A subgroup
of a group
is
commensurated if
the commensurator of
in
is the
entire group
.
Our main result is that a finitely generated group
containing an infinite, finitely generated, commensurated subgroup
of infinite index
in
is one-ended
and semistable at
.
Furthermore, if
and
are finitely presented
and either
is one-ended
or the pair
has one
filtered end, then
is
simply connected at
.
A normal subgroup of a group is commensurated, so this result is a generalization
of M Mihalik’s result [Trans. Amer. Math. Soc. 277 (1983) 307–321] and
of B Jackson’s result [Topology 21 (1982) 71–81]. As a corollary, we give
an alternate proof of V M Lew’s theorem that a finitely generated group
containing
an infinite, finitely generated, subnormal subgroup of infinite index is semistable at
.
So several previously known semistability and simple connectivity at
results for group extensions follow from the results in this paper. If
is a monomorphism of a finitely generated group and
has finite
index in
,
then
is
commensurated in the corresponding ascending HNN extension, which in turn is semistable
at
.
Keywords
commensurator, semistability, simply connected at infinity
