Section: New Results
Can you hear the homology of 3-dimensional drums?
Participant : Aurel Page.
In , A. Bartel and A. Page describe all possible actions of groups of automorphisms on the homology of 3-manifolds, and prove that for every prime , there are 3-dimensional drums that sound the same but have different -torsion in their homology. This completes previous work  by proving that the behaviour observed by computer experimentation was indeed a general phenomenon.
More precisely: if is a manifold with an action of a group , then the homology group is naturally a -module, where denotes the rational group ring. Bartel and Page prove that for every finite group , and for every -module , there exists a closed hyperbolic 3-manifold with a free -action such that the -module is isomorphic to . They give an application to spectral geometry: for every finite set of prime numbers, there exist hyperbolic 3-manifolds and that are strongly isospectral such that for all , the -power torsion subgroups of and of have different orders. They also show that, in a certain precise sense, the rational homology of oriented Riemannian 3-manifolds with a -action "knows" nothing about the fixed point structure under , in contrast to the 2-dimensional case. The main geometric techniques are Dehn surgery and, for the spectral application, the Cheeger-Müller formula, but they also make use of tools from different branches of algebra, most notably of regulator constants, a representation theoretic tool that was originally developed in the context of elliptic curves.