Finitely Presented Subgroups of Automatic Groups and their Isoperimetric Functions

Abstract

Abstract. We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgroups satisfy, at best, an exponential isoperimetric inequality.

Fibre products, non-positive curvature, and decision problems

“…Contracting a maximal tree if necessary, we may assume that Y has a single vertex, and thus the 2-skeleton Y (2) corresponds to a finite presentation P Y of Q, with generators given by the 1-cells and relators given by the attaching maps of the 2-cells. Because Y (3) is finite, π 2 P Y := π 2 Y (2) is finitely generated as a Q-module (the attaching maps of the finitely many 3-cells of Y give a generating set). This finite generation property is useful because it admits the following algebraic interpretation.…”

“…In the case of the word problem [3] our discussion focused on the complexity of solutions because existence was a trivial matter (any finitely presented subgroup of a group with a solvable word problem obviously has a solvable word problem). In contrast, solvability of the conjugacy problem is not inherited by finitely pre-sented subgroups in general (see [18] -for an example of such a finite index subgroup see [8]).…”

“…In [3] we showed that the finitely presented subgroups of products of hyperbolic groups can be rather complicated by concentrating on the complexity of the word problem: the optimal isoperimetric inequality satisfied by finitely presented subgroups may be significantly worse than the quadratic isoperimetric inequality satisfied by the ambient group. In the present article we turn our attention to the other basic decision problems: the conjugacy problem, the isomorphism problem and the generalized word problem.…”

Incoherent negatively curved groups

“…Thus, the applications of Rips' construction in [2], [3], and [10] mentioned above may be extended to finding examples of fundamental groups of compact negatively curved 2-complexes (or where appropriate, non-positively curved complexes) exhibiting the various pathologies. For example, Corollary 1.5 provides certain pathological examples of negatively curved groups analogous to the pathological examples of small-cancellation groups given in [10].…”

“…In [3], Rips' construction is used to show that various problems about small-cancellation groups are recursively unsolvable. In [2], Rips' construction is used to find Automatic groups with non-Automatic finitely-presented subgroups.…”