439 research outputs found
Torsion homology and regulators of isospectral manifolds
Given a finite group G, a G-covering of closed Riemannian manifolds, and a
so-called G-relation, a construction of Sunada produces a pair of manifolds M_1
and M_2 that are strongly isospectral. Such manifolds have the same dimension
and the same volume, and their rational homology groups are isomorphic. We
investigate the relationship between their integral homology. The
Cheeger-Mueller Theorem implies that a certain product of orders of torsion
homology and of regulators for M_1 agrees with that for M_2. We exhibit a
connection between the torsion in the integral homology of M_1 and M_2 on the
one hand, and the G-module structure of integral homology of the covering
manifold on the other, by interpreting the quotients Reg_i(M_1)/Reg_i(M_2)
representation theoretically. Further, we prove that the p-primary torsion in
the homology of M_1 is isomorphic to that of M_2 for all primes p not dividing
#G. For p <= 71, we give examples of pairs of isospectral hyperbolic
3-manifolds for which the p-torsion homology differs, and we conjecture such
examples to exist for all primes p.Comment: 21 pages; minor changes; included a data file; to appear in J.
Topolog
The Bianchi groups are subgroup separable on geometrically finite subgroups
We show that for certain arithmetic groups, geometrically finite subgroups
are the intersection of finite index subgroups containing them. Examples are
the Bianchi groups and the Seifert-Weber dodecahedral space. In particular, for
manifolds commensurable with these groups, immersed incompressible surfaces
lift to embeddings in a finite sheeted covering.Comment: 19 page
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