Oswald Teichmüller

How to use this course

Oswald Teichmüller worked for about eight years and published 34 papers. Two bodies of work survive him. In algebra, his multiplicative representatives and his 3-cocycle are standard tools in number theory. In complex analysis, his use of quasiconformal maps turned the classical problem of moduli of Riemann surfaces into the geometry of a space that now bears his name.

He was also an early and active member of the Nazi party and the SA. In November 1933 he organized the student boycott that ended Edmund Landau’s teaching at Göttingen. This course treats both facts plainly. Part I and Part III give the historical record with sources. Part II teaches the mathematics.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04–06 cover his algebraic work of 1936–1940. Lessons 07–10 cover the function theory of 1938–1944, which is his lasting fame.

Timeline

Nordhausen, 1913–1931

Paul Julius Oswald Teichmüller was born on 18 June 1913 in Nordhausen, at the southern edge of the Harz mountains. His father, Paul Teichmüller, was an independent weaver; his mother was Gertrude Dinse. He was their only child, and the family home was in the small mining town of Sankt Andreasberg.

His mother later said he had taught himself to read and count by the age of three. His father, wounded in the First World War, died when the boy was twelve. Oswald then moved to an aunt in Nordhausen and attended the Gymnasium there, taking his Abitur in 1931.

Little else is documented about his childhood. What the records do show is a gifted, solitary young man arriving at university at the moment when the Weimar Republic was collapsing and the Nazi movement was recruiting heavily among students.

Göttingen and the Landau boycott, 1931–1934

Teichmüller entered the University of Göttingen in the summer semester of 1931 to study mathematics and physics. Göttingen was then the strongest mathematics department in the world. His teachers included Richard Courant, Gustav Herglotz, Edmund Landau, Otto Neugebauer and Hermann Weyl. Hans Lewy, then a young lecturer, later recalled anecdotes of the student’s brilliance.

Within months of arriving he joined the NSDAP and then the SA, the party’s paramilitary wing. Party records cited by his biographers date both memberships to 1931, before Hitler came to power. He became deputy leader of the mathematics and natural-sciences section of the Nazi student organization.

The boycott

In April 1933 the new government’s civil-service law began removing Jewish scholars. Courant was suspended and Weyl, whose wife was Jewish, soon left. Landau, a number theorist of world rank, was protected for the moment by an exemption for those appointed before 1914.

On 2 November 1933, when Landau arrived to lecture, Teichmüller and other students blocked the entrance and turned away anyone who tried to attend. Landau left. Afterwards he asked Teichmüller to explain himself in writing. Teichmüller’s letter, preserved and later published, justified the boycott on explicitly racial grounds: German students, he argued, should not be taught by a Jewish teacher. Landau never lectured at Göttingen again and was retired early in 1934. He died in Berlin in 1938.

Teichmüller also agitated against Courant. He persuaded Landau’s own assistant, Werner Weber, to join the party. The historian Sanford Segal, who studied these events in detail, summed him up as a gifted and seminal mathematician who was also “a dedicated Nazi.” The cryptographer Friedrich Bauer, who knew him in the war, called him both a genius and a fanatic. There is no evidence that he ever repudiated these views.

Hasse, the doctorate, Berlin, 1934–1939

In 1934 Helmut Hasse was appointed to the Göttingen chair vacated by Weyl. Hasse, a German nationalist, tried to rebuild the gutted institute. Teichmüller chose him as advisor; MacTutor suggests the choice was in part political. His dissertation, Operatoren im Wachsschen Raum, extended the spectral theory of operators to Hilbert spaces over the quaternions. He passed the oral examination, with Hasse, Herglotz and the physicist Robert Pohl, in June 1935, and the degree was formally conferred in November. Hasse described him as having extraordinary mathematical gifts and made him an assistant.

Under Hasse he turned to algebra and number theory: valuation theory, \(p\)-adic fields, and the theory of simple algebras that Emmy Noether, Richard Brauer and Hasse had built. These papers are the subject of Lessons 04 and 05.

The turn to function theory

In 1936–37 the Finnish analyst Rolf Nevanlinna lectured at Göttingen on value distribution theory. Teichmüller attended, and his interest shifted to complex analysis. In April 1937 he moved to the University of Berlin to work with Ludwig Bieberbach, the most prominent Nazi among German mathematicians and the founder of the journal Deutsche Mathematik, created to promote what Bieberbach called a German style of mathematics. Teichmüller published 21 of his 34 papers there, which made his work hard for later readers to find.

He habilitated in Berlin in March 1938 with a thesis on conformal and quasiconformal mappings and became a Dozent in 1939. The two Berlin years before the war were his most productive. They produced the long memoir of 1939 on extremal quasiconformal mappings and quadratic differentials, the foundation of Teichmüller theory.

Teichmüller representatives

For the general reader. In the \(p\)-adic numbers, every number has a “digit” modulo \(p\). Teichmüller showed that these digits can be chosen in a uniquely good way: so that multiplying the representatives agrees with multiplying the digits. The chosen representatives are roots of unity, and they let one write every \(p\)-adic number as a power series in \(p\) with canonical coefficients.

Let \(R\) be a complete discrete valuation ring with maximal ideal \(\mathfrak m\), and suppose its residue field \(k=R/\mathfrak m\) is perfect of characteristic \(p\). Examples: \(\mathbb Z_p\) with \(k=\mathbb F_p\), and \(\mathbb F_q[[t]]\) with \(k=\mathbb F_q\).

Proof sketch

Key lemma. If \(x\equiv y \pmod{\mathfrak m^j}\) with \(j\ge1\), then \(x^p\equiv y^p\pmod{\mathfrak m^{j+1}}\). Indeed, write \(x=y+z\) with \(z\in\mathfrak m^j\). Then \(x^p-y^p=\sum_{i=1}^{p-1}\binom pi y^{p-i}z^i+z^p\); the middle terms lie in \(p\,\mathfrak m^j\subseteq\mathfrak m^{j+1}\) because \(p\in\mathfrak m\), and \(z^p\in\mathfrak m^{pj}\subseteq\mathfrak m^{j+1}\).

Construction. Since \(k\) is perfect, \(a\) has a unique \(p^n\)-th root \(a^{p^{-n}}\). Choose any lift \(x_n\in R\) of it. Then \(x_{n+1}^p\equiv x_n\pmod{\mathfrak m}\), and applying the lemma \(n\) times gives \(x_{n+1}^{p^{n+1}}\equiv x_n^{p^n}\pmod{\mathfrak m^{n+1}}\). So \(x_n^{p^n}\) is Cauchy; set \(\omega(a)=\lim x_n^{p^n}\). The same lemma shows the limit does not depend on the lifts, and multiplicativity follows by lifting \(ab\) with products of lifts.

Uniqueness. If \(\omega'\) is another multiplicative section, then \(\omega'(a)=\omega'(a^{p^{-n}})^{p^n}\), and \(\omega'(a^{p^{-n}})\) is a lift of \(a^{p^{-n}}\), so \(\omega'(a)\) equals the same limit.

For \(k=\mathbb F_p\), \(\omega(a)=\lim_{n\to\infty}a^{p^n}\) in \(\mathbb Z_p\), and for \(a\ne0\) it is the unique \((p-1)\)-th root of unity congruent to \(a\). The map \(\omega:\mathbb F_p^\times\to\mathbb Z_p^\times\) is the Teichmüller character, which appears throughout Iwasawa theory and the theory of \(p\)-adic \(L\)-functions.

Witt vectors

The same year, Ernst Witt published his construction of the ring \(W(k)\) of Witt vectors, the unique complete discrete valuation ring of characteristic 0 with uniformizer \(p\) and residue field \(k\). In \(W(k)\) the vector \((a,0,0,\dots)\) is the Teichmüller representative of \(a\), and every element is uniquely \(\sum_n \omega(a_n^{p^{-n}})\,p^n\). Addition of Witt vectors is defined through the ghost components

by requiring each \(w_n\) to be additive and multiplicative. The resulting structure theorem, that a complete unramified discrete valuation ring of mixed characteristic is determined by its perfect residue field, came out of the joint ideas of Witt and Teichmüller in Hasse’s circle.

Simple algebras and the Teichmüller cocycle

For the general reader. A Galois group acts on a field. Suppose an algebra is built over the larger field, and each symmetry of the field can be extended to the algebra. Do these extensions fit together into an algebra over the smaller field? Teichmüller found that the answer is measured by a cohomology class of degree three, an object no one had yet named.

Let \(L/K\) be a finite Galois extension with group \(G\). Noether’s crossed products show that the central simple \(K\)-algebras split by \(L\) are classified by \(H^2(G,L^\times)\). Teichmüller’s 1940 paper in Deutsche Mathematik, on the so-called noncommutative Galois theory, asks a question one degree higher.

The construction

Let \(A\) be a central simple algebra over \(L\) such that every \(\sigma\in G\) extends to a ring automorphism \(\varphi_\sigma\) of \(A\). By the Skolem–Noether theorem, \(\varphi_\sigma\) is unique up to an inner automorphism. Hence \(\varphi_\sigma\varphi_\tau\varphi_{\sigma\tau}^{-1}\), which is \(L\)-linear, is inner:

Computing \(\varphi_\sigma\varphi_\tau\varphi_\rho\) in two ways gives \(\operatorname{Inn}\big(\varphi_\sigma(u_{\tau,\rho})\,u_{\sigma,\tau\rho}\big)=\operatorname{Inn}\big(u_{\sigma,\tau}\,u_{\sigma\tau,\rho}\big)\). Two units with the same inner automorphism differ by a central unit, so

Samuel Eilenberg and Saunders Mac Lane gave \(c\) its name in their 1948 paper on cohomology and Galois theory, where they recognized it as one of the first natural occurrences of \(H^3\). For a Galois extension of global fields, \(H^3(G,L^\times)\) is cyclic and generated by Teichmüller classes. The same pattern, an obstruction in \(H^3\) to lifting a structure defined up to inner automorphism, returns in the theory of group extensions with nonabelian kernel and in modern gerbes.

Finite character and the axiom of choice

For the general reader. Algebraists constantly need maximal objects: a basis of a vector space, a maximal ideal, an algebraic closure. All rely on the axiom of choice. In 1939 Teichmüller asked whether algebraists really need it, and he gave a form of the axiom that is especially convenient to use. John Tukey found the same principle independently a year later.

A family \(\mathcal F\) of subsets of a set \(S\) has finite character if, for every \(X\subseteq S\), \(X\in\mathcal F\) exactly when every finite subset of \(X\) belongs to \(\mathcal F\).

Proof of equivalence with Zorn’s lemma

Zorn ⇒ Teichmüller–Tukey. Order \(\mathcal F\) by inclusion. If \(\mathcal C\subseteq\mathcal F\) is a chain, let \(U=\bigcup\mathcal C\). A finite subset of \(U\) lies in finitely many members of \(\mathcal C\), hence in the largest of them, hence in \(\mathcal F\). By finite character \(U\in\mathcal F\), and \(U\) bounds \(\mathcal C\). Zorn gives a maximal element.

Teichmüller–Tukey ⇒ Zorn. Let \(P\) be a poset in which every chain has an upper bound. The family of chains of \(P\) has finite character (a set is a chain iff each two-element subset is comparable). Take a maximal chain \(C\) and an upper bound \(m\). If \(m'>m\), then \(C\cup\{m'\}\) is a larger chain, a contradiction. So \(m\) is maximal.

The paper, Braucht der Algebraiker das Auswahlaxiom? (Deutsche Mathematik 4, 1939), also discusses where choice can be avoided in algebra. Tukey’s version appeared in his 1940 monograph on convergence and uniformity in topology.

Quasiconformal maps and Grötzsch’s problem

For the general reader. A conformal map preserves angles and sends tiny circles to tiny circles. Most shapes cannot be mapped conformally onto each other in a prescribed way: a square cannot be mapped conformally onto a long rectangle with corners to corners. A quasiconformal map is allowed to distort tiny circles into ellipses of bounded eccentricity. The natural question is how little distortion is needed.

For an orientation-preserving \(C^1\) map \(f\) with Jacobian \(J=|f_z|^2-|f_{\bar z}|^2>0\), where \(f_z=\tfrac12(f_x-if_y)\) and \(f_{\bar z}=\tfrac12(f_x+if_y)\), define

The derivative maps circles to ellipses with axis ratio \(K_f\). The map is \(K\)-quasiconformal if \(K_f\le K\) everywhere; the general definition requires only weak derivatives with \(\|\mu_f\|_\infty\le\frac{K-1}{K+1}\). Herbert Grötzsch introduced these maps in 1928, and Ahlfors used them in 1935. Teichmüller made them the central tool of the theory of Riemann surfaces.

Proof of the bound (length–area)

Each horizontal segment \(y=\text{const}\) is mapped to a curve joining the two vertical sides of \(R'\), so

Integrate in \(y\) and apply Cauchy–Schwarz with weight \(J\):

Here \((|f_z|+|f_{\bar z}|)^2/J=K_f\), and \(\iint J\) is the area of \(R'\). Hence \(K\ge a'b/(ab')=m'/m\). Teichmüller’s great idea was to run this argument on a Riemann surface with a quadratic differential supplying the “horizontal” direction (Lesson 08).

Extremal maps and quadratic differentials

For the general reader. Two Riemann surfaces of the same genus look the same topologically but differ in their conformal shape. Teichmüller asked for the least distorted map between them in a given topological class. His answer: the best map is always a pure stretch in one direction, and the direction field is supplied by a holomorphic quadratic differential. This gave the field its central theorem.

Let \(X\) be a closed Riemann surface of genus \(g\ge2\). A holomorphic quadratic differential is \(\varphi=\varphi(z)\,dz^2\), with \(\varphi(z)\) holomorphic in each chart and transforming by \(\varphi(z)=\tilde\varphi(w)\,(dw/dz)^2\). The space \(Q(X)\) has complex dimension \(3g-3\). Away from zeros, a natural coordinate \(\zeta=\int\sqrt{\varphi(z)}\,dz\) puts \(\varphi\) in the form \(d\zeta^2\). The lines \(\operatorname{Im}\zeta=\text{const}\) are the horizontal trajectories, and \(\operatorname{Re}\zeta=\text{const}\) the vertical ones. At a zero of order \(n\), \(n+2\) horizontal rays meet (the picture at the top of this page has \(n=1\)).

Given \(\varphi\ne0\) and \(K>1\), a Teichmüller map is a homeomorphism \(f:X\to Y\) that in natural coordinates of \(\varphi\) on \(X\) and of some \(\psi\in Q(Y)\) on \(Y\) has the form

Teichmüller proved uniqueness in the 1939 memoir, by a length–area argument in the flat metric \(|\varphi|^{1/2}|dz|\) that generalizes Lesson 07. He proved existence in 1943 by a continuity method. Many of his arguments were heuristic, and the first complete proof is due to Ahlfors (1953–54); Lipman Bers gave another in 1960.

The Beltrami coefficient

In the natural coordinate the stretch is \(\zeta\mapsto\tfrac{K^{1/2}+K^{-1/2}}{2}\zeta+\tfrac{K^{1/2}-K^{-1/2}}{2}\bar\zeta\), whose Beltrami coefficient is \(k=\frac{K-1}{K+1}\). Changing to the coordinate \(z\) with \(\zeta_z=\sqrt{\varphi}\) gives

So extremal Beltrami coefficients have constant modulus and argument dictated by a holomorphic quadratic differential. Maps with \(\mu=k\bar\varphi/|\varphi|\) are now called Teichmüller differentials, and the pairing \(\int_X\mu\varphi\) between Beltrami differentials and quadratic differentials is the duality between tangent and cotangent vectors of Teichmüller space.

Teichmüller space and its metric

For the general reader. Bernhard Riemann counted in 1857 that a surface of genus \(g\ge2\) has \(3g-3\) complex “moduli,” numbers describing its shape. Teichmüller built the space whose points are these shapes, with the topology remembered, and measured distances by how much stretching is needed to go from one shape to another. The space turned out to be a ball.

Fix a closed oriented surface \(S\) of genus \(g\). A marked Riemann surface is a pair \((X,h)\) with \(h:S\to X\) an orientation-preserving homeomorphism. Two pairs are equivalent if \(h_2\circ h_1^{-1}\) is homotopic to a biholomorphism. The set of classes is the Teichmüller space \(\mathcal T_g\). The Teichmüller distance is

Teichmüller proved that this map is defined and injective, using his uniqueness theorem, and argued for surjectivity, which is the existence theorem of Lesson 08. The mapping class group \(\mathrm{Mod}_g\) of homotopy classes of orientation-preserving homeomorphisms of \(S\) acts on \(\mathcal T_g\) by changing the marking. It acts by isometries and properly discontinuously, and \(\mathcal T_g/\mathrm{Mod}_g\) is the moduli space \(\mathcal M_g\) of Riemann surfaces. In 1971 Halsey Royden proved that the Teichmüller metric equals the Kobayashi metric of \(\mathcal T_g\), and deduced that every holomorphic automorphism of \(\mathcal T_g\) comes from \(\mathrm{Mod}_g\).

The Teichmüller metric is a Finsler metric, not a Riemannian one, and it is not negatively curved in any strong sense. It is still the natural metric for dynamics. William Thurston’s classification of surface homeomorphisms, Howard Masur’s and William Veech’s ergodicity of the Teichmüller geodesic flow, and the work of Curtis McMullen, Alex Eskin and Maryam Mirzakhani all take place in this space.

Variable Riemann surfaces

For the general reader. In his last paper, published in 1944 after his disappearance, Teichmüller sketched how Teichmüller space is itself a complex manifold and how all marked surfaces fit together into one holomorphic family. The sketch was right. Making it rigorous took the work of Ahlfors and Bers over the following two decades.

In Veränderliche Riemannsche Flächen (Deutsche Mathematik, 1944), Teichmüller described a complex-analytic structure on \(\mathcal T_g\) of dimension \(3g-3\) and a universal family over it, characterized by a universal property. The modern route goes through the Beltrami equation.

Represent \(X=\mathbb H/\Gamma\) for a Fuchsian group \(\Gamma\). A point of \(\mathcal T_g\) is given by a \(\Gamma\)-invariant Beltrami coefficient \(\mu\) on \(\mathbb H\). Extend \(\mu\) by \(0\) to the lower half-plane \(\mathbb H^*\); then \(f^\mu\) is conformal on \(\mathbb H^*\), and its Schwarzian derivative

is a holomorphic quadratic differential on \(\mathbb H^*/\Gamma\), the mirror surface \(X^*\). The Bers embedding \([\mu]\mapsto S(f^\mu|_{\mathbb H^*})\) maps \(\mathcal T_g\) injectively onto a bounded open subset of \(Q(X^*)\cong\mathbb C^{3g-3}\). By the theorems of Kraus, Nehari, and Ahlfors–Weill, the image contains the ball of radius \(\tfrac12\) and lies in the ball of radius \(\tfrac32\) for the norm \(\sup_{z\in\mathbb H^*}|\operatorname{Im}z|^2\,|\varphi(z)|\). This realizes Teichmüller’s vision of \(\mathcal T_g\) as a bounded domain of holomorphy in \(\mathbb C^{3g-3}\).

War, cryptography, disappearance, 1939–1943

Teichmüller was called up on 18 July 1939 for what should have been eight weeks of training. The war began in September and he remained a soldier. In April 1940 his unit took part in the invasion of Norway, and he continued to write mathematics there.

He was then brought back to Berlin to work in the cipher department of the high command of the armed forces, where Ernst Witt and other mathematicians were also assigned. Bieberbach requested his release for teaching, and in the 1942–43 academic year he lectured at the University of Berlin while continuing the decryption work. In this period he wrote the existence proof for extremal maps on closed surfaces (1943) and the paper on variable Riemann surfaces.

After the German defeat at Stalingrad in February 1943, he gave up his Berlin position and volunteered for the Eastern Front. His unit fought in the Kursk offensive that began on 5 July 1943. In early August he was on leave when the unit reached the area of Kharkov, where it was encircled and largely destroyed. In early September he set out to rejoin it. He was last known to be east of the Dnieper and was probably near Poltava when he disappeared, during the German retreat. His death is usually dated 11 September 1943. The circumstances are unknown, and the Neue Deutsche Biographie records him simply as missing. He was 30.

Legacy and the problem of judgment

Teichmüller’s function theory was ahead of its readers. It appeared mostly in Deutsche Mathematik, in dense German with arguments that were often heuristic. After the war Lars Ahlfors read the memoirs closely. His 1953–54 paper on quasiconformal mappings gave the first complete proof of Teichmüller’s theorem. Lipman Bers, a Jewish refugee from Nazi-occupied Europe, made Teichmüller spaces the centre of his own research from the late 1950s and built, with Ahlfors, the complex-analytic theory.

What bears his name

• Geometry and dynamics: Teichmüller space, the Teichmüller metric, Teichmüller maps, the Teichmüller geodesic flow, Teichmüller curves. These are basic objects in low-dimensional topology, complex dynamics and string theory.
• Number theory: Teichmüller representatives and the Teichmüller character in \(p\)-adic analysis.
• Algebra: the Teichmüller cocycle in Galois cohomology; \(p\)-bases of inseparable extensions.
• Set theory: the Teichmüller–Tukey lemma.

His collected papers, Gesammelte Abhandlungen, were edited by Ahlfors and Frederick Gehring in 1982. Between 2007 and 2020 the Handbook of Teichmüller Theory, edited by Athanase Papadopoulos, published English translations of his main papers with detailed commentaries. They show that several of his results were later rediscovered without credit because so few people had read him.

Judging the man

Historians agree on the facts. Teichmüller joined the Nazi movement before it held power, organized the action that drove Landau from teaching, put his reasons in racial terms in writing, and volunteered for combat late in the war. Norbert Schappacher and Erhard Scholz, in their 1992 study, and Segal, in his 2003 book, place his mathematics in this context without separating it from it. The mathematical community has kept his name on the objects he discovered, while making the record of his actions part of how he is taught.

Connections

People linked to Oswald Teichmüller across this collection. A name without a link has no page here yet. See Oswald Teichmüller in the network of all 337 lives →

Teachers and advisors

• Helmut Hasse 1935

Influenced by

• Bernhard Riemann
• Emmy Noether
• Herbert Grötzsch
• Rolf Nevanlinna 1936-1937

Influenced

• Howard Masur
• Lars Ahlfors
• Maryam Mirzakhani
• William Thurston
• Lipman Bers

Colleagues and collaborators

• Ernst Witt 1936-1943
• Gustav Herglotz
• Ludwig Bieberbach 1937-1943

Rivals

• Edmund Landau 1933

Also mentioned in this life

• Eugenio Beltrami
• Richard Brauer
• Augustin-Louis Cauchy
• Richard Courant
• Alex Eskin
• Leonhard Euler
• Évariste Galois
• David Hilbert
• Curtis T. McMullen
• August Ferdinand Möbius
• Hermann Weyl

Fields

• complex analysis 13 lives
• riemann surfaces 3 lives
• algebra 29 lives
• number theory 45 lives

On CoursesHub

• Complex Analysis · Mathematics for Physics

Sources and further reading

Biography and history

1. J. J. O’Connor and E. F. Robertson, Paul Julius Oswald Teichmüller, MacTutor History of Mathematics.
2. Oswald Teichmüller, Wikipedia.
3. M. Schottenloher, Teichmüller, Oswald, Neue Deutsche Biographie 26 (2016).
4. Teichmüller, Dictionary of Scientific Biography (MacTutor copy).
5. N. Schappacher and E. Scholz, Oswald Teichmüller: Leben und Werk, Jahresbericht der DMV 94 (1992).
6. S. L. Segal, Mathematicians under the Nazis, Princeton University Press, 2003.
7. W. Abikoff, Oswald Teichmüller, Mathematical Intelligencer 8 (1986).
8. M. R. Chowdhury, Landau and Teichmüller, Mathematical Intelligencer 17 (1995).

Mathematics

1. O. Teichmüller, Gesammelte Abhandlungen, ed. L. V. Ahlfors and F. W. Gehring, Springer, 1982.
2. O. Teichmüller, Extremale quasikonforme Abbildungen und quadratische Differentiale, Abh. Preuss. Akad. Wiss. (1939, published 1940).
3. O. Teichmüller, Braucht der Algebraiker das Auswahlaxiom? Deutsche Mathematik 4 (1939), 567–577 (review in J. Symbolic Logic).
4. A. Papadopoulos, V. Alberge, W. Su, A commentary on Teichmüller’s paper “Extremale quasikonforme Abbildungen und quadratische Differentiale”, arXiv:1511.01313.
5. A commentary on Teichmüller’s paper “Veränderliche Riemannsche Flächen”, arXiv:1209.1882.
6. A. Papadopoulos (ed.), Handbook of Teichmüller Theory, 7 vols., EMS, 2007–2020.
7. L. V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, 1966; 2nd ed. AMS, 2006.
8. F. P. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, AMS, 2000.
9. K. Strebel, Quadratic Differentials, Springer, 1984.
10. Teichmüller cocycle and Witt vector, Wikipedia.
11. S. Eilenberg and S. Mac Lane, Cohomology and Galois theory I: Normality of algebras and Teichmüller’s cocycle, Trans. AMS 64 (1948).