• The Inria's Research Teams produce an annual Activity Report presenting their activities and their results of the year. These reports include the team members, the scientific program, the software developed by the team and the new results of the year. The report also describes the grants, contracts and the activities of dissemination and teaching. Finally, the report gives the list of publications of the year.

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## Section: New Results

### Can you hear the homology of 3-dimensional drums?

Participant : Aurel Page.

In [16], 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 $p$, there are 3-dimensional drums that sound the same but have different $p$-torsion in their homology. This completes previous work  [42] by proving that the behaviour observed by computer experimentation was indeed a general phenomenon.

More precisely: if $M$ is a manifold with an action of a group $G$, then the homology group ${H}_{1}\left(M,ℚ\right)$ is naturally a $ℚ\left[G\right]$-module, where $ℚ\left[G\right]$ denotes the rational group ring. Bartel and Page prove that for every finite group $G$, and for every $ℚ\left[G\right]$-module $V$, there exists a closed hyperbolic 3-manifold $M$ with a free $G$-action such that the $ℚ\left[G\right]$-module ${H}_{1}\left(M,ℚ\right)$ is isomorphic to $V$. They give an application to spectral geometry: for every finite set $P$ of prime numbers, there exist hyperbolic 3-manifolds $N$ and ${N}^{\text{'}}$ that are strongly isospectral such that for all $p\in P$, the $p$-power torsion subgroups of ${H}_{1}\left(N,ℤ\right)$ and of ${H}_{1}\left({N}^{\text{'}},ℤ\right)$ have different orders. They also show that, in a certain precise sense, the rational homology of oriented Riemannian 3-manifolds with a $G$-action "knows" nothing about the fixed point structure under $G$, 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.