9] about Teichmüller translation length. We also prove a complementary result that explains the necessity of removing level curves by producing new estimates for the Weil–Petersson translation length of compositions of pseudo-Anosov mapping classes and arbitrary powers of a Dehn twist.

Random veering triangulations are not geometric

Every pseudo-Anosov mapping class \phi defines an associated veering triangulation \tau_\phi of a punctured mapping torus. We show that generically, \tau_\phi is not geometric. Here, the word "generic" can be taken either with respect to random walks in mapping class groups or with respect to counting geodesics in moduli space. Tools in the proof include Teichmüller theory, the Ending Lamination Theorem, study of the Thurston norm, and rigorous computation.

Pseudo-Anosov maps with small stretch factors on punctured surfaces

Consider the problem of estimating the minimum entropy of pseudo-Anosov maps on a surface of genus $g$ with $n$ punctures. We determine the behaviour of this minimum number for a certain large subset of the $(g,n)$ plane, up to a multiplicative constant. In particular it has been shown that for fixed $n$, this minimum value behaves as $\frac{1}{g}$, proving what Penner speculated in 1991.

McShane-Type Identities for Quasifuchsian Representations of Nonorientable Surfaces

Abstract We show that Norbury’s McShane identity for nonorientable cusped hyperbolic surfaces $N$ generalizes to quasifuchsian representations of $\pi _1(N)$ as well as pseudo-Anosov mapping Klein bottles with singular fibers given by $N$.

Finiteness of mapping class groups: locally large strongly irreducible Heegaard splittings

By Namazi and Johnson’s results, for any distance at least 4 Heegaard splitting, its mapping class group is finite. In contrast, Namazi showed that for a weakly reducible Heegaard splitting, its mapping class group is infinite; Long constructed an irreducible Heegaard splitting where its mapping class group contains a pseudo anosov map. Thus it is interesting to know that for a strongly irreducible but distance at most 3 Heegaard splitting, when its mapping class group is finite. In [19], Qiu and Zou introduced the definition of a locally large distance 2 Heegaard splitting. Extending their definition into a locally large strongly irreducible Heegaard splitting, we proved that its mapping class group is finite. Moreover, for a toroidal 3-manifold which admits a locally large distance 2 Heegaard splitting in [19], we prove that its mapping class group is finite.

Minimal pseudo-Anosov stretch factors on nonoriented surfaces

We determine the smallest stretch factor among pseudo-Anosov maps with an orientable invariant foliation on the closed nonorientable surfaces of genus 4, 5, 6, 7, 8, 10, 12, 14, 16, 18 and 20. We also determine the smallest stretch factor of an orientation-reversing pseudo-Anosov map with orientable invariant foliations on the closed orientable surfaces of genus 1, 3, 5, 7, 9 and 11. As a byproduct, we obtain that the stretch factor of a pseudo-Anosov map on a nonorientable surface or an orientation-reversing pseudo-Anosov map on an orientable surface does not have Galois conjugates on the unit circle. This shows that the techniques that were used to disprove Penner's conjecture on orientable surfaces are ineffective in the nonorientable cases.

Ruelle spectrum of linear pseudo-Anosov maps

The Ruelle resonances of a dynamical system are spectral data describing the precise asymptotics of correlations. We classify them completely for a class of chaotic two-dimensional maps, the linear pseudo-Anosov maps, in terms of the action of the map on cohomology. As applications, we obtain a full description of the distributions which are invariant under the linear flow in the stable direction of such a linear pseudo-Anosov map, and we solve the cohomological equation for this flow.

Pseudo–Anosov mapping classes from pure mapping classes

We study types of mapping classes which arise as a product of a given mapping class and powers of certain pure mapping classes. We derive an explicit constant depending only on a surface such that almost all above pure mapping classes give rise to pseudo–Anosov type whenever their powers are larger than the constant. Throughout this study, we also give various ways of constructing pseudo–Anosov mapping classes. Furthermore, we are able to capture the stable lengths of all pseudo–Anosov mapping classes constructed by our methods.

Recurrence of quadratic differentials for harmonic measure

AbstractWe consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum of quadratic differentials. We show that a Teichmüller geodesic typical with respect to the harmonic measure for such random walks, is recurrent to the thick part of the principal stratum. As a consequence, the vertical foliation of such a random Teichmüller geodesic has no saddle connections.

Minimal Penner dilatations on nonorientable surfaces

For any nonorientable closed surface, we determine the minimal dilatation among pseudo-Anosov mapping classes arising from Penner’s construction. We deduce that the sequence of minimal Penner dilatations has exactly two accumulation points, in contrast to the case of orientable surfaces where there is only one accumulation point. One of our key techniques is representing pseudo-Anosov dilatations as roots of Alexander polynomials of fibered links and comparing dilatations using the skein relation for Alexander polynomials.

Quantum representations and monodromies of fibered links

Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level r). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants TVr of hyperbolic 3-manifolds. We show that if the r-growth of |TVr(M)| for a hyperbolic 3-manifold M that fibers over the circle is exponential, then the monodromy of the fibration of M satisfies the AMU conjecture. Building on earlier work [11] we give broad constructions of (oriented) hyperbolic fibered links, of arbitrarily high genus, whose SO(3)-Turaev-Viro invariants have exponential r-growth. As a result, for any g>n⩾2, we obtain infinite families of non-conjugate pseudo-Anosov mapping classes, acting on surfaces of genus g and n boundary components, that satisfy the AMU conjecture.We also discuss integrality properties of the traces of quantum representations and we answer a question of Chen and Yang about Turaev-Viro invariants of torus links.

Classification of one-dimensional attractors of diffeomorphisms of surfaces by means of pseudo-anosov homeomorphisms

In the present paper axiom diffeomorphisms of closed 2-manifolds of genus whose nonwandering set contains perfect spaciously situated one-dimensional attractor are considered. It is shown that such diffeomorphisms are topologically semiconjugate to pseudo-Anosov homeomorphism with the same induced automorphism of fundamental group. The main result of the paper is the following. Two diffeomorphisms from the given class are topologically conjugate on attractors if and only if corresponding pseudo-Anosov homeomorphisms are topologically conjugate by means of homeomorphism that maps a certain subset of one pseudo-Anosov map onto the certain subset of the other pseudo-Anosov map.

Small asymptotic translation lengths of pseudo-Anosov maps on the curve complex

Let M be a hyperbolic fibered 3-manifold whose first Betti number is greater than 1 and let S be a fiber with pseudo-Anosov monodromy \psi . We show that there exists a sequence (R_n, \psi_n) of fibers and monodromies contained in the fibered cone of (S,\psi) such that the asymptotic translation length of \psi_n on the curve complex \mathcal C(R_n) behaves asymptotically like 1=j .Rn/j2. As applications, we can reprove the previous result by Gadre–Tsai that the minimal asymptotic translation length of a closed surface of genus g asymptotically behaves like 1/|\chi(R_n)|^2 . We also show that this holds for the cases of hyperelliptic mapping class group and hyperelliptic handlebody group.

Asymptotic expansions of the Witten–Reshetikhin–Turaev invariants of mapping tori I

In this paper we engage in a general study of the asymptotic expansion of the Witten–Reshetikhin–Turaev invariants of mapping tori of surface mapping class group elements. We use the geometric construction of the Witten–Reshetikhin–Turaev topological quantum field theory via the geometric quantization of moduli spaces of flat connections on surfaces. We identify assumptions on the mapping class group elements that allow us to provide a full asymptotic expansion. In particular, we show that our results apply to all pseudo-Anosov mapping classes on a punctured torus and show by example that our assumptions on the mapping class group elements are strictly weaker than hitherto successfully considered assumptions in this context. The proof of our main theorem relies on our new results regarding asymptotic expansions of oscillatory integrals, which allows us to go significantly beyond the standard transversely cut out assumption on the fixed point set. This makes use of the Picard–Lefschetz theory for Laplace integrals.

Degrees of Stretch Factors and Counterexamples to Farb's Conjecture

Franks and Rykken proved that any pseudo-Anosov map with quadratic stretch factor and orientable invariant foliations, can be obtained by lifting a torus map by a branched or unbranched covering map. Farb conjectured that this generalizes to higher degree stretch factors. We construct counterxamples to this conjecture. The proof relies on studying the Perron-Frobenius degrees of our constructed stretch factors and a careful analysis of their Galois conjugates.

Normalized entropy versus volume for pseudo-Anosovs

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 fixed topology, and a proof of a slightly weaker version of the result by Farb, Leininger and Margalit first, and by Agol later, on finiteness of cusped manifolds generating surface automorphisms with small normalized entropies. Also, we present an analogous linear inequality between the Weil-Petersson translation distance of a pseudo-Anosov map (normalized by multiplying the square root of the area of a surface) and the volume of its mapping torus, which leads to a better bound.

The Mathematics of Taffy Pullers

We describe a number of devices for pulling candy, called taffy pullers,that are related to pseudo-Anosov maps of punctured spheres. Though the mathematical connection has long been known for the two most common taffy puller models, we unearth a rich variety of early designs from the patent literature, and introduce a new one.

Lifts of pseudo-Anosov homeomorphisms of nonorientable surfaces have vanishing SAF invariant

We show that any pseudo-Anosov map that is a lift of pseudo-Anosov homeomorphism of a nonorientable surface has vanishing SAF invariant. We also provide a criterion to certify that a pseudo-Anosov map is not such a lift.

Rel leaves of the Arnoux–Yoccoz surfaces

We analyze the rel leaves of the Arnoux-Yoccoz translation surfaces. We show that for any genus $g \geq 3$, the leaf is dense in the connected component of the stratum $H(g -1 , g -1)$ to which it belongs, and the one-sided imaginary-rel trajectory of the surface is divergent. For one surface on this trajectory, namely the Arnoux-Yoccoz surface itself, the horizontal foliation is invariant under a pseudo-Anosov map (and in particular is uniquely ergodic), but for all other surfaces, the horizontal foliation is completely periodic. The appendix proves a field theoretic result needed for denseness of the leaf: for any $n \geq 3$, the field extension of the rationals obtained by adjoining a root of $X^n-X^{n-1}-\ldots-X-1$ has no totally real subfields other than the rationals.

Is a typical bi-Perron algebraic unit a pseudo-Anosov dilatation?

In this note, we deduce a partial answer to the question in the title. In particular, we show that asymptotically almost all bi-Perron algebraic units whose characteristic polynomial has degree at most $2n$ do not correspond to dilatations of pseudo-Anosov maps on a closed orientable surface of genus $n$ for $n\geq 10$. As an application of the argument, we also obtain a statement on the number of closed geodesics of the same length in the moduli space of area-one abelian differentials for low-genus cases.