#
Volume entropy rigidity

June 15 to June 19, 2009
at the

American Institute of Mathematics,
Palo Alto, California

organized by

Marc Burger,
Francois Ledrappier,
Seonhee Lim,
and Gabriele Link

## Original Announcement

This workshop will be devoted to
volume entropy rigidity for higher rank symmetric spaces and piecewise
Riemannian manifolds such as buildings and polyhedral complexes. We
want to bring together mathematicians from different related fields
such as differential geometry, ergodic theory and
geometric group theory to attack the problem of characterizing the
"symmetric" metrics
by normalized volume entropy.
For any piecewise Riemannian manifold the volume entropy is
defined as the exponential growth rate of volumes of balls in the
universal cover.
It seems to be a rather coarse asymptotic invariant, but it is related
to Gromov's
simplicial volume, the bottom of the spectrum of Laplacian, the
Cheeger isoperimetric constant, the growth of fundamental groups, etc.
The entropy rigidity conjecture due to Gromov and Katok states that
among all Riemannian metrics on a closed Riemannian manifold of
non-positive curvature the locally symmetric metric minimizes the
normalized entropy. This conjecture was first shown by Katok for
surfaces. Later, Besson, Courtois and Gallot proved that for a
manifold which carries a rank one symmetric metric the normalized
entropy is minimal if and only if the metric is rank one symmetric.
The conjecture is still open for higher rank symmetric spaces,
although some work has already been carried out in this direction by
Connell and Farb.

Remarkably enough, not much attention has been paid so far to the
natural question whether there exists an entropy rigidity for singular
spaces. Besides analyzing the question of volume entropy rigidity in
higher rank symmetric spaces, one of our goals in this workshop is to
consider buildings, hyperbolic or Euclidian, which are piecewise
Riemannian manifolds and yet
have ``a lot of symmetries". In this case the combinatorics of the
space can give us some information about the volume entropy and its
rigidity.

Related problems such as the question when Liouville measure,
harmonic measure and Bowen-Margulis measure coincide will also be addressed.

We hope that the collaboration and exchange of ideas in the frame of
the workshop will lead to an improved understanding of volume entropy
rigidity.

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.