Howard Masur

How to use this course

Howard Masur has spent five decades on one object, the Teichmüller space of a surface, and has approached it from several sides. From complex analysis he took quadratic differentials. From dynamics he took ergodic theory, and applied it to interval exchanges and billiards. From geometric group theory he took coarse hyperbolicity, and with Yair Minsky applied it to the complex of curves. Each of these threads is now a field with its own community, and Masur’s papers sit near the root of each.

Each technical lesson opens with an italic paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow the order of his work: Teichmüller geodesics (1974–1979), dynamics of flat surfaces (1982–2003), and coarse geometry (from 1995). Readers interested only in the career can read Parts I and III and the opening paragraphs of Part II. The published record of Masur’s life outside mathematics is short, and this course keeps to it.

Timeline

The inheritance of Teichmüller

Masur’s subject began with Oswald Teichmüller in the late 1930s. Bernhard Riemann had counted that a compact Riemann surface of genus \(g\ge2\) depends on \(3g-3\) complex parameters. Teichmüller made this precise by measuring how far one surface is from another: the distance is half the logarithm of the least distortion of a quasiconformal map between them. He found that the extremal maps are described by holomorphic quadratic differentials, and that the space of marked surfaces is a ball of dimension \(6g-6\).

Teichmüller’s proofs were incomplete, and he died in 1943. Lars Ahlfors and Lipman Bers rebuilt the theory in the 1950s and 1960s, with full proofs, a complex structure on Teichmüller space, and the Bers embedding. By 1970 Teichmüller space was a well-established object of complex analysis. Its geometry, though, was poorly understood. Was it negatively curved, like the hyperbolic plane? How did its geodesics behave at infinity? These were the questions Masur took up.

His doctoral advisor, Albert Marden, was a student of Ahlfors at Harvard and worked on Kleinian groups and hyperbolic 3-manifolds at the University of Minnesota. Through Marden, Masur belongs to the Ahlfors school of geometric function theory.

Minnesota and the thesis, 1974

Masur received his PhD from the University of Minnesota in Minneapolis in 1974. His thesis, The Curvature of Teichmüller Space, took up the question of whether the Teichmüller metric is negatively curved. The relevant notion, due to Herbert Busemann, asks whether the distance between two geodesics is a convex function, so that geodesics from a common point spread apart. Teichmüller space looks negatively curved in many respects: it is contractible, uniquely geodesic, and its mapping class group acts much like a lattice in a Lie group. Masur showed that the answer is no.

The decisive argument, published in the Annals of Mathematics in 1975, was to study a special class of Teichmüller geodesics: those defined by Jenkins–Strebel differentials, whose trajectories are all closed and fill a family of cylinders. Along such a geodesic the surface is stretched in one direction and compressed in the other, and the cylinders become thin. Masur showed that two different geodesic rays of this type can remain a bounded distance apart for all time. In a space of negative curvature in Busemann’s sense, distinct rays from a common point must diverge, so the Teichmüller metric does not have that property (Lesson 04).

The thesis set the pattern for his career: a concrete question about Teichmüller space, answered by close analysis of the flat geometry of quadratic differentials.

The first decade, 1975–1986

The late 1970s were a turning point for the subject. William Thurston had introduced measured foliations and his compactification of Teichmüller space, and had classified surface homeomorphisms into periodic, reducible and pseudo-Anosov types. His methods were topological and hyperbolic-geometric. Masur’s work in this period linked them to the complex-analytic theory.

In 1976 he studied the Weil–Petersson metric, the other natural metric on Teichmüller space, and showed how it extends to the boundary where curves are pinched. In 1979, with John Hubbard, he proved that every measured foliation is realized by a unique quadratic differential on each Riemann surface (Lesson 05). This gave Thurston’s space of measured foliations a complex-analytic model and became a standard tool.

He received a Sloan Research Fellowship in 1980. In 1982 the Annals published back to back his paper and one by William Veech that independently proved a conjecture of Michael Keane: almost every interval exchange transformation is uniquely ergodic (Lesson 06). Both proofs went through the dynamics of the Teichmüller geodesic flow, and in the process both established that the flow preserves a finite measure on each stratum of quadratic differentials. Four years later, with Steven Kerckhoff and John Smillie, he applied the same ideas to billiards in polygons (Lesson 07). By the mid-1980s he had become, with Veech, one of the founders of Teichmüller dynamics.

Teichmüller space and its metric

For the general reader. A surface with a hole, like a doughnut, can be given many different shapes. Teichmüller space is the collection of all such shapes, with a way of measuring how different two shapes are: the least amount of stretching needed to turn one into the other. Masur’s thesis showed that this space, which resembles a negatively curved space in many ways, is not negatively curved in the classical sense.

Definitions

Let \(S\) be a closed oriented surface of genus \(g\ge2\). A point of the Teichmüller space \(\mathcal T(S)\) is a Riemann surface \(X\) together with a marking, a homotopy class of homeomorphisms \(S\to X\). The Teichmüller distance is

where \(f:X\to Y\) ranges over quasiconformal maps in the marked homotopy class and \(K(f)\ge1\) is the maximal dilatation. The mapping class group \(\mathrm{Mod}(S)\) acts by changing the marking, isometrically and properly discontinuously; the quotient is the moduli space \(\mathcal M(S)\).

A holomorphic quadratic differential on \(X\) is given locally by \(q=\varphi(z)\,dz^2\) with \(\varphi\) holomorphic. They form a complex vector space \(Q(X)\) of dimension \(3g-3\) (Riemann–Roch). Away from its \(4g-4\) zeros, counted with multiplicity, \(q\) has a natural coordinate \(\zeta=\int\sqrt{\varphi}\,dz\), unique up to \(\zeta\mapsto\pm\zeta+c\). The metric \(|d\zeta|\) is flat, with cone points of angle \((k+2)\pi\) at zeros of order \(k\). Horizontal lines \(\operatorname{Im}\zeta=\text{const}\) form the horizontal foliation, and vertical lines the vertical foliation.

The Teichmüller geodesic through \(X\) in direction \(q\) is therefore \(t\mapsto X_t\), where \(X_t\) is obtained by stretching the flat structure of \(q\) horizontally by \(e^{t}\) and compressing vertically by \(e^{-t}\). This is the Teichmüller geodesic flow; on unit-area differentials it is the action of \(g_t=\mathrm{diag}(e^t,e^{-t})\subset\mathrm{SL}(2,\mathbb R)\).

Not negatively curved

The idea: when all vertical trajectories are closed, the surface is a union of flat cylinders. Along the geodesic each cylinder becomes long and thin, and in the limit only the ratios of circumferences matter. Two rays whose cylinders differ by twists, or by heights that the stretching renders irrelevant, approach each other. Masur’s 1982 work on the Thurston boundary complements this: when the vertical foliation is uniquely ergodic, the ray converges to a single point of Thurston’s boundary.

The Hubbard–Masur theorem

For the general reader. Thurston described a surface’s possible “grain patterns”, foliations decorated with a way of measuring width, as a purely topological space. Complex analysts had a different object, quadratic differentials, attached to a particular shape of surface. Hubbard and Masur proved that on any given shape each grain pattern is realized exactly once. The two languages became one.

Measured foliations

A measured foliation on \(S\) is a singular foliation with prong singularities (at least three prongs) and a transverse measure invariant under holonomy. Two are equivalent if they differ by isotopy and Whitehead moves, which collapse or expand leaves joining singularities. Each measured foliation \(\mathcal F\) defines a function on simple closed curves, \(\gamma\mapsto i(\mathcal F,\gamma)=\inf\) of the transverse measure over curves homotopic to \(\gamma\). Thurston showed that this embeds the space \(\mathcal{MF}(S)\) of equivalence classes in \(\mathbb R^{\mathcal S}\), and that \(\mathcal{MF}(S)\) is homeomorphic to \(\mathbb R^{6g-6}\setminus\{0\}\), with the empty foliation added as a point.

Every non-zero \(q\in Q(X)\) has a vertical foliation \(\mathcal V(q)\), measured by \(|\operatorname{Re}\sqrt q|\). This defines a map \(\mathcal V:Q(X)\setminus\{0\}\to\mathcal{MF}(S)\).

Strategy of the proof

Uniqueness follows from a length–area comparison of the two flat metrics, a technique going back to Teichmüller and Grötzsch: if two differentials had equivalent vertical foliations, each would have to be strictly shorter than the other in a suitable averaged sense.

For existence, Hubbard and Masur showed that \(\mathcal V\) is continuous, injective and proper between spaces of the same dimension \(6g-6\). Invariance of domain then makes \(\mathcal V\) open, properness makes its image closed, and connectedness of \(\mathcal{MF}(S)\) makes it onto. The heart of the proof is properness, a quantitative estimate that controls the norm \(\|q\|\) by the intersection numbers of \(\mathcal V(q)\) with finitely many curves. The theorem extends an earlier result of Kurt Strebel, which realizes any system of disjoint curves with prescribed weights by a unique Jenkins–Strebel differential; such foliations, with all leaves closed, are dense in \(\mathcal{MF}(S)\).

Consequences

• Fixing the horizontal foliation as well, a pair of transverse measured foliations determines a unique quadratic differential and hence a point of \(\mathcal T(S)\) (Gardiner–Masur). So Teichmüller geodesics can be labelled by pairs of foliations.
• Every point of Thurston’s boundary \(\mathcal{PMF}(S)\) is the direction of a unique Teichmüller ray from each base point.

Interval exchanges and unique ergodicity

For the general reader. Cut a ribbon into a few pieces, shuffle them, and glue them back. Repeat the same shuffle again and again, and follow one point. Does it visit every part of the ribbon equally often? Michael Keane conjectured that for almost every choice of cut lengths the answer is yes. Masur and Veech proved it in 1982 by moving the problem into Teichmüller space and running the Teichmüller flow.

Interval exchanges

Let \(\lambda=(\lambda_1,\dots,\lambda_d)\) be positive lengths with \(\sum\lambda_i=1\), and \(\pi\) a permutation of \(\{1,\dots,d\}\). The interval exchange transformation \(T_{\lambda,\pi}\) cuts \([0,1)\) into intervals of lengths \(\lambda_1,\dots,\lambda_d\) and rearranges them in the order \(\pi\), translating each piece. It preserves Lebesgue measure. With \(d=2\) and \(\pi=(2\,1)\) it is the rotation \(x\mapsto x+\lambda_2\bmod1\). The permutation is irreducible if \(\pi\{1,\dots,k\}\ne\{1,\dots,k\}\) for \(k<d\).

Interval exchanges are first-return maps of vertical flows on flat surfaces to a horizontal segment. Conversely, every IET with irreducible \(\pi\) is the first-return map of the vertical flow of an abelian differential on some surface, by the zippered-rectangle construction of Veech.

The theorem is not automatic. Examples from the 1970s, by Harvey Keynes and Donald Newton and by Keane, showed that a minimal interval exchange can have several ergodic invariant measures.

The proof through Teichmüller dynamics

Both proofs rest on two facts about the Teichmüller flow \(g_t\) on a stratum \(\mathcal Q_1\) of unit-area quadratic differentials (Masur 1982, Veech 1982):

1. The natural measure on \(\mathcal Q_1\), given by Lebesgue measure in period coordinates and normalized to the unit-area hypersurface, is finite.
2. The flow \(g_t\) is ergodic with respect to this measure on each connected component.

By Poincaré recurrence and (1), almost every trajectory \(g_tq\) returns infinitely often to a compact part of the moduli space. Masur then proved the following criterion, in its final form in 1992.

The intuition: if the vertical foliation carried two different ergodic measures, the surface could be cut along a curve that the flow \(g_t\) shrinks, so that \(g_tq\) leaves every compact set. Recurrence prevents this.

With John Smillie (1991), Masur showed that the set of measured foliations that are not uniquely ergodic, although of measure zero, has positive Hausdorff dimension. In 1992 he proved that for any fixed quadratic differential, the set of directions whose flow is not uniquely ergodic has Hausdorff dimension at most \(1/2\).

Rational billiards

For the general reader. A billiard ball moves in straight lines on a table shaped like a polygon and bounces off the walls. If the angles of the polygon are rational multiples of \(\pi\), the motion in a fixed direction can be unfolded into a straight-line flow on a flat surface. Kerckhoff, Masur and Smillie used this to show that in almost every direction the ball eventually visits every region of the table in proportion to its area.

Unfolding

Let \(P\) be a polygon whose angles are \(\pi m_i/n_i\) with \(\gcd(m_i,n_i)=1\), and let \(N=\operatorname{lcm}(n_i)\). The reflections in the sides of \(P\) generate a dihedral group \(D_N\) of order \(2N\) acting on directions. Glue the \(2N\) reflected copies of \(P\) along corresponding sides (the Katok–Zemlyakov construction). The result is a translation surface \(X_P\), and billiard trajectories in \(P\) lift to straight lines on \(X_P\). A billiard direction \(\theta\) takes at most \(2N\) values along the trajectory, and the trajectory is the projection of a single straight-line flow on \(X_P\).

Idea of proof. Rotate \(X\) so that \(\theta\) becomes vertical. The family of rotations \(r_\theta X\) is a circle in the stratum, and the Teichmüller flow \(g_t\) applied to it produces larger and larger circles. KMS showed that for almost every \(\theta\), the trajectory \(g_tr_\theta X\) is recurrent to a compact set of moduli space, by a quantitative argument that uses the fact that short saddle connections are rare: a circle cannot spend much of its length near infinity. Masur’s recurrence criterion, already implicit in his 1982 paper (Lesson 06), then gives unique ergodicity.

The corollary uses approximation of an arbitrary polygon by rational ones, on which almost every direction is ergodic, and a Baire category argument. Ergodicity of the billiard in a specific irrational polygon, such as most triangles, remains open.

Counting saddle connections and cylinders

For the general reader. On a flat surface, a cylinder is a band of parallel closed straight lines, like a tube made from a strip of paper. How many cylinders are there with circumference at most \(L\)? Masur proved that the number grows like \(L^2\), exactly as the number of lattice points in a disc does. Later, with Eskin, he showed that for almost every surface the count is asymptotic to a constant times \(L^2\), and that the constant can be computed.

Definitions

Let \((X,q)\) be a surface with a quadratic differential and its flat metric. A saddle connection is a straight segment joining two cone points with no cone point in its interior. A cylinder is a maximal family of parallel closed geodesics; its circumference is their common length. Let \(N_{\mathrm{cyl}}(X,L)\) and \(N_{\mathrm{sc}}(X,L)\) count cylinders and saddle connections of length at most \(L\).

The lower bound includes the existence of at least one cylinder, which is itself non-trivial. The upper bound uses the Teichmüller flow: a saddle connection of length about \(L\) in direction \(\theta\) becomes short on \(g_{\log L}r_\theta X\), and the number of short saddle connections on any surface is bounded.

Exact asymptotics

For lattice surfaces, William Veech proved in 1989 that \(N_{\mathrm{cyl}}(X,L)\sim cL^2\) with an explicit \(c\). The general statement is due to Eskin and Masur.

The constants are Siegel–Veech constants. The proof combines Veech’s Siegel-type formula, which expresses averages over the stratum of counting functions in terms of a single constant, with an ergodic theorem for circle averages \(\frac1{2\pi}\int f(g_tr_\theta X)\,d\theta\) as \(t\to\infty\), proved using the method of Eskin, Margulis and Mozes. In 2003 Eskin, Masur and Zorich computed the constants in terms of the Masur–Veech volumes of boundary strata, the volumes themselves having been computed by Eskin and Andrei Okounkov.

The curve complex is hyperbolic

For the general reader. Draw all the simple loops on a surface, and join two loops when they can be drawn without crossing. The result is a huge network, the curve complex, with infinitely many connections at every point. Masur and Minsky proved in 1999 that, seen from far away, this network looks like a tree. That fact gave a new handle on the symmetry group of the surface and became a key step in the proof of Thurston’s ending lamination conjecture.

Definitions

Let \(S\) be a surface of genus \(g\) with \(n\) punctures and \(\xi(S)=3g-3+n\ge2\). The curve complex \(\mathcal C(S)\), introduced by William Harvey in 1978, has a vertex for each isotopy class of essential, non-peripheral simple closed curves, and a simplex for each set of classes with pairwise disjoint representatives. Its dimension is \(\xi(S)-1\). Give its 1-skeleton the path metric with edges of length 1. A geodesic metric space is \(\delta\)-hyperbolic if every side of every geodesic triangle lies in the \(\delta\)-neighbourhood of the other two.

Strategy. The proof uses Teichmüller geometry. To each point \(X\in\mathcal T(S)\) assign the set of curves of shortest extremal length on \(X\), a bounded set in \(\mathcal C(S)\). Masur and Minsky showed that this map sends Teichmüller geodesics to unparametrized quasi-geodesics, and that the resulting family of paths satisfies a contraction property: balls disjoint from a path project to sets of bounded diameter. A general criterion then gives hyperbolicity. The proof explains a paradox: Teichmüller space is not hyperbolic (Lesson 10) because of the regions where some curve is very short, and \(\mathcal C(S)\) is exactly what remains when those regions are collapsed.

Hierarchies (2000)

For each essential subsurface \(Y\subset S\) there is a subsurface projection \(\pi_Y\) from \(\mathcal C(S)\) to subsets of \(\mathcal C(Y)\), given by intersecting curves with \(Y\). Masur and Minsky proved the bounded geodesic image theorem: if a geodesic in \(\mathcal C(S)\) avoids the curves disjoint from \(Y\), then its projection to \(\mathcal C(Y)\) has diameter bounded by a constant \(M(S)\). Building on this, they organized geodesics in all the curve complexes \(\mathcal C(Y)\) into hierarchies and proved a distance formula for the mapping class group.

As a corollary, they showed that conjugate pseudo-Anosov elements are conjugate by an element of word length linear in theirs, giving an effective solution to the conjugacy problem. Hierarchies were the combinatorial engine of the proof of the ending lamination conjecture by Minsky, Jeffrey Brock and Richard Canary.

Coarse geometry of Teichmüller space

For the general reader. After the curve complex, Masur returned to Teichmüller space itself and asked how it looks from very far away. With Michael Wolf he showed it is not hyperbolic. With Eskin and Rafi he showed that its largest flat regions have exactly the dimension one expects from the number of independent curves. With Dowdall and Duchin he showed that, on average, it still behaves like a hyperbolic space.

Not hyperbolic

The reason is the thin parts. If a pants decomposition \(\{\alpha_1,\dots,\alpha_{3g-3}\}\) consists of very short curves, Teichmüller space near that point resembles a product of hyperbolic planes, one for each curve (the length and twist of \(\alpha_i\)). Minsky’s product region theorem (1996) makes this precise. A product of two unbounded spaces contains quasi-flats and is not hyperbolic.

Brian Bowditch proved the same result independently.

Statistical hyperbolicity

Dowdall, Duchin and Masur (2014) proved that Teichmüller space is statistically hyperbolic: if two points are chosen independently and uniformly on a large sphere about a base point, the expected distance between them is asymptotic to twice the radius, as in a tree or the hyperbolic plane, and unlike Euclidean space. Farb and Masur (2010) studied the large-scale geometry of moduli space itself, which seen from far away looks like a cone over a finite complex obtained from the curve complex modulo the mapping class group. With Brock and Minsky, Masur analysed geodesics of the Weil–Petersson metric, showing that their behaviour at infinity is governed by the same subsurface projections that control the curve complex.

Chicago and collaborators

Masur was for many years a professor at the University of Illinois at Chicago, where the department was a centre for geometry, topology and dynamics. He later moved to the University of Chicago. The Chicago area gave him long collaborations with Alex Eskin and Benson Farb at the University of Chicago, and he supervised doctoral students including Jing Tao, who works on the geometry of mapping class groups and Teichmüller space.

Masur’s work is unusually collaborative. His major papers are joint with John Hubbard, Steven Kerckhoff, John Smillie, Yair Minsky, Michael Wolf, Alex Eskin, Anton Zorich, Benson Farb, Jeffrey Brock, Kasra Rafi, Spencer Dowdall and Moon Duchin. He also wrote, with Serge Tabachnikov, the survey Rational billiards and flat structures (2002), which introduced a generation of students to the subject, and the chapter on ergodic theory of translation surfaces in the Handbook of Dynamical Systems (2006).

In 1994 he was an invited speaker at the International Congress of Mathematicians in Zürich, on Teichmüller space, dynamics and probability. In 2009 a conference at CIRM, Luminy, marked his 60th birthday, bringing together the communities of Teichmüller dynamics and geometric group theory that his work connects.

Honours and influence

Honours

• Sloan Research Fellowship (1980).
• Invited speaker, International Congress of Mathematicians, Zürich (1994).
• Fellow of the American Mathematical Society.
• Conference in his honour, CIRM Luminy (2009), for his 60th birthday.

Influence

• Teichmüller dynamics. The finiteness of Masur–Veech volumes and the ergodicity of the Teichmüller flow are the starting point of the field. It led to the classification of orbit closures by Alex Eskin, Maryam Mirzakhani and Amir Mohammadi (2015), to Curtis McMullen’s work in genus 2, and to the computation of Masur–Veech volumes by Eskin and Okounkov and, later, through intersection theory on moduli spaces.
• Billiards and flat surfaces. The Kerckhoff–Masur–Smillie theorem and the quadratic growth bounds are the basic tools for rational billiards. The weak mixing of almost every interval exchange, proved by Artur Avila and Giovanni Forni, continues the line of the Masur–Veech theorem.
• Geometric group theory. Hyperbolicity of the curve complex, subsurface projections and the distance formula underlie the theory of hierarchically hyperbolic spaces, developed by Jason Behrstock, Mark Hagen and Alessandro Sisto, and much of the modern study of mapping class groups.
• Hyperbolic 3-manifolds. Hierarchies were a central ingredient of the proof of Thurston’s ending lamination conjecture.
• Complex analysis. The Hubbard–Masur theorem is the standard bridge between Thurston’s measured foliations and quadratic differentials.

Sources and further reading

Biography

1. Howard Masur, Wikipedia.
2. Howard Masur, German Wikipedia (with selected publications).
3. Albert Marden, Wikipedia.
4. University of Chicago, Department of Mathematics, faculty profile.

Mathematics

1. H. Masur, On a class of geodesics in Teichmüller space, Annals of Math. 102 (1975).
2. H. Masur, Extension of the Weil–Petersson metric to the boundary of Teichmüller space, Duke Math. J. 43 (1976).
3. J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math. 142 (1979).
4. H. Masur, Interval exchange transformations and measured foliations, Annals of Math. 115 (1982).
5. W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Annals of Math. 115 (1982).
6. S. Kerckhoff, H. Masur and J. Smillie, Ergodicity of billiard flows and quadratic differentials, Annals of Math. 124 (1986).
7. H. Masur and M. Wolf, Teichmüller space is not Gromov hyperbolic, Ann. Acad. Sci. Fenn. 20 (1995).
8. H. Masur and Y. Minsky, Geometry of the complex of curves I: Hyperbolicity, Invent. Math. 138 (1999).
9. H. Masur and Y. Minsky, Geometry of the complex of curves II: Hierarchical structure, GAFA 10 (2000).
10. A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems 21 (2001).
11. H. Masur and S. Tabachnikov, Rational billiards and flat structures, in Handbook of Dynamical Systems 1A, Elsevier, 2002.
12. A. Eskin, H. Masur and A. Zorich, Moduli spaces of abelian differentials: the principal boundary, counting problems, and the Siegel–Veech constants, Publ. Math. IHÉS 97 (2003).
13. A. Zorich, Flat surfaces, arXiv:math/0609392.
14. S. Dowdall, M. Duchin and H. Masur, Statistical hyperbolicity in Teichmüller space, GAFA 24 (2014).
15. A. Eskin, H. Masur and K. Rafi, Large-scale rank of Teichmüller space, Duke Math. J. 166 (2017).