Clow, Andrew David Menzies (2000) Ends of groups : a computational approach. PhD thesis, University of Warwick.

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Abstract

We develop
ways in which we can
find the number of ends of automatic
groups and groups with solvable word problem.
In
chapter 1, we provide an
introduction to ends, splittings, and computa-
tion in
groups.
We
also remark that the `JSJ
problem'
for finitely
presented
groups
is
not solvable.
In
chapter
2,
we prove some geometrical properties of
Cayley
graphs
that
underpin
later
computational results.
In
chapter
3,
we study coboundaries
(sets
of edges which
disconnect the
Cayley
graph), and show
how Stallings' theorem gives us
finite
objects
from
which we can calculate splittings.
In
chapter
4,
we
draw the
results of previous chapters
together to
prove
that
we can
detect
zero,
two,
or
infinitely
many ends
in
groups with
`good'
automatic structures.
We
also prove
that
given an automatic group or a
group with solvable word problem,
if the group splits over a
finite
subgroup,
we can
detect this,
and explicitly calculate a
finite
subgroup over which
it
splits.
In
chapter 5 we give an exposition of
Gerasimov's
result
that one-
endedness can be detected in hyperbolic
groups.
In
chapter
6,
we give an exposition of
Epstein's boundary
construction
for
graphs.
We
prove
that
a
testable
condition
for
automatic groups
implies
that this boundary is
uniformly path-connected, and also prove
that infinitely
ended groups
do
not
have
uniformly path-connected
boundary. As
a result
we are able to
sometimes
detect
one endedness
(and thus
solve
the
problem
of
how
many ends the
group
has).

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Geometry, Cayley graphs, Group theory
Official Date:	September 2000
Dates:	Date Event September 2000 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Epstein, D. B. A.
Extent:	189 leaves
Language:	eng

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