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.