By Alfred Barnard Basset

This quantity is made from electronic pictures from the Cornell college Library old arithmetic Monographs assortment.

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Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. 121 (1985), 169-186. [Sz1] Z. Szabo, Spectral theory for operator families on Riemannian manifolds, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990), vol. 54, part 3, AMS, Providence, RI, 1993, pp. 615-665. [Sz2] _ _ , Locally non isometric yet super isospectral spaces, preprint. [U] H. Urakawa, Bounded domains which are isospectral but not congruent, Ann. Scient. Bc. Norm. Sup. 15 (1982),441-456. E Vigneras, Varietes Riemanniennes isospectrales et non isometriques, Ann.

Let r 2 be the cocompact discrete subgroup of G generated by The cocompact discrete subgroups r] and r2 are representation equivalent subgroups of G, however r] and r 2 are not isomorphic as groups. 9. 5. Remark. Recently C. Vijayan [Vj] has shown that every three-step nilpotent Lie group G with flat coadjoint orbits contains a pair of nonisomorphic, representation equivalent, cocompact discrete subgroups r], r2. 4 occurs surprisingly often. Note that strictly nonsingular three-step nilpotent Lie groups necessarily have flat coadjoint orbits.

The following are equivalent for * E Aut(G): (i) is almost inner; (ii) for each X E g. there exists a E G such that *(X) = Ad(a)(X); (iii) for each ,\ E g*. there exists a E G such that ,\ 0 * = ,\ 0 Ad(a). Analogous statements hold for f -almost inner automorphisms. 3 [GWI], [GI]. Let G be a simply-connected nilpotent Lie group, f a cocompact discrete subgroup of G, and *

*(r) are unitarily equivalent. In particular, if g is a left-invariant Riemannian metric on G, then p- spec (f\ G, g) = p- spec (*