Charles F. Miller

Decision Problems for Groups — Survey and Reflections

This is a survey of decision problems for groups, that is of algorithms for answering various questions about groups and their elements. The general objective of this area can be formulated as follows: Objective: To determine the existence and nature of algorithms which decide local properties — whether or not elements of a group have certain properties or relationships; global properties— whether or not groups as a whole possess certain properties or relationships.

Structure and finiteness properties of subdirect products of groups

We investigate the structure of subdirect products of groups, particularly their finiteness properties. We pay special attention to the subdirect products of free groups, surface groups and HNN extensions. We prove that a finitely presented subdirect product of free and surface groups virtually contains a term of the lower central series of the direct product or else fails to intersect one of the direct summands. This leads to a characterization of the finitely presented subgroups of the direct product of 3 free or surface groups, and to a solution to the conjugacy problem for arbitrary finitely presented subgroups of direct products of surface groups. We obtain a formula for the first homology of a subdirect product of two free groups and use it to show there is no algorithm to determine the first homology of a finitely generated subgroup.

Computable algebra and group embeddings

This work grew out of our investigations of the infinitely generated subgroups of finitely presented groups. A useful technique for embedding countable groups with sufficiently nice local properties into finitely presented groups was introduced in [l]. This technique was then applied in [ 1, 21 to prove that every countable locally polycyclic group and every countable metabelian group can be embedded in a finitely presented group. It became apparent from [2] that effective methods in commutative algebra could be extended to group rings of polycyclic groups and then used to embed certain groups in finitely presented groups. In this paper these extensions are carried out with the consequent application that every countable locally abelian-by-nilpotent group can be embedded in a finitely presented group. Much of the effective commutative algebra we require seems to be known. (See for instance [6, 9, 14, 171.) Unfortunately a systematic account is not available. In particular we need a proof of the Hilbert basis theorem which

On the finite presentation of subdirect products and the nature of residually free groups

We establish {\it virtual surjection to pairs} (VSP) as a general criterion for the finite presentability of subdirect products of groups: if

Algorithms and classification in combinatorial group theory

\Gamma_1,\ldots,\Gamma_n

Finitely Presented Subgroups of Automatic Groups and their Isoperimetric Functions

are finitely presented and

Recognition of subgroups of direct products of hyperbolic groups

S<\Gamma_1\times\cdots\times\Gamma_n

Reflections on the residual finiteness of one-relator groups

projects to a subgroup of finite index in each

Finite presentation of fibre products of metabelian groups

\Gamma_i\times\Gamma_j

VIRTUAL PROPERTIES OF CYCLICALLY PINCHED ONE-RELATOR GROUPS

, then