Various aspects of the geometry and the thermodynamics of black holes in gauged supergravities are explored. In such theories, where the gauge coupling or the cosmological constant can be thought of as an integration constant arising from a higher-dimensional origin, it becomes appropriate to think of it as an additional thermodynamic variable associated with a pressure. The conjugate variable defines a volume for the black hole, although its geometric interpretation becomes quite subtle if the black hole is rotating. Further geometric properties of the black holes are also explored, including an intriguing universal structure for the product of the horizon areas, which is suggestive of a possible dual field theory explanation for the microscopic entropy. Isoperimetric and hoop inequalities are also discussed.

The global symmetry groups that result from compactifying eleven-dimensional supergravity on an n-dimensional torus play a central role in our understanding of U-dualities in string and M-theory. The mechanism leading to the En Lie algebra upon compactification on Tn is well-known for n less than 9, but the situation for n=9, corresponding to the compactification to 2 dimensions, has been described much less clearly in the literature. We give an elementary, and completely explicit, description of the infinite-dimensional symmetries of all symmetric-space coset models in 2-dimensional gravitational backgrounds, including symmetries of both Kac-Moody and Virasoro type.