A subgroup
of a group
is
commensurated in
if for each
,
has finite index
in both
and
. If there is a sequence
of subgroups
where
is
commensurated in
for all , then
is
subcommensurated
in
.
In this paper we introduce the notion of the simple connectivity at
of a finitely generated group (in analogy with that for finitely
presented groups). Our main result is this: if a finitely generated group
contains an infinite finitely generated subcommensurated subgroup
of infinite
index in
, then
is one-ended and semistable
at
. If, additionally,
is recursively presented
and
is finitely presented
and one-ended, then
is
simply connected at
.
A normal subgroup of a group is commensurated, so this result is a strict
generalization of a number of results, including the main theorems in works of
G Conner and M Mihalik, B Jackson, V M Lew, M Mihalik, and J Profio. We also
show that Grigorchuk’s group (a finitely generated infinite torsion group) and a
finitely presented ascending HNN extension of this group are simply connected at
,
generalizing the main result of a paper of L Funar and D E Otera.
Keywords
semistability, simple connectivity at infinity,
commensurated, subcommensurated
