Grigory Margulis

How to use this course

Grigory Margulis is one of the few mathematicians to have received the Fields Medal, the Wolf Prize and the Abel Prize. His work has an unusual shape. A handful of central objects recur, above all lattices in Lie groups and flows on the spaces they define, yet the problems he solved with them come from far away: arithmetic groups, the design of communication networks, the values of quadratic forms at integer points, and finitely additive measures on spheres.

Each technical lesson opens with an italic paragraph for the general reader, then states results precisely, sketches proofs and ends with exercises whose solutions are folded. The lessons follow the chronology of his work: geodesic flows (1969–1970), lattices and rigidity (1968–1978), combinatorial applications (1973–1988), and homogeneous dynamics and number theory (from 1986). Readers interested only in the life can read Parts I and III and the opening paragraphs of Part II.

Timeline

Moscow, 1946–1962

Grigory Aleksandrovich Margulis was born in Moscow on 24 February 1946, the year after the end of the Second World War. His family was of Lithuanian Jewish descent. Friends and colleagues know him as Grisha; in English his first name is often written Gregory.

He grew up in the Moscow of the 1950s, when the city’s mathematical culture was unusually rich. Andrei Kolmogorov, Israel Gelfand and their students ran seminars, specialized schools and mathematical circles for teenagers, and the olympiad system picked out talented pupils early. Margulis came through this system. He finished secondary school in 1962, and in the same year, aged 16, he won a silver medal at the International Mathematical Olympiad.

The Soviet setting matters for the rest of this course. It gave him an outstanding education and a community of strong mathematicians. It also limited where he could work and whether he could travel, and those limits shaped his career for almost thirty years.

Moscow University and Sinai, 1962–1970

Margulis entered the Faculty of Mechanics and Mathematics of Moscow State University in 1962 and graduated in 1967. He stayed for graduate work under Yakov Sinai, then a young but already central figure in ergodic theory and dynamical systems. Sinai would himself receive the Abel Prize in 2014.

Moscow was at that moment the world centre of hyperbolic dynamics. Dmitri Anosov had just introduced the flows that bear his name, which Russian authors then called U-systems, and he had proved that geodesic flows on compact manifolds of negative curvature are of this type and are ergodic. Margulis’s thesis, written in 1969 and defended in 1970, studied these flows. In it he built an invariant measure from the stable and unstable foliations, now called the Margulis measure or the Bowen–Margulis measure, and used it to count closed geodesics (Lesson 04). The thesis circulated in Russian for decades and was published in English only in 2004.

In 1968, while still a graduate student, he received the Moscow Mathematical Society’s prize for young mathematicians. In the same year he and David Kazhdan, another young member of the Moscow school, published their proof of a conjecture of Atle Selberg about lattices (Lesson 05). This paper turned Margulis toward the subject he would transform: discrete subgroups of Lie groups.

The information institute, 1970–1978

After his doctorate Margulis did not obtain a position at Moscow University. The Abel Prize biography attributes this to discrimination against his Jewish origin, which was common in Soviet academic hiring at the time. He joined instead the Institute for Problems of Information Transmission of the Academy of Sciences, a research institute devoted to communication theory and coding. He was a junior scientific worker from 1970 to 1974, a senior scientific worker from 1974, and a leading scientific worker from 1986.

The setting proved fertile. Colleagues at the institute told him about expander graphs, which were wanted for building efficient switching networks. Within a short time he gave the first explicit construction, using representation theory of Lie groups (Lesson 08). The rest of his output in these years concerned lattices: the non-divergence of unipotent flows (1971), superrigidity and arithmeticity (1974), and the normal subgroup theorem (1978). Taken together they settled the main structural questions about lattices in Lie groups of higher rank.

The Fields Medal

In 1978 the International Mathematical Union awarded him the Fields Medal at the Congress in Helsinki, mainly for his work on discrete subgroups of Lie groups. He was 32. The Soviet authorities did not allow him to travel to Finland. Jacques Tits, who presented his work, spoke of his regret at the laureate’s absence and described a mathematician who had repeatedly surprised the experts by solving questions that seemed out of reach.

Counting closed geodesics

For the general reader. On a curved surface shaped like a pretzel, a closed geodesic is a loop that a light ray could travel forever, like a racing line that closes up. There are infinitely many of them. Margulis’s thesis answered a basic question: how many have length at most \(T\)? The answer has the same form as the prime number theorem, with geodesics playing the role of primes.

Geodesic flows

Let \(M\) be a compact Riemannian manifold and \(SM\) its unit tangent bundle. The geodesic flow \(g_t\) moves a unit vector \(v\) for time \(t\) along the geodesic it determines. A flow on a compact manifold is Anosov if the tangent bundle splits invariantly as \(E^s\oplus E^0\oplus E^u\), where \(E^0\) is the flow direction, vectors in \(E^s\) contract exponentially under \(g_t\) as \(t\to+\infty\), and vectors in \(E^u\) contract exponentially as \(t\to-\infty\). Anosov proved that the geodesic flow of a compact manifold with negative sectional curvature is Anosov. Closed orbits of \(g_t\) correspond to closed geodesics, with period equal to length.

The theorem

The topological entropy \(h\) of the flow measures the exponential rate at which orbits become distinguishable. For a compact hyperbolic \(n\)-manifold (constant curvature \(-1\)), \(h=n-1\).

For surfaces of constant curvature \(-1\), the counting result had been obtained by Heinz Huber in 1959 using the Selberg trace formula, which requires constant curvature. Margulis’s method uses no spectral theory and works for variable curvature.

Sketch of the method

Margulis constructed a measure \(\mu\) on \(SM\) from two families of measures: measures \(\mu^u\) on the unstable leaves, which the flow multiplies by exactly \(e^{ht}\), and measures \(\mu^s\) on the stable leaves, which it multiplies by \(e^{-ht}\). The local product of \(\mu^s\), \(\mu^u\) and \(dt\) is a flow-invariant measure, which turns out to be the unique measure of maximal entropy. Rufus Bowen found the same measure independently in the early 1970s, by a different construction based on periodic orbits. Mixing of \(g_t\) with respect to \(\mu\) then lets one count: a small box \(B\) in \(SM\), flowed for time \(t\), meets \(B\) again in pieces whose measure tends to \(\mu(B)^2\), and each piece contains, by the Anosov closing lemma, exactly one closed orbit of period near \(t\). This gives \(\sum_{\ell(\gamma)\le T}\ell(\gamma)\sim e^{hT}/h\), since a closed orbit of period \(\ell\) is met once per unit of its length. Removing the weight \(\ell(\gamma)\approx T\) yields \(\pi(T)\sim e^{hT}/(hT)\).

Lattices and the Kazhdan–Margulis theorem

For the general reader. The integer points of the plane form a lattice: a discrete pattern that tiles the plane with a finite-area cell. The same idea makes sense inside groups of matrices, and the most important example is the group of integer matrices of determinant 1 inside all real matrices of determinant 1. Kazhdan and Margulis showed that such patterns cannot be arbitrarily fine, and that lattices with a non-compact quotient always contain special elements called unipotents.

Definitions

Let \(G\) be a connected semisimple Lie group with finite centre, with Haar measure \(\mu\). A lattice is a discrete subgroup \(\Gamma\) such that \(G/\Gamma\) carries a finite \(G\)-invariant measure; its covolume is that measure. The lattice is uniform (cocompact) if \(G/\Gamma\) is compact. Examples: \(\mathbb Z^n\subset\mathbb R^n\); \(\mathrm{SL}(n,\mathbb Z)\subset\mathrm{SL}(n,\mathbb R)\), which is non-uniform; the fundamental group of a compact hyperbolic surface, which is a uniform lattice in \(\mathrm{PSL}(2,\mathbb R)\). An element \(u\) is unipotent if \(\mathrm{Ad}(u)-1\) is nilpotent, for instance \(\begin{pmatrix}1&1\\0&1\end{pmatrix}\).

Selberg had conjectured that a lattice is non-uniform exactly when it contains a non-trivial unipotent element. One direction is elementary: a unipotent element can be conjugated as close to the identity as one likes, while compactness of \(G/\Gamma\) keeps every conjugate of \(\Gamma\) away from the identity by a uniform amount. The difficult direction is to produce a unipotent element from non-compactness.

Consequences

Covolumes are bounded below. Choose a neighbourhood \(W\) of \(e\) with \(W^{-1}W\subset U\). If \(g\Gamma g^{-1}\cap U=\{e\}\), the projection \(W\to G/g\Gamma g^{-1}\) is injective, so every lattice has covolume at least \(\mu(W)>0\), a constant depending only on \(G\). For hyperbolic surfaces this was known through Gauss–Bonnet; for general \(G\) it was new.

Selberg’s conjecture. A refined form of the theorem shows that if \(\Gamma\) has no unipotent elements, then the conjugates \(g\Gamma g^{-1}\) avoid a fixed neighbourhood of \(e\) uniformly in \(g\), which, by a compactness criterion of Mahler type, forces \(G/\Gamma\) to be compact. Hence a non-uniform lattice contains non-trivial unipotent elements.

The Margulis lemma

The same circle of ideas gives the Margulis lemma: for each \(n\) there is \(\varepsilon_n>0\) such that, for any discrete group \(\Gamma\) of isometries of hyperbolic \(n\)-space and any point \(x\), the subgroup generated by the elements moving \(x\) less than \(\varepsilon_n\) has a nilpotent subgroup of finite index. Consequently a finite-volume hyperbolic manifold splits into a thick part, where the injectivity radius is at least \(\varepsilon_n/2\), and a thin part made of cusps and tubes around short closed geodesics, the Margulis tubes. This decomposition is a basic tool in three-dimensional topology.

Superrigidity and arithmeticity

For the general reader. The integer matrices \(\mathrm{SL}(n,\mathbb Z)\) are the model lattice: they are defined by arithmetic. Are all lattices built from arithmetic in some such way? In rank one, the answer is no. Atle Selberg and Ilya Piatetski-Shapiro conjectured that in higher rank the answer is yes. Margulis proved it in 1974, by first showing that lattices in higher rank are “superrigid”: every way of representing them by matrices comes from the big group.

Definitions

The real rank of \(G\) is the dimension of a maximal subgroup diagonalizable over \(\mathbb R\): \(\mathrm{SL}(n,\mathbb R)\) has rank \(n-1\), \(\mathrm{SO}(n,1)\) and \(\mathrm{SU}(n,1)\) have rank 1. A lattice in a product \(G_1\times G_2\) is irreducible if its projection to each factor is dense. A lattice \(\Gamma\subset G\) is arithmetic if there are a semisimple algebraic group \(\mathbf H\) defined over \(\mathbb Q\) and a surjective homomorphism \(\pi:\mathbf H(\mathbb R)^\circ\to G\) with compact kernel such that \(\Gamma\) is commensurable with \(\pi(\mathbf H(\mathbb Z))\). Armand Borel and Harish-Chandra proved in 1962 that such \(\pi(\mathbf H(\mathbb Z))\) are lattices.

When \(k\) is totally disconnected, such as \(\mathbb Q_p\), the only continuous homomorphism from the connected group \(G\) is trivial, so superrigidity says that \(\rho(\Gamma)\) is relatively compact.

From superrigidity to arithmeticity

Embed \(\Gamma\) in \(G\subset\mathrm{GL}(N,\mathbb R)\). One can arrange (by local rigidity) that the matrix entries of \(\Gamma\) generate a finitely generated field \(K\). Then:

1. If \(K\) contained a transcendental number, an automorphism of \(\mathbb C\) moving it would give a representation of \(\Gamma\) with unbounded image that does not extend to \(G\), contradicting superrigidity. So \(K\) is a number field.
2. For each prime \(\mathfrak p\) of \(K\), the embedding \(\Gamma\to\mathrm{GL}(N,K_{\mathfrak p})\) has bounded image, since \(K_{\mathfrak p}\) is totally disconnected. Bounded at every prime means the entries are algebraic integers, up to finite index.
3. For each non-identity archimedean embedding \(\sigma\) of \(K\), the image \(\sigma(\Gamma)\) is either bounded or comes from \(G\). Collecting these, \(\Gamma\) sits as a lattice in the real points of the restriction of scalars of a \(K\)-group, with compact factors for the bounded embeddings. This is the definition of arithmetic.

Idea of the proof of superrigidity

Margulis’s proof was inspired by George Mostow’s strong rigidity theorem (1968) and its use of boundary maps. Let \(P\subset G\) be a minimal parabolic subgroup, so that \(G/P\) is the Furstenberg boundary. Suppose \(\rho(\Gamma)\) is Zariski dense in a simple group \(H\) and unbounded. Because \(P\) is amenable, there is a \(\Gamma\)-equivariant measurable map \(\varphi:G/P\to H/Q\) for a proper parabolic \(Q\subset H\). Higher rank enters here: \(G/P\) fibres over many smaller boundaries \(G/P'\), and ergodicity of the action of lattices on these fibrations (a consequence of the Howe–Moore theorem) forces \(\varphi\) to agree almost everywhere with a rational map. A rational equivariant map between boundaries can be integrated to a homomorphism \(G\to H\) extending \(\rho\). In rank one there is only one boundary and no fibrations, and the argument has nothing to grip.

Rank one

The hypothesis is necessary. Surface groups have uncountably many inequivalent embeddings in \(\mathrm{PSL}(2,\mathbb R)\), and many lattices there are not arithmetic. Mikhail Gromov and Ilya Piatetski-Shapiro constructed non-arithmetic lattices in \(\mathrm{SO}(n,1)\) for every \(n\), and Mostow and Pierre Deligne found some in \(\mathrm{SU}(2,1)\) and \(\mathrm{SU}(3,1)\). Superrigidity, and hence arithmeticity, was later extended by Kevin Corlette (for \(\mathrm{Sp}(n,1)\)) and by Gromov and Richard Schoen (for the exceptional rank-one group) using harmonic maps.

The normal subgroup theorem

For the general reader. A group can often be simplified by dividing out a normal subgroup, much as one reduces integers modulo \(n\). For a higher-rank lattice, Margulis showed that this produces nothing new: every quotient is either finite or essentially the whole group. The proof combines two opposite properties of groups, amenability and Kazhdan’s property (T), and shows that the quotient must have both.

Two properties

A countable group \(\Lambda\) is amenable if it has a Følner sequence: finite sets \(F_n\) with \(|gF_n\,\triangle\,F_n|/|F_n|\to0\) for every \(g\). Abelian and solvable groups are amenable; free groups are not. A locally compact group has property (T), introduced by Kazhdan in 1967, if there are a compact set \(S\) and \(\varepsilon>0\) such that every unitary representation with a unit vector \(v\) satisfying \(\|sv-v\|<\varepsilon\) for all \(s\in S\) has a non-zero invariant vector. Kazhdan proved that \(\mathrm{SL}(n,\mathbb R)\) for \(n\ge3\) has (T), that lattices inherit (T) from the ambient group, and that groups with (T) have finite abelianization.

Structure of the proof

Suppose \(N\) is not central and set \(\Lambda=\Gamma/N\). Margulis proved two things.

1. \(\Lambda\) is amenable. This is the measure-theoretic half. Margulis proved a factor theorem: every \(\Gamma\)-equivariant measurable quotient of the boundary \(G/P\) is of the form \(G/P'\) for a parabolic \(P'\supseteq P\). Using it, he showed that any affine action of \(\Lambda\) on a compact convex set has a fixed point, which characterizes amenability.
2. \(\Lambda\) has property (T). When every simple factor of \(G\) has (T), which holds for all simple groups of real rank at least 2, \(\Gamma\) has (T) by Kazhdan’s theorem, and (T) passes to quotients. Products with rank-one factors require a separate argument.

Then \(\Lambda\) is finite by the following lemma.

Proof. Let \(\Lambda\) act on \(\ell^2(\Lambda)\) by translation, and let \(v_n=1_{F_n}/|F_n|^{1/2}\) for a Følner sequence. Then \(\|gv_n-v_n\|^2=|gF_n\triangle F_n|/|F_n|\to0\), so there are almost invariant unit vectors. By (T) there is a non-zero invariant vector in \(\ell^2(\Lambda)\). An invariant function is constant, and a non-zero constant is square-summable only if \(\Lambda\) is finite. \(\square\)

Expanders and invariant means

For the general reader. An expander is a network in which every node has only a few links, yet no group of nodes is poorly connected to the rest. Random networks have this property, but engineers need explicit ones they can build. In 1973 Margulis gave the first explicit family, using the rigidity of matrix groups. The same property of groups solved a 1920s question about measuring sets on spheres.

Expansion

For a finite \(d\)-regular graph \(X\) on \(n\) vertices, the edge expansion is \(h(X)=\min_{|A|\le n/2}|\partial A|/|A|\), where \(\partial A\) is the set of edges leaving \(A\). A family of \(d\)-regular graphs with \(n\to\infty\) is an expander family if \(h\) is bounded below by a fixed \(\epsilon>0\). If \(\lambda_2\) is the second largest eigenvalue of the adjacency matrix, the discrete Cheeger inequalities give

so expansion is equivalent to a uniform spectral gap. Mark Pinsker had shown by counting that random regular graphs are expanders, but no explicit family was known.

Proof. Let \(\varepsilon\) be a Kazhdan constant for \(S\). The group \(\Gamma\) acts on \(\ell^2_0(\Gamma/N_i)\), the functions of mean zero, without invariant vectors, so every \(v\) has some \(s\in S\) with \(\|sv-v\|\ge\varepsilon\|v\|\). Given \(A\) with \(|A|\le n/2\), take \(v=1_A-\tfrac{|A|}{n}\mathbf 1\). Then \(\|v\|^2=|A|(1-|A|/n)\ge|A|/2\) and \(\|sv-v\|^2=|sA\triangle A|=2|sA\setminus A|\). Each \(a\in A\) with \(sa\notin A\) gives an edge leaving \(A\), so

Hence \(h\ge\varepsilon^2/4\) for every \(i\). \(\square\)

The Margulis graphs

In the same paper Margulis gave a second family, on finite tori. In the symmetric form now usually taught, following the simplification by Gabber and Galil, the vertex set is \(\mathbb Z_m\times\mathbb Z_m\), and \((x,y)\) is joined to

an 8-regular graph. The proof uses the relative property (T) of the pair \((\mathrm{SL}(2,\mathbb Z)\ltimes\mathbb Z^2,\mathbb Z^2)\). Margulis’s argument gave no explicit constant; Ofer Gabber and Zvi Galil obtained one in 1981 by Fourier analysis on the torus. Expanders are now used throughout computer science, in error-correcting codes, derandomization and network design. In 1982 Margulis also gave explicit families of graphs of large girth, again from arithmetic groups.

Ramanujan graphs

The Alon–Boppana bound says that \(\lambda_2\ge2\sqrt{d-1}-o(1)\) in any large \(d\)-regular family. Graphs with all non-trivial eigenvalues of absolute value at most \(2\sqrt{d-1}\) are called Ramanujan graphs. In 1988 Margulis, and independently Alexander Lubotzky, Ralph Phillips and Peter Sarnak, constructed infinite families of them as Cayley graphs of \(\mathrm{PGL}(2,\mathbb F_q)\), using the Ramanujan conjecture proved by Pierre Deligne.

The Ruziewicz problem

Stefan Ruziewicz asked whether Lebesgue measure is the only finitely additive, rotation-invariant measure on the Lebesgue measurable subsets of the sphere \(S^n\) with total mass 1. Stefan Banach had shown that the answer is no for the circle. In 1980 Margulis, and independently Dennis Sullivan, proved uniqueness for \(n\ge4\), by finding a finitely generated subgroup of \(\mathrm{SO}(n+1)\) with a spectral gap on \(L^2_0(S^n)\), built from arithmetic groups with property (T). Vladimir Drinfeld settled \(n=2,3\) in 1984. Margulis also treated the analogous question for Euclidean spaces.

The Oppenheim conjecture

For the general reader. Take an expression like \(x^2+y^2-\sqrt2\,z^2\) and plug in whole numbers. Can the result be made as close to zero as you like without all three numbers being zero? Alexander Oppenheim conjectured in 1929 that the answer is yes whenever the expression is genuinely irrational and takes both signs. For nearly sixty years number theorists could prove it only with many variables. Margulis proved it in 1986 by studying how a group of symmetries moves lattices around, and with that proof created the field now called homogeneous dynamics.

Oppenheim conjectured this for \(n\ge5\) and Harold Davenport extended the conjecture to \(n\ge3\). Analytic methods had established it for large \(n\). Shrikrishna Dani and Margulis then showed that the values of \(Q\) at primitive integer vectors are dense in \(\mathbb R\).

Reduction to dynamics

It suffices to treat \(n=3\), since one can restrict to a suitable rational 3-dimensional subspace. Let \(X_3=\mathrm{SL}(3,\mathbb R)/\mathrm{SL}(3,\mathbb Z)\) be the space of unimodular lattices in \(\mathbb R^3\). Write \(Q=c\,Q_0\circ g\) with \(Q_0(x)=2x_1x_3-x_2^2\) and \(g\in\mathrm{SL}(3,\mathbb R)\); let \(H=\mathrm{SO}(Q_0)^\circ\), a group locally isomorphic to \(\mathrm{SL}(2,\mathbb R)\). For \(h\in H\) and \(v\in\mathbb Z^3\), \(Q_0(hgv)=Q_0(gv)=Q(v)/c\).

If the orbit \(Hg\mathbb Z^3\) is not relatively compact, Mahler gives \(h\in H\) and \(v\ne0\) with \(hgv\) tiny, so \(|Q(v)|=|c\,Q_0(hgv)|\) is tiny. Margulis proved that every relatively compact \(H\)-orbit on \(X_3\) is compact. A compact orbit means \(H\cap g\mathrm{SL}(3,\mathbb Z)g^{-1}\) is a lattice in \(H\); by the Borel density theorem it is then Zariski dense in \(H\), which forces \(Q\) to be proportional to a rational form. So for irrational \(Q\) the orbit is unbounded and small values exist.

The proof that relatively compact orbits are compact studies the unipotent subgroups of \(H\). Unipotent motions are slow, polynomial in time, so nearby orbits drift apart only in controlled directions. Margulis exploited this to show that a minimal closed invariant set must be invariant under a larger group, and then must be a whole compact orbit.

Unipotent flows

Margulis had already proved in 1971, answering a question of Piatetski-Shapiro, that unipotent orbits on \(X_n\) never escape to infinity: for a unipotent one-parameter group \(u_t\) and any \(x\in X_n\), the orbit \(\{u_tx:t\ge0\}\) does not diverge, so it returns to a fixed compact set at arbitrarily large times. Dani later showed that it spends most of its time in a suitable compact set. Madabhushi Raghunathan had conjectured more: that orbit closures of unipotent groups are always homogeneous. Dani and Margulis proved cases of this conjecture, and Marina Ratner proved it in full in 1990–1991.

Margulis and Georgi Tomanov later gave a different proof of Ratner’s measure classification using entropy.

Diophantine approximation and affine actions

For the general reader. After Oppenheim, Margulis and his students turned the dynamics of lattices into a general tool for number theory. Two highlights are a proof that typical points on curved surfaces cannot be approximated unusually well by rationals, and an exact count of integer points where a quadratic form takes values in a given range. A separate line of work produced strange flat spacetimes that now carry his name.

Approximation on manifolds

By Dirichlet’s theorem, for every \(y\in\mathbb R^n\) there are infinitely many \(q\in\mathbb Z^n\) and \(p\in\mathbb Z\) with \(|q\cdot y+p|<\|q\|^{-n}\). Call \(y\) very well approximable if for some \(\varepsilon>0\) there are infinitely many solutions of \(|q\cdot y+p|<\|q\|^{-n-\varepsilon}\). Almost every \(y\in\mathbb R^n\) is not very well approximable. Kurt Mahler conjectured that the same holds for almost every point of the curve \((x,x^2,\dots,x^n)\); Vladimir Sprindžuk proved this in 1964 and conjectured an extension to all suitably curved manifolds.

The proof translates approximation into the behaviour of orbits \(g_tu_{f(x)}\mathbb Z^{n+1}\) on the space of lattices and proves a quantitative version of non-divergence: the proportion of \(x\) for which the orbit enters a neighbourhood of infinity is small, with explicit polynomial bounds.

Counting values of quadratic forms

The answer is what the volume predicts. For signatures \((2,1)\) and \((2,2)\) the statement can fail, because of forms very well approximated by rational ones.

Margulis spacetimes

In 1977 John Milnor asked whether a group acting properly discontinuously by affine transformations on \(\mathbb R^n\) must be virtually solvable. In 1983 Margulis answered no: there are free groups of rank 2 acting properly on \(\mathbb R^3\) by affine maps whose linear parts lie in \(\mathrm{SO}(2,1)\). The quotients, flat Lorentzian 3-manifolds with free fundamental group, are now called Margulis spacetimes. His tool was an invariant \(\alpha(g)\) measuring the signed Lorentzian translation of an affine map along its invariant line; a necessary condition for properness is that \(\alpha\) not take both signs on the group. The Auslander conjecture, that groups acting properly and cocompactly are virtually solvable, remains open in general.

From Bonn to Yale, 1979–2020

In 1979, as Soviet restrictions on academic travel eased slightly, Margulis received permission to go abroad and spent three months at the University of Bonn. For most of the 1980s he remained at the Institute for Problems of Information Transmission. It was there that he proved the Oppenheim conjecture in 1986. The first announcement appeared in the Comptes Rendus of the Paris Academy in 1987, with a full account in 1989.

Between 1988 and 1991, as the Soviet Union opened, he made long visits to the Max Planck Institute for Mathematics in Bonn, the Institut des Hautes Études Scientifiques, the Collège de France, Harvard and the Institute for Advanced Study in Princeton. In 1990 he gave an invited address at the International Congress of Mathematicians in Kyoto on the dynamics of subgroup actions on homogeneous spaces and their applications to number theory. In 1991 he published his monograph Discrete Subgroups of Semisimple Lie Groups in Springer’s Ergebnisse series, still the standard reference on rigidity and arithmeticity, and he accepted a professorship at Yale University, where he later held the Erastus L. De Forest chair and is now emeritus. He returned to the Institute for Advanced Study as a member in 2006.

The Yale school

At Yale Margulis built a school in homogeneous dynamics and its applications. The Mathematics Genealogy Project lists 22 doctoral students, among them Dmitry Kleinbock (1996), Hee Oh (1997), Emmanuel Breuillard (2004), Mikhail Ershov (2005), Alireza Salehi Golsefidy (2006) and Amir Mohammadi (2009); his earliest student, Gregory Soifer, completed his degree in Minsk in 1979. Many of his later results were joint work with students and collaborators: with Kleinbock on Diophantine approximation, with Alex Eskin and Shahar Mozes on quantitative Oppenheim, with Elon Lindenstrauss on effective versions for ternary forms (2014), with William Goldman and François Labourie on proper affine actions, and with Herbert Abels and Gregory Soifer on the Auslander conjecture. Hee Oh has described him as one of the most original mathematicians she has met.

Honours and influence

Honours

• Fields Medal (1978), Helsinki, for his work on discrete subgroups of Lie groups.
• Medal of the Collège de France (1991) and honorary membership of the American Academy of Arts and Sciences (1991).
• Humboldt Research Award (1995) and the Lobachevsky Prize of the Russian Academy of Sciences (1996).
• US National Academy of Sciences (2001); Fellow of the American Mathematical Society (2012).
• Wolf Prize in Mathematics (2005), shared with Sergei Novikov, for contributions to the theory of lattices and their applications to ergodic theory, representation theory, number theory, combinatorics and measure theory.
• AMS Colloquium Lectures (2009) and honorary membership of the London Mathematical Society.
• Abel Prize (2020), shared with Hillel Furstenberg, for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics. The award was announced on 18 March 2020; the Oslo ceremony planned for May was postponed because of the pandemic.

He is one of a handful of mathematicians to have received the Fields Medal, the Wolf Prize and the Abel Prize.

Influence

• Rigidity theory. Superrigidity and arithmeticity made higher-rank lattices into arithmetic objects. Robert Zimmer’s program on actions of such lattices on manifolds grew directly from Margulis’s methods.
• Homogeneous dynamics. The Oppenheim proof showed that orbit closures of subgroups on spaces of lattices answer number-theoretic questions. Ratner’s theorems, measure rigidity for diagonal actions and its work on the Littlewood conjecture, and the dynamics on moduli spaces of Alex Eskin and Maryam Mirzakhani all build on that template.
• Combinatorics and computer science. Explicit expanders and Ramanujan graphs started a line of work connecting group theory, number theory and theoretical computer science.
• Geometry. The Margulis lemma, Margulis tubes, the Margulis measure and Margulis spacetimes are standard objects in hyperbolic geometry, dynamics and Lorentzian geometry.

In 2022 his colleagues published Dynamics, Geometry, Number Theory: The Impact of Margulis on Modern Mathematics, a volume of surveys covering each of these directions.

Connections

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

Teachers and advisors

• Yakov Sinai 1970

Students

• Alireza Salehi Golsefidy 2006
• Amir Mohammadi 2009
• Dmitry Kleinbock 1996
• Emmanuel Breuillard 2004
• Gregory Soifer 1979
• Hee Oh 1997
• Mikhail Ershov 2005

Influenced by

• Atle Selberg
• Jacques Tits
• Dmitri Anosov
• George Mostow
• Ilya Piatetski-Shapiro
• Madabhushi Raghunathan

Influenced

• Maryam Mirzakhani
• Robert Zimmer

Colleagues and collaborators

• Alex Eskin
• Dennis Sullivan
• Jacques Tits
• Marina Ratner
• David Kazhdan 1968
• Elon Lindenstrauss
• François Labourie
• Georgi Tomanov
• Herbert Abels
• Hillel Furstenberg
• Shahar Mozes
• Shrikrishna Dani
• William Goldman

Also mentioned in this life

• Stefan Banach
• Arthur Cayley
• Pierre Deligne
• Peter Gustav Lejeune Dirichlet
• Vladimir Drinfeld
• Joseph Fourier
• Évariste Galois
• Carl Friedrich Gauss
• Israel Gelfand
• Mikhail Gromov
• Harish-Chandra
• Alexander von Humboldt
• Andrey Kolmogorov
• Henri Lebesgue
• Sophus Lie
• Nikolai Lobachevsky
• John Milnor
• Max Planck
• Carl Ludwig Siegel
• Oscar Zariski

Fields

• lie groups 4 lives
• ergodic theory 7 lives
• homogeneous dynamics 3 lives
• number theory 45 lives
• combinatorics 9 lives

On CoursesHub

• Lie Groups & Algebras · Mathematics for Physics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Gregori Aleksandrovic Margulis, MacTutor History of Mathematics.
2. Abel Prize, Biography of Gregory Margulis (2020).
3. Grigory Margulis, Wikipedia.
4. Yale News, 2020 Abel Prize in mathematics goes to Yale’s Gregory Margulis, 18 March 2020.
5. Institute for Advanced Study, 2020 Abel Prize to be shared by Gregory Margulis.
6. London Mathematical Society, Honorary membership citation.
7. Gregory Aleksandrovic Margulis, Mathematics Genealogy Project.
8. J. Tits, The work of Gregori Aleksandrovitch Margulis, Proceedings of the ICM, Helsinki 1978.

Mathematics

1. G. A. Margulis, On Some Aspects of the Theory of Anosov Systems, Springer, 2004 (the 1970 thesis, with a survey by R. Sharp).
2. D. A. Kazhdan and G. A. Margulis, A proof of Selberg’s hypothesis, Mat. Sbornik 75 (1968).
3. G. A. Margulis, Explicit constructions of concentrators, Problemy Peredachi Informatsii 9 (1973).
4. G. A. Margulis, Discrete groups of motions of manifolds of nonpositive curvature, Proceedings of the ICM, Vancouver 1974.
5. G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer, 1991.
6. G. A. Margulis, Oppenheim conjecture, in Fields Medallists’ Lectures, World Scientific, 1997.
7. D. Kleinbock and G. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Annals of Math. 148 (1998).
8. A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Annals of Math. 147 (1998).
9. D. Fisher, Superrigidity, arithmeticity, normal subgroups: results, ramifications and directions, arXiv:2003.02956.
10. Survey of Margulis’s contributions to homogeneous dynamics and number theory, hosted by A. Eskin (University of Chicago).
11. S. Hoory, N. Linial and A. Wigderson, Expander graphs and their applications, Bull. AMS 43 (2006).
12. D. Fisher, D. Kleinbock and G. Soifer (eds.), Dynamics, Geometry, Number Theory: The Impact of Margulis on Modern Mathematics, University of Chicago Press, 2022.
13. D. W. Morris, Introduction to Arithmetic Groups, arXiv:math/0106063.