On the difficulty of presenting finitely presentable groups

Abstract

Abstract. We exhibit classes of groups in which the word problem is uniformly solvable but in which there is no algorithm that can compute finite presentations for finitely presentable subgroups. Direct products of hyperbolic groups, groups of integer matrices, and right-angled Coxeter groups form such classes. We discuss related classes of groups in which there does exist an algorithm to compute finite presentations for finitely presentable subgroups. We also construct a finitely presented group that has a polynomial Dehn function but in which there is no algorithm to compute the first Betti number of its finitely presentable subgroups.

The homology of groups, profinite completions, and echoes of Gilbert Baumslag

“…It extends the theme of [7,8], which demonstrated the wildness that is to be found among the finitely presented subgroups of automatic groups. It also reinforces the point made in [26] about the necessity of including the full input data in the effective version of the 1-2-3 Theorem [23]. The proof that the ambient biautomatic group is residually finite relies on deep work of Wise [50,52] and Agol [1] as well as Serre's insights into the connection between residual finiteness and cohomology with finite coefficient modules [48,Section I.2.6].…”

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

“…There does not exist an algorithm that, on input a finite presentation of a group of type FP ∞ can output a finite set of module generators for π 2 of the presentation, so the last piece of input data in the above theorem cannot easily be dispensed with. In fact, Bridson and Wilton [16] have proved that it is essential: there exists a recursive sequence of maps φ n : Γ n → Q n , with Γ n and Q n given by finite presentations, such that each Q n is of type FP ∞ and each kernel ker φ n is finitely generated, but the first Betti number of the associated (finitely presentable) fibre product P n < Γ n × Γ n cannot be determined algorithmically (whereas it could be if one had a finite presentation in hand).…”

The isomorphism problem for profinite completions of finitely presented, residually finite groups

“…The algorithm in [6] requires as input a finite presentation for H, a finite generating set for N, a finite presentation Q for Q, and a set of elements of π 2 Q n that generate it as a ZQ n -module. (Our results in [8] show that this last piece of data is essential. )…”

“…But in his construction, although the Γ n are given by explicit finite presentations, the subgroups P n ãÑ Γ are given by specifying a finite generating set Σ n Ă Γ (with a guarantee that P n " xΣ n y is finitely presentable). In [8] we explored in detail the question of when such data is sufficient to allow the algorithmic construction of a finite presentation for P n , and in this situation it is not. Indeed, the lack of an algorithm to present P n is an essential feature of the proof of [2,Theorem B]: what is actually proved is that one cannot decide ifι n is injective (note the contrast with our Theorem A) because there is no algorithm that can determine if the map H 1 pP, Zq Ñ H 1 pΓ, Zq induced by ι n is injective; if one had a finite presentation for both groups, it would be easy to determine if H 1 pP, Zq Ñ H 1 pΓ, Zq were injective.…”