# 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 |
## Contents |

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 Topology Extending the action Cocompactness |

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. |

### References

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.