Symmetric spaces play a central role in geometry, analysis, and arithmetic, due to their inherent rich and beautiful structure. A key question is the fascinating interplay between spectral properties of their Laplace operators on the one hand, and geometric, topological, and arithmetic properties on the other.

Analytic torsion, and L^{2}-analytic torsion, are refined spectral invariants which
contain (partly conjecturally) a lot of deep, also arithmetic information. Other approaches
use ergodic theory and what is called measurable group theory.

The summer school aims to provide an introduction to this highly active area, with four lecture series and one special talk as follows:

## Lecture series:

Various recent works show that certain arithmetic groups — that generalize the modular group —
can have ‘a lot’ of torsion in their homology. Among these groups are the finite index
(congruence) subgroups of SL_{3}(ℤ) or SL_{2}(ℤ[i]).
In the latter case, homology reduces to abelianization. In particular, for

_{0}(N) = { (

a | b |

c | d |

_{2}(ℤ[i]) : N | c } where N ∈ ℤ[i],

one may ask about the structure of the finitely generated ℤ-module
Γ_{0}(N)^{ab}
= Γ_{0}(N) / [Γ_{0}(N) : Γ_{0}(N)].
It has a finite torsion part Γ_{0}(N)^{ab}_{tors} and Akshay Venkatesh and I
have conjectured that, as N tends to infinity among primes, we have:

lim |

N → ∞ |

log |Γ_{0}(N)^{ab}_{tors}| |

|N|^{2} |

λ |

18π |

_{ℚ[i]}) = 1 –

1 |

9 |

1 |

25 |

1 |

49 |

The right hand side turns out to be related to the L^{2}-torsion of the hyperbolic 3-space.
More generally, one may ask: * How does the amount of torsion in the homology of an arithmetic group
grow with the level N? *
The study of L^{2}-torsion quite naturally leads to a conjectural partial answer.
In my lectures, I will present the relation between homology torsion and L^{2}-torsion and
discuss recent progress towards the general conjecture.

Measured Group Theory studies groups through their measurable actions. It investigates classification, rigidity and invariants of (actions of) discrete countable or locally compact groups and the ways these are encoded in the orbit structures of the actions.

The subject has roots in ergodic theory and its connections with operator algebras. Some milestones are Ornstein-Weiss' theorem on amenability and Zimmer's work on super-rigidity for lattices in higher rank Lie groups.

The subject gained a new trend when Gromov suggested measured group theory as a relative to geometric group theory and as a common framework to better understand the family of (cocompact and non-cocompact) lattices in a fixed Lie group.

In the last 20 years, measured group theory has known dramatic developments. It has also established new and sometimes quite unexpected connections between various areas, including arithmetic groups, geometric group theory, von Neumann algebras, asymptotic group theory, Borel dynamics and combinatorics, percolation and graph convergence.

Arithmetic groups are, roughly speaking, groups of matrices with integer entries. For example,
the groups SL_{4}(ℤ) and SL_{2}(ℤ[i]) are
arithmetic groups, and as such they are lattices in the associated simple Lie groups
SL_{4}(ℝ) and SL_{2}(ℂ), respectively.

It is a beautiful theorem of Margulis that in a higher-rank simple Lie group, all lattices are in fact arithmetically defined. Moreover, arithmetic subgroups of semi-simple Lie groups act on symmetric spaces, which makes it possible to study their cohomology as the cohomology of a locally symmetric space. This gives rise to a nice interplay between group theory, geometry and number theory.

In this course, we introduce arithmetic groups and discuss basic properties and examples. We introduce the associated locally symmetric spaces and discuss how much cohomology arithmetic groups can asymptotically have.

Starting with the work of Bergeron and Venkatesh, analytic torsion has turned out to be a very useful analytic tool to study torsion in the cohomology of locally symmetric spaces, which are defined as quotients of symmetric spaces by arithmetic groups. This is of great interest, because the Langlands program conjecturally relates cohomology of arithmetic groups to Galois representations. An emerging extension of this program predicts that such a correspondence exists not only for the characteristic zero cohomology, but also for the torsion part.

State-of-the-art techniques to study torsion in the cohomology of locally symmetric spaces
are based on the approximation of L^{2}-torsion of a locally symmetric space by the
renormalized logarithm of the analytic torsions for a tower of finite coverings. With a few
exceptions, the existing analytic methods are restricted to the case of compact locally symmetric
spaces. Since many arithmetic groups are non-compact, it is an important problem to develop the
analytic tools for non-compact locally symmetric spaces of finite volume.

In this course, I will give an introduction to the subject, explain some of the main techniques and results, discuss open problems and give an outlook on possible future developments.

## Special talk:

The cohomology of arithmetic groups is a module under the action of the Hecke
algebra. This module is built up from irreducible Hecke modules *π _{f}*,
and to these irreducible Hecke modules we can attach

*L*-functions

*L*(

*π*) which are holomorphic in the complex variable

_{f},r,s*s*provided

*s*≫ 0. In some favourable cases, it can be shown that these

*L*-functions have meromorphic or even holomorphic continuations into the entire complex plane.

The structure of the cohomology is influenced by the special values
*L*(*π _{f},r,ν*

_{0}) which these functions attain at certain critical arguments

*ν*

_{0}∈ ℤ. This has implications in two directions. Since the cohomology groups are finite-dimensional ℚ-vector spaces, we get rationality results for these critical values. On the other hand, the arithmetic of these values has influence on the structure of the integral cohomology groups.