Subgroups of finitely presented simple groups
by Claas Röver
Abstract
This thesis is concerned with the possible structure of subgroups of
finitely presented infinite simple groups. We survey the finitely presented simple groups that were
known prior to this thesis and prove that they are all torsion locally
finite except possibly those for which the conjugacy problem is unsolvable. A group is called torsion locally finite if every finitely
generated torsion subgroup is finite.
Then we generalise
the old methods for constructing finitely presented simple
groups. This, in turn, enables us to describe constructive
embeddings of certain recursively presented groups into finitely presented
groups which in theory exist by Higman's Embedding Theorem.
What is more, we can construct a class of finitely presented
simple groups $\ch{f,p}'$ that are not torsion locally finite. More precisely,
they have subgroups isomorphic to Grigorchuk-Gupta-Sidki
groups which are finitely generated infinite torsion
groups under suitable assumptions. We also prove that each group $\ch{f,p}'$ is generated by two elements.
In the last two chapters we investigate algorithmic decision
problems for the groups $\ch{f,p}$, thereby obtaining a positive
answer to the order problem. We also prove that the conjugacy problem
is solvable for all elements with `flat symbols' if $p$ is a prime. This includes all
periodic elements, so in particular the conjugacy problem for periodic
elements is shown to be solvable. In addition, we describe an
effective procedure to decide whether an element has a flat symbol.
We also show that the family $\ch{f,p}'$ of finitely
presented simple groups contains a countable infinity of isomorphism classes.
The whole thesis (97 pages) is downloadable as gzipped dvi (370 kB), post-script (319 kB) or pdf (610 kB) files. Beware that the dvi version might not contain the pictures.
Those who don't want to read that much might be interested in the published version of the first half of this thesis.
Leagan Gaeilge
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