Type of Document Dissertation Author Muranov, Alexey Author's Email Address email@example.com URN etd-05102006-160631 Title On geometry and combinatorics of van Kampen diagrams Degree PhD Department Mathematics Advisory Committee
Advisor Name Title Alexander Olshanskiy Committee Chair Mark Sapir Committee Member Michael Mihalik Committee Member Ralph McKenzie Committee Member Thomas Kephart Committee Member Keywords
- van Kampen diagrams
- group presentations
- bounded generation
Date of Defense 2006-05-03 Availability unrestricted AbstractThe subject of this work is application of combinatorial
group theory to the problem of constructing groups with
It is shown how certain groups can
be presented by generators and defining relations,
thus proving their existence.
Several existence theorems proved in this paper are
based on one approach:
van Kampen diagrams over group presentations are used
to derive algebraic properties of the groups from combinatorial properties
of their presentations.
The focus of this paper is on boundnely generated and
boundedly simple groups.
It is proved that there exist an infinite simple boundedly generated
a torsion-free group with a finite regular file basis and
with a free non-cyclic subgroup, and
a boundedly simple finitely generated group
with a free non-cyclic subgroup.
In particular, a question of Vasiliy Bludov, which has been
open since 1995, is settled.
The question was whether every torsion-free group
with a finite regular file basis
has to be virtually polycyclic,
and this question is answered negatively by
providing a counterexample.
The groups in question, or rather their presentations,
are constructed by imposing relations
that force the group to be boundedly simple, or boundedly generated,
or have a ``regular file basis,' accordingly,
while in the same time choosing those relations so that certain
small-cancellation-type conditions are satisfied.
These conditions are used to establish other properties
of the groups.
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