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Title page for ETD etd-05102006-160631

Type of Document Dissertation
Author Muranov, Alexey
Author's Email Address alexey.muranov@vanderbilt.edu
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
  • van Kampen diagrams
  • group presentations
  • bounded generation
Date of Defense 2006-05-03
Availability unrestricted
The subject of this work is application of combinatorial

group theory to the problem of constructing groups with

prescribed properties.

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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