Math 201A: Strong and local rigidity.

Winter 2017

 Lectures: T-Th 12:30 PM--1:50 PM APM 5829 Office Hour: Send me an e-mail.
Send me an e-mail, we can meet and discuss math (possibly in a coffee shop).

Course description:

The following is a tentative list of topics to be covered:

• Poincare's Recurrence theorem.
• Chevalley's theorem on closed subgroups of an algebraic group.
• Borel's density theorem.
• Selberg's proof of local rigidity of cocompact lattices in SL(n,R).
• Basic properties of the symmetric space X of a semisimple group G with no compact factors:
• They are CAT(0).
• Any compact subgroup of G fixes a point in X.
• Neighborhood of a convex subset of a CAT(0) space is convex.
• The distance function of points in a geodesic from a convex subset of a CAT(0) space is a convex function.
• Understanding flats and chambers.
• Maximal boundary of X: from metric point of view.
• The higher rank case of Mostow's strong rigidity.
• Along the way, we cover most of the needed materials on the structure of semisimple Lie groups. We might present some of the proofs only for SL(n,R).
• N-x C(A) x N+ decomposition.
• A version of Iwasawa decomposition.

Prerequisite:

will be kept to a minimum.

Resources:

We will mostly follow Mostow's book Strong rigidity of locally symmetric spaces. But I will be using other resources. I will post my lecture notes, and mention what resources have been used for each individual lecture.

Here are a few related references:

• M. S. Raghunathan, Discrete Subgroups of Lie Groups, Springer, New York, 1972.
• R. Zimmer, Ergodic theory and semisimple Lie groups.
• G. D. Mostow, Lectures on discrete subgroups of Lie groups, Tata Institute publication. Here is a link to this lecture note.
• G. A. Margulis, Discrete subgroups of semisimple Lie groups.
• A. Eskin, B. Farb, Quasi-flats and rigidity in higher rank symmetric spaces JAMS 10, no 3, (1997) 653-692.
• R. J. Spatzier, An invitation to rigidity theory.
• S. G. Dani, A simple proof of Borel's density theorem, Mathematische Zeitschrift, 174, no 1, (1980) 81-94. (This article has a mistake, nevertheless it has the right ideas.)
• M. R. Bridson, A. Hafliger, Metric Spaces of Non-Positive Curvature.

Notes related to lectures:
• Lecture 1:

defined what a lattice is; recalled Zariski-topology; mentioned Borel-Harish-Chandra theorem, and saw a few examples; mentioned examples coming from fundamental groups of a finite volume hyperbolic manifold; constructed a cocompact (arithmetic) lattice of SL(n,R); mentioned cocompactness criteria for arithmetic groups; mentioned what local rigidity means; showed that, if a group is locally rigid in SL(n,R), then after conjugation all of its entries are algebraic numbers.

Here is my note for lecture 1.

• Lecture 2:

defined what a unipotent flow is; proved that SL(n,R) is generated by its unipotent subgroups; Proved Poincare recurrence theorem; proved Borel's density theorem using Chevalley's theorem.

Here is my note for lecture 2. (I used the above mentioned article by Dani.)

• Lecture 3:

Proved Chevalley's theorem; along the way defined ring of regular functions of a Zariski-closed subset of an affine space (We do not do the justice to algebraic groups to save time for other topics); defined the wedge powers of a vector space and mentioned its basic (needed) properties.

Here is my note for lecture 3. (Any standard book on linear algebraic groups, e.g. Humphreys' or Borel's or Springer's.)

• Lecture 4:

mentioned the outline of Selberg's proof of local rigidity of cocompact lattices of SL(n,R) for n>2:

• Step 1: trace rigidity implies local rigidity.
• Step 2: trace rigidity of R-regular elements implies local rigidity.
• Step 3: assuming Step 5, ratio of log of eigenvalues of an R-regular element is constant along a deformation.
• Step 4: assuming Step 5, eigenvalues of an R-regular element are preserved along a deformation.
• Step 5: local deformation gives us an isomorphic cocompact lattice.
Step 1 is proved; R-regularity is defined and we started exploring the properties of the set of R-regualr elements.

Here is my note for lecture 4. (The above mentioned article by Selberg, and Mostow's Lectures at Tata Institute.)

• Lecture 5:

formulated a statement regarding R-regular elements; proving NX D X N+ is diffeomorphic to a Zariski-open subset of SL(n,R) (with concrete relations for its complement); proving openness of the set of R-regular elements; along the way we discussed how to think about tangent bundle of an algebraic group, and compute differential of a morphism.

Here is my note for lecture 5. (The above mentioned Mostow's Lectures at Tata Institute.)

• Lecture 6:

proved a proper subcone of positive Weyl chamber times a small ball consists of R-regular elements, and a bit more.

Here is my note for lecture 6.

• Lecture 7:

proved that a typical traslate of large powers of an R-regular element are R-regular; found lots of R-regular elements in a lattice; proved that to get trace rigidity it is enough to show it for R-regular elements of a lattice.

Here is my note for lecture 7.

• Lecture 8:

We studied: centralizer of elements of a cocompact lattice; centralizer of an R-regular element of a cocompact lattice in the lattice; preserving the chambers in a flat; linear rigidity of the combinatorial structure of chambers in an apartment;

Here is my note for lecture 8.

• Lecture 9:

Either eigenvalues of an R-regular element is presrved under the deformation or all the Weyl chamber walls are Gamma-compact; getting a deformation in SL(2) by going to neighbors;

Here is my note for lecture 9.

• Lecture 10:

Proved that the mentioned deformations in SL(2) scale all the traces of R-regular elements by the same constant; Used trace equalities to get the triviality of these deformations; Summarized the whole proof;

Here is my note for lecture 10.

• Lecture 11:

Defined the Riemannian metric on P(n) and defined the symmetric space of a semisimple Lie group with no compact factors;

Here is my note for lecture 11.

• Lecture 12:

Proved that X is a unqiue geodesic space; recognized the geodesics passing through I; proved that the Euclidean length of the log of a curve C in P(n) is larger than the Riemannian length of C; proved Cos law inequality;

Here is my note for lecture 12.

• Lecture 13:

flats; sum of angles in triangles; defined CAT(0), and proved convexity of the distant funtion of points of a geodesic from a convex set in a CAT(0) space; proved convexity of a neighborhood of a convex set;

Here is my note for lecture 13.

• Lecture 14:

Proved that P(n) is CAT(0); defined, and proved the existence and the uniquness of the center of mass of a bounded subset of X which has positive volume; proved that any compact subgroup of G fixes a point in X; proved that any two maximal compact subgroups of G are conjugate; understood the connection between flats and polar subgroups (connected component of R-split tori) of G;

Here is my note for lecture 14.

• Lecture 15:

Reviewed a little bit of sturcture theory of semisimple Lie groups: root systems; Weyl chambers; parabolic elements; Proved parts of Iwasawa decomposition; defined the maximal boundary of X.

Here is my note for lecture 15.

• Lecture 16:

Proved that the maximal boundary is homeomorphic to G/P; studied the displacement function of elements of G.

Here is my note for lecture 16.

• Lecture 17:

Quasi-isometric embedding is defined; The Svarc-Milnor lemma is proved; the stronger needed version is formulated.

Here is my note for lecture 17.

• Lecture 17, 18:

The existence of a Gamma-equivariant QI embedding is proved; Uniform injectivity radius for compact locally symmetric non-positive curvature manifolds is proved; space of flats is introduced and it is proved that the set of Gamma-compact flats is dense.

Here is my note for lecture 17 and 18.

• Lecture 19:

The equality of R-ranks is proved; there is a unique Gamma_2-compact flat in a bounded distance from the image of a Gamma_1-compact flat.

Here is my note for lecture 19.

• Lecture 20:

For any flat F, there is a unique flat in a bounded distance from phi(F), and this defines a homeomorphism between the spaces of flats; why the space of flats is important: rigidity of Tits Geometry.

Here is my note for lecture 20.

• Lecture 21:

What splices are; getting a map from the space of splices to the space of splices; getting a map between the maximal boundaries; proving that this map preserves the ordering and so we get an isomorphism between the Tits Geometries, which finishes the proof of Mostow rigidity for higher rank simple groups.

Here is my note for lecture 21.