Title:

Encoding and detecting properties in finitely presented groups

In this thesis we study several properties of finitely presented groups, through the unifying paradigm of encoding soughtafter group properties into presentations and detecting group properties from presentations, in the context of Geometric Group Theory. A group law is said to be detectable in power subgroups if, for all coprime m and n, a group G satisfies the law if and only if the power subgroups G(^{m}) and G(^{n}) both satisfy the law. We prove that for all positive integers c, nilpotency of class at most c is detectable in power subgroups, as is the kEngel law for k at most 4. In contrast, detectability in power subgroups fails for solvability of given derived length: we construct a finite group W such that W(^{2}) and W(^{3}) are metabelian but W has derived length 3. We analyse the complexity of the detectability of commutativity in power subgroups, in terms of finite presentations that encode a proof of the result. We construct a census of twogenerator onerelator groups of relator length at most 9, with complete determination of isomorphism type, and verify a conjecture regarding conditions under which such groups are automatic. Furthermore, we introduce a family of onerelator groups and classify which of them act properly cocompactly on complete CAT(0) spaces; the nonCAT(0) examples are counterexamples to a variation on the aforementioned conjecture. For a subclass, we establish automaticity, which is needed for the census. The deficiency of a group is the maximum over all presentations for that group of the number of generators minus the number of relators. Every finite group has nonpositive deficiency. For every prime p we construct finite pgroups of arbitrary negative deficiency, and thereby complete Kotschick's proposed classification of the integers which are deficiencies of Kähler groups. We explore variations and embellishments of our basic construction, which require subtle Schur multiplier computations, and we investigate the conditions on inputs to the construction that are necessary for success. A wellknown question asks whether any two nonisometric finite volume hyperbolic 3manifolds are distinguished from each other by the finite quotients of their fundamental groups. At present, this has been proved only when one of the manifolds is a oncepunctured torus bundle over the circle. We give substantial computational evidence in support of a positive answer, by showing that no two manifolds in the SnapPea census of 72 942 finite volume hyperbolic 3manifolds have the same finite quotients. We determine examples of sizeable graphs, as required to construct finitely presented nonhyperbolic subgroups of hyperbolic groups, which have the fewest vertices possible modulo mild topological assumptions.
