General Group Actions Research Articles

Analytic torsion for group actions

The Reidemeister torsion is a classical topological invariant for nonsimply-connected manifolds [14]. Let M be a closed oriented smooth manifold with the fundamental group πλ(M) nontrivial. Let p:πx{M) -• O(N) be an orthogonal representation of nx(M). If the twisted real cohomology H*(M, Ep) vanishes, then one can define the Reidemeister torsion τp GR , which is a homeomorphism invariant of M. The original interest of τp was that it is not a homotopy invariant, and so can distinguish spaces which are homotopy equivalent but are not homeomorphic. Ray and Singer asked whether, as for many other topological quantities, one can compute τp by analytic methods [17], Given a Riemannian metric g on M, they defined an analytic torsion T' e R as a certain combination of the eigenvalues of the Laplacian acting on twisted differential forms. They showed that under the above acyclicity condition, T is independent of the metric g, and they conjectured that the analytic expression T equals the combinatorial expression τ . This conjecture was proven to be true independently by Cheeger [4] and Mϋller [15]. One can look at the above situation in the following way. The group nχ{M) acts freely on the universal cover M, and so one has an invariant for free group actions. A natural question is whether the Reidemeister torsion can be extended to an invariant for more general group actions. For a finite group acting (not necessarily freely) on a closed oriented PL manifold X, a Reidemeister torsion was defined algebraically by Rothenberg [19]. One can then ask whether there is a corresponding analytic torsion when X is smooth, and whether the analytic torsion equals the combinatorial torsion. In §11 we define the analytic torsion Tp for a finite group action and show that if the relevant cohomology groups vanish, then it is independent of the G-invariant metric used in its definition. (This was shown previously in unpublished work by Cheeger [5].) The analysis involved to

Orbit equivalence and rigidity of ergodic actions of Lie groups

AbstractThe rigidity theorem for ergodic actions of semi-simple groups and their lattice subgroups provides results concerning orbit equivalence of the actions of these groups with finite invariant measure. The main point of this paper is to extend the rigidity theorem on one hand to actions of general Lie groups with finite invariant measure, and on the other to actions of lattices on homogeneous spaces of the ambient connected group possibly without invariant measure. For example, this enables us to deduce non-orbit equivalence results for the actions of SL (n, ℤ) on projective space, Euclidean space, and general flag and Grassman varieties.

A pólya interpretation of the schur function

When the Schur function is written as a linear combination of products of symmetric power sums, it takes the form of a character-weighted cycle index polynomial. A Pólya-like interpretation is given to this formula and a purely combinatorial proof is given. Some observations concerning general group actions are made.

The Hausdorff Paradox for general group actions

A well-known criterion for the amenability of a locally compact group (or more generally a group action) is applied to demonstrate that analogues of the familiar “Hausdorff Paradox” exist if and only if the group (group action) is nonamenable. Moreover, related and stronger formulations are conjectured and discussed.

Real Borel cohomology of locally compact groups

enemy. Some of the theorems in the first two sections are modeled on those of Snapper, in [18], who considered these problems for everything discrete, the powerful techniques of homological algebra being available in this case. Unfortunately, there is no machinery of projective or injective resolutions immediately at our disposal for the cohomology theories based on Borel cochains. There is a type of resolution, which leans toward the injective, but its full homological character is not understood as yet. The entire idea of considering the cohomology of general group actions was inspired by the paper of Snapper. These things have been considered in the continuous case by Mostow in [17], where equivariant cohomology groups are defined and studied. Almost nothing being known about the higher cohomology groups, it was desirable to ferret out those isolated pockets of triviality. This is done in the last section, where it is shown that for G abelian locally compact separable metrizable, R the real line, G operating trivially on R, Hn(G, R), the bounded Borel groups defined in [16], do not depend upon the topology of G, nor its torsion. In fact, R seems to ignore everything about G except a certain torsion-free discrete group D associated to G. The further fact that these groups are isomorphic to the group of alternating multilinear functions An (G, R) was suggested by the results of Kleppner on H2(G, T), where G is locally compact abelian and T the circle group. He showed that for all G except a certain intractable class of 2-primary groups, whose fate is yet undecided, that H2(G, T)=L2(G, T)/S2(G, T), the continuous bihomomorphisms modulo the symmetric ones. We have not resolved this open question,