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We study algebraic structures of certain submonoids of the monoid of homology
cylinders over a surface and the homology cobordism groups, using Reidemeister
torsion with non-commutative coefficients. The submonoids consist of ones whose
natural inclusion maps from the boundary surfaces induce isomorphisms on higher
solvable quotients of the fundamental groups. We show that for a surface whose first
Betti number is positive, the homology cobordism groups are other enlargements of
the mapping class group of the surface than that of ordinary homology cylinders.
Furthermore we show that for a surface with boundary whose first Betti number
is positive, the submonoids consisting of irreducible ones as 3–manifolds
trivially acting on the solvable quotients of the surface group are not finitely
generated.
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