Finitely presented infinite simple groups

Thesis

This thesis is mainly concerned with the development of a procedure to construct finitely presented infinite simple groups.

The finitely presented groups, G_n,i, constructed by G.Higman [5) are extended by certain subgroups, H, of the inverse limit of wreath products, S_n-1(wr s_n-1)^w. It is shown that if H is defined by a finite set of expansion rules (see Chapter 2) then the group n,1,H> is finitely presented if H is finitely presented. The commutator subgroup, n,1,H>, is shown to be a simple group and it is proved that an integer m and a set of expansion rules defining H can be chosen such that is a finitely presented infinite simple group.

Let GL(n,𝕫) be the general linear group of nxn dimensional matrices over the integers. The finitely presented simple group, m,1,𝕫ⁿ.GL(n,𝕫)¹, is constructed, where 𝕫ⁿ is the free Abelian group of finite rank, n. So GL(n,𝕫) can be embedded in a finitely presented simple group, for all n.

Let X_p be the ring of rationals whose denominators are divisible only by primes belonging to the finite set P. If A is a countable Abelian group whose torsion factor group is a subgroup of a finitely generated torsion free X_p-module, then it is shown that A can be embedded in a finitely presented simple group.

It is also noted, at the end of the discussion of linear groups, that if there exists a finitely presented linear group, over the integers, with unsolvable conjugacy or orbit problem, then a finitely presented infinite simple group with unsolvable conjugacy problem can be constructed.

Authors: Scott, EA

• DOI: 10.5287/ora-nyeyavqae
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English