**Brief description**. In this class we will study n-manifolds that are locally modeled on the hyperbolic n-space.

These locally symmetric spaces are especially important in low dimensional geometry and topology (n=2 and 3) where they exhibit the highest degree of flexibility.

In higher dimensions (n>2), closed hyperbolic manifolds diplay, instead, a strong rigidity phenomenon (volume rigidity), which will be the one of the main goals of the class.

- Tuesday 11:15 - 12:50
- Wednesday 16:15 - 17:50

Notes from the class will be made available here, but the lectures will not be recorded.

- Basic topology (manifolds, fundamental group, covering spaces).
- Basic differential geometry (connection, curvature).

- Lecture 1: Introduction to the class.
- Lecture 2: Hyperboloid model of the hyperbolic space. Computation of the isometry group. Hyperplanes, k-Subspaces and geodesics.
- Lecture 3: Geodesics and path metric. The boundary at infinity. Classification of isometries: Elliptic, loxodromic and parabolic.
- Lecture 4: Geometry of loxodromic and parabolic motions. Horospheres and their properties. The Poincaré disk model, definition and first properties.
- Lecture 5: The Poincaré and the upper half space models. Normal forms of isometries.
- Lecture 6: The hyperbolic plane. Isometries and PSL_2(R). Triangles and computation of area. Existence and uniqueness of triangles with prescribed angles. Addendum
- Lecture 7: Definition of a hyperbolic manifold. Classification of surfaces, surfaces with punctures. Ideal triangulations. Construction of hyperbolic surfaces: Geometric ideal triangulations. Local charts around interior and edge points. Shear parameters. Referece: [T] see Section 3.9.
- Lecture 8: Geometric ideal triangulations. Review of the construction: Ideal triangles, basepoints, shear parameters, local charts. Completeness equations. Space of parameters.
- Lecture 9: Independence of the vetrex completeness equations and dimensions of the space of complete solutions. Metric completion of an incomplete gluing, local description: The completion is a surface with boundary. The Riemannian metric extends and the boundary is totally geodesic with respect to the extension. The natural extension of the distance coincides with path metric induced by the extended Riemannian metric. Computation of the length of the boundary with respect to the shear parameters. Explicit example of a gluing of two triangles: Heuristic description.
- Lecture 10: Review and comments on the completions of incomplete gluings. Exercise session: North-south dynamics of loxodromic motions. Ping-pong arguments for groups generated by a pair of loxodromic motions.
- Lecture 11: Exercise session: PSL(2,Z). Fundamental domain, identifications, volume. Modular curve. Congruence subgroups, index, loxodromics and systole (as minimal translation length of loxodromic motions).
- Lecture 12: The hyperbolic 3-space. Isometries in the upper half space model and PSL(2,C). Ideal tetrahedra: Classification via associated similarity classes of Euclidean triangles, dihedral angles and complex moduli. Finiteness of the volumes of n-simplices. Statement of the exact computation of the volume via Lobachevsky functions. References: [M] Section 13.1 for the exact computation of the volume of an ideal tetrahedron.
- Lecture 13: Uniqueness of ideal tetrahedra of maximal volume. Topology and geometry of ideal triangulations in dimension 3: Once the vertices are removed, every oriented gluing is homeomorphic to the interior of an oriented compact manifold with boundary. Dunfield-Thurston: Generically there is a component of the boundary which is a high genus oriented surface. Geometric ideal triangulations: They are locally modeled on the hyperbolic 3-space away from the edges. Problematic gluings at the edges, non-trivial shearing and total dihedral angles. Topological restrictions on the gluing if the total dihedral angle around each edge is 2\pi.
- Lecture 14: Review of the contructions of geometric ideal triangulations in dimension 3. Consistency equations and existence of an extension of the hyperbolic metric on the edges. Explicit expression in terms of the tetrahedra parameters and the combinatorics of the gluing. The space of solutions is either empty or has dimension equal to the number of vertices of the gluing (statement only, no proof). References: Chapter 14 of [M], Section 14.1, until 14.1.4. For an in-depth discussion of the space of solutions see Chapter 15 of [M], Section 15.2, until 15.2.5.
- Lecture 15+16: Completeness of ideal triangulations near the ideal vertices: Coherent choices of horosections and equivalent algebraic statement. An example: Full discussion for the figure-8 knot. References: [M] See Section 14.1.5 for the completeness equations and Section 15.1.4 for a discussion of the figure-8 knot. [BP] See Section E.6-i for a complete discussion on consistency and completeness equations.
- Lecture 17: Correspondence between complete hyperbolic manifolds and discrete, torsion free subgroups of the isometry group of hyperbolic space. Reference: [M] See Section 3.1.
- Lecture 18: Arithmetic constructions, main ingredients: Admissible quadratic forms, Selberg's Lemma and Mahler Compactness Criterion. References: [BP] See examples on pages 186-190.
- Lecture 19: Arithmetic constructions: End of the discussion of the case of the square root of 2 and general statement. Fundamental domains, Voronoi tassellations (see Sections 3.3.4-3.4.2 of [M]).
- Lecture 20: Malcev's Lemma and residual finiteness: Closed hyperbolic manifolds have many finite coverings. Dirichlet polyhedron for a discrete subgroup of isometries: The quotient manifold can be obtained as a quotient of a hyperbolic polyhedron with respect to an isometric pairing of its faces. Statement of Mostow Rigidity and road map for the proof.
- Lecture 21: Mostow Rigidity. First step of the proof: Construction of an equivariant quasi-isometry via a Milnor-Schwarz argument. Word metric on a finitely generated group. References: For the proof of the Milnor-Schwarz Lemma see Lemma 8.18 and Proposition 8.19 of [BH] (Part I.8).
- Lecture 22: Boundary extension of quasi-isometries. Stability of quasi-geodesics in hyperbolic n-space. References: See Proposition 1.6 and Theorem 1.7 of [BH] (Part III.H.1). For another approach see also Section C.1 of [BP] and Section 5.2 of [M].
- Lecture 23: Boundary extension of quasi-isometries: The extension is well-defined and provides a homeomorphism between the boundaries (see also Section C.1 of [BP] and Section 5.2 of [M]). If n is at least 3, a homeomorphism that sends the vertices of regular ideal simplices to vertices of regular ideal simplices is the restriction of an isometry.
- Lecture 24: Why the boundary map sends the vertices of regular ideal simplices to vertices of regular ideal simplices, heuristic picture: Use ideal (regular=maximal volume) triangulations+equality of the volumes of the quotients. Statement of the proportionality principle. Gromov norm on homology and simplicial volume of a closed orientable manifold: Definition and general properties. A simple construction of a homotopy equivalence between closed hyperbolic manifolds with isomorphic deck groups. References: For Gromov norm and simplicial volume see Section C.3 of [BP] and Sections 13.2.1-13.2.2 of [M]. For singular homology a standard reference freely available is Hatcher's book [H].
- Lecture 25: Gromov-Thurston proportionality principle: Statement of Haagerup-Munkholm. Volume of singular n-chains. Straightening in hyperbolic manifolds. The hyperbolic volume is a lower bound for the simplicial volume. Efficient representations of the volume and efficient cycles. Idea for the construction of an almost regular ideal representation of the fundamental class.
- Lecture 26: Construction of efficient cycles and end of the proof of the proportionality principle. End of the proof of Mostow rigidity: The boundary map must send regular ideal simplices to regular ideal simplices. References: [M] Section 13.2.6 and 13.2.2. [BP] Section C.4.

- [M]
*An introduction to geometric topology*, B. Martelli. Available here. - [BP]
*Lectures on hyperbolic geometry*, R. Benedetti and C. Petronio. - [T]
*Geometry and topology of 3-manifolds*, W. Thurston. Available here. - [BH]
*Metric spaces of non-positive curvature*, M. Bridson and A. Haefliger. - [H]
*Algebraic Topology*, A. Hatcher. Available here