We show that the existence of a nonparabolic local cut point in the Bowditch boundary
of a relatively
hyperbolic group
implies that
splits
over a
–ended
subgroup. This theorem generalizes a theorem of Bowditch from the setting of
hyperbolic groups to relatively hyperbolic groups. As a consequence we are able to
generalize a theorem of Kapovich and Kleiner by classifying the homeomorphism type of
–dimensional
Bowditch boundaries of relatively hyperbolic groups which satisfy certain properties, such as no
splittings over
–ended
subgroups and no peripheral splittings.
In order to prove the boundary classification result we require a notion of ends of a
group which is more general than the standard notion. We show that if a finitely generated
discrete group acts properly and cocompactly on two generalized Peano continua
and
, then
is homeomorphic
to
.
Thus we propose an alternative definition of
which increases the
class of spaces on which
can act.
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