Decision Problems for Groups — Survey and Reflections

Miller, Charles F.

doi:10.1007/978-1-4613-9730-4_1

Cited by 104 publications

(81 citation statements)

References 37 publications

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“…In the special case when Γ is free and Q is a finitely presented group with undecidable word problem, the corresponding P will have a number of undecidable problems (see [18]). In that case, however, P is almost never finitely presented (see [5], [14]).…”

Section: Corollary C There Exists a Closed Non-positively Curved Manmentioning

confidence: 99%

“…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]). Thus, although products of biautomatic and related groups have a solvable conjugacy problem [11], it is possible that their finitely presented subgroups may not.…”

Section: The Proof Of Theorem Amentioning

confidence: 99%

“…It is rather straightforward to construct finitely generated subgroups with unsolvable decision problems inside apparently well-behaved groups (see [18], [17], [20], [4]); constructing finitely presented subgroups with unsolvable decision problems is a much more delicate matter. 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.…”

Section: Introductionmentioning

confidence: 99%

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Fibre products, non-positive curvature, and decision problems

Baumslag

Bridson²,

Miller

et al. 2000

Commentarii Mathematici Helvetici

107

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Abstract. We give a criterion for fibre products to be finitely presented and use it as the basis of a construction that encodes the pathologies of finite group presentations into pairs of groups P ⊂ G where G is a product of hyperbolic groups and P is a finitely presented subgroup. This enables us to prove that there is a finitely presented subgroup P in a biautomatic group G such that the generalized word problem for P ⊂ G is unsolvable and P has an unsolvable conjugacy problem. An additional construction shows that there exists a compact non-positively curved polyhedron X such that π 1 X is biautomatic and there is no algorithm to decide isomorphism among the finitely presented subgroups of π 1 X.

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Section: Corollary C There Exists a Closed Non-positively Curved Manmentioning

confidence: 99%

Section: The Proof Of Theorem Amentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Fibre products, non-positive curvature, and decision problems

Baumslag

Bridson²,

Miller

et al. 2000

Commentarii Mathematici Helvetici

107

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“…Hydra groups are subject of several investigations in the last years, because of their role in the famous decision problems of M. Dehn (see [26]). The following result illustrates this: the proof is omitted, because it involves specific techniques of combinatorial group theory (Van Kampen diagrams and generalizations of Seifert-Van Kampen's Theorem, see [17] for feedback and definitions).…”

Section: S K E T C H O F T H E P R O O F Given K N > 0 We May Defimentioning

confidence: 99%

Some considerations on Hydra groups and a new bound for the length of words

Otera

Russo

2014

Mathematica Slovaca

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ABSTRACT. After a survey on some recent results of Riley and others on Ackermann functions and Hydra groups, we make an analogy between DNA sequences, whose growth is the same of that of Hydra groups, and a musical piece, written with the same algorithmic criterion. This is mainly an aesthetic observation, which emphasizes the importance of the combinatorics of words in two different contexts. A result of specific mathematical interest is placed at the end, where we sharpen some previous bounds on deterministic finite automata in which there are languages with hairpins. Hydra groupsThe combinatorics of words is increased significatively in the last years, since it is a powerful tool for the description of several processes in pure and applied sciences. A classic reference on the topic is [24] and will be used for the basic notions. We recall that a nonempty set A (which will be always assumed to be finite) is said to be an alphabet and its elements are called letters. By A + we denote the free semigroup generated by A, i.e. the set of all finite sequences of letters with the operation of concatenation. Elements of A + will be called words. It is well known that each word can be written in a unique way as a sequence of letters, so it is possible to define for each word w its length |w|:It is useful (though not necessary) to introduce the formal empty word ε: that is a word of length 0 such that2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 57M50; Secondary 68R05, 68Q15. K e y w o r d s: isoperimetric functions, finitely presented groups, lengths of words, counterpoint, Myhill-Nerode's theorem.

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“…Local properties are those pertaining to the elements of a group, while global properties speak about groups as a whole. The three most important decision problems, formulated by Max Dehn in 1911, are the word, conjugacy, and isomorphism problems [Mil91]. The …rst two problems talk about local properties, while the third problem concerns a global relationship.…”

mentioning

confidence: 99%

The complexity of ascendant sequences in locally nilpotent groups

Conidis

Shore

2014

Int. J. Algebra Comput.

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Abstract. We analyze the complexity of ascendant sequences in locally nilpotent groups, showing that if G is a computable locally nilpotent group and x 0 ; x 1 ; : : : ; x N 2 G, N 2 N, then one can always …nd a uniformly computably enumerable (i.e. uniformly 0 1 ) ascendant sequence of order type ! + 1 of subgroups in G beginning with hx 0 ; x 1 ; : : : ; x N i G , the subgroup generated by x 0 ; x 1 ; : : : ; x N in G. This complexity is surprisingly low in light of the fact that the usual de…nition of ascendant sequence involves arbitrarily large ordinals that index sequences of subgroups de…ned via a trans…nite recursion in which each step is incomputable. We produce this surprisingly low complexity seqeunce via the e¤ective algebraic commutator collection process of P. Hall, and a related purely algebraic Normal Form Theorem of M. Hall for nilpotent groups.

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Decision Problems for Groups — Survey and Reflections

Cited by 104 publications

References 37 publications

Fibre products, non-positive curvature, and decision problems

Fibre products, non-positive curvature, and decision problems

Some considerations on Hydra groups and a new bound for the length of words

The complexity of ascendant sequences in locally nilpotent groups

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