For a group
of
type
for a ring
and a homomorphism
, we show that
if the top-dimensional
cohomology of
with coefficients
in the Novikov rings
vanishes. This criterion is applied to show that if
is a finitely generated RFRS group of cohomological dimension
, then
is virtually free-by-cyclic
if and only if
.
This answers a question of Wise, and generalises and gives a significantly
shorter proof of a recent theorem of Kielak and Linton, where
the same result is obtained under the additional hypotheses that
is virtually compact special and hyperbolic. A consequence of the result
is that all virtually RFRS groups of rational cohomological dimension
with vanishing
second
-Betti
number are coherent. More generally, we show that if
is a RFRS group of
cohomological dimension
and of type
, then
admits a virtual map to
with kernel of rational
cohomological dimension
if and only if
.
Keywords
free-by-cyclic groups, coherence, group cohomology, Novikov
rings, L2 invariants
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