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Compactifications of Moduli Spaces of Tori and Graphs

Course description

The quotient of a symmetric space by the action of a non-uniform lattice is not compact, by definition. However, there is a larger space, called the Borel-Serre bordification, whose quotient is compact and which has the same homotopy type. In the first part of this course we will construct this bordification in the special case of the action of SL(n,Z) on the symmetric space SL(n,R)/SO(n). This case illustrates the ideas while avoiding many of the technical difficulties of the general case.

The symmetric space SL(n,R)/SO(n) can also described as a space of marked flat tori, and from this point of view invites generalization to the space of marked metric graphs. The latter space is known as Outer space because it has a natural action of the group of outer automorphisms of a free group, and the analog of the Borel-Serre bordification for Out(F_n) was constructed by Bestvina and Feighn. We will explore this in the second part of the course.

Lecture Notes

Link to notes
Date of lecture
Lecture 1 January 19, 2016 The hyperbolic plane H^2
Compactifications of the SL(2,Z) quotient
Lecture 2 January 26 H^2 as moduli space of lattices, forms, graphs, flat tori, elliptic curves
Symmetric space for GL_n
Properties of the Borel-Serre bordification
Parabolic subgroups
Lecture 3 February 2 Boundary space e(P) for a parabolic P
Levi component of P
Geodesic action of A_P on symmetric space
Detailed construction and pictures for n=3
Lecture 4 February 9 Gluing the e(P) to X
Extending the action
Lecture 5 February 23 Bieri-Eckmann duality and cohomology with compact supports
Tits building paramaterizes BS boundary
Proof of Solomon-Tits theorem
Definition of Outer space and action
Combinatorial structure of Outer space
Lecture 6 March 1 Correction to Lec. 5: correspondence between Tits bldg and BS boundary
Closed cell associated to an open simplex of Outer space
Construction of Bestvina-Feighn bordification
Action of Out(F_n) is proper and cocompact
Lecture 7 March 8 Review of cohomology with compact supports
The simplicial bordification of Outer space
Definition of combinatorial Morse function
Reduction to local problem: connectivity of upper link
Lecture 8 March 15 Vertices of the bordification
Realizing the bordification as a deformation retract
Proof that the Morse function well-orders the vertices
The upper link and the star graph
Proof that the upper link is non-empty for all n
Comments   Remainder of proof that the upper link
is (2n-5)-connected is not contained in
these notes.


Borel, A. and Serre, J.-P (1973). Corners and arithmetic groups. Commentarii Mathematici Helvetici 4, 436-491.

Bestvina, M. and Feighn, M. (2000). The topology at infinity of Out(F_n), Inventionnes Math. 140 (3), 651-692.