Friends and relatives of BS(1,2)

2014 ◽  
Vol 6 (2) ◽  
Author(s):  
Charles F. Miller III
Keyword(s):  
Group Theory ◽  
Word Problem ◽  
Teaching Experience ◽  
Geometric Group Theory ◽  
Point Of View ◽  
Fundamental Importance ◽  
Finitely Presented Groups ◽  
Historical Essay ◽  
Central Interest ◽  
Finitely Presented

AbstractAlgorithms, constructions and examples are of central interest in combinatorial and geometric group theory. Teaching experience and, more recently, preparing a historical essay have led me to think the familiar group BS(1,2) is an example of fundamental importance. The purpose of this note is to make a case for this point of view. We recall several interesting constructions and important examples of groups related to BS(1,2), and indicate why certain of these groups played a key role in showing the word problem for finitely presented groups is unsolvable.

The word problem for division rings

1973 ◽  
Vol 38 (3) ◽  
pp. 428-436 ◽  
Author(s):  
Angus Macintyre
Keyword(s):  
Group Theory ◽  
Recent Work ◽  
Word Problem ◽  
Equivalent Formulation ◽  
Finitely Presented Groups ◽  
First Order ◽  
Division Rings ◽  
Group Presentations ◽  
Finitely Presented ◽  
Horn Sentences

In this paper we prove that the word problem for division rings is recursively unsolvable. Our proof relies on the corresponding result for groups [7], [28], and makes essential use of P. M. Cohn's recent work [11], [13], [15], [16] on division rings.The word problem for groups is usually formulated in terms of group presentations or finitely presented groups, as in [7], [24], [28], [30]. An equivalent formulation, in terms of the universal Horn sentences of group theory, is mentioned in [32]. This formulation makes sense for arbitrary first-order theories, and it is with respect to this formulation that we show that the word problem for division rings has degree 0′.

A. A. Fridman. Stépéni nérazréšimosti problémy toždéstva v konéčno oprédélénnyh gruppah. Doklady Akadémii Nauk SSSR, vol. 147 (1962), pp. 805–808. - A. A. Fridman. Degrees of insolvability of the word problem in finitely defined groups. English translation of the preceding by Sue Ann Walker. Soviet mathematics, vol. 3 no. 6 (1962), pp. 1733–1737. - C. R. J. Clapham. Finitely presented groups with word problems of arbitrary degrees of insolubility. Proceedings of the London Mathematical Society, ser. 3 vol. 14 (1964), pp. 633–676. - William W. Boone. Finitely presented group whose word problem has the same degree as that of an arbitrarily given Thue system (an application of methods of Britton). Proceedings of the National Academy of Sciences, vol. 53 (1965), pp. 265–269. - William W. Boone. Word problems and recursively enumerable degrees of unsolvability. A first paper on Thue systems. Annals of mathematics, ser. 2 vol. 83 (1966), pp. 520–571. - William W. Boone. Word problems and recursively enumerable degrees of unsolvability. A sequel on finitely presented groups. Annals of mathematics, ser. 2 vol. 84 (1966), pp. 49–84.

1968 ◽  
Vol 33 (2) ◽  
pp. 296-297
Author(s):  
J. C. Shepherdson
Keyword(s):  
Word Problem ◽  
Word Problems ◽  
Nauk Sssr ◽  
London Mathematical Society ◽  
National Academy Of Sciences ◽  
Doklady Akademii Nauk Sssr ◽  
Finitely Presented Groups ◽  
Recursively Enumerable ◽  
Finitely Presented ◽  
Degrees Of Unsolvability

Separating Classes of Groups by First-Order Sentences

2003 ◽  
Vol 13 (03) ◽  
pp. 287-302 ◽  
Author(s):  
André Nies
Keyword(s):  
Nilpotent Group ◽  
Word Problem ◽  
Finitely Generated ◽  
Finitely Presented Groups ◽  
First Order ◽  
Solvable Word Problem ◽  
Rank 2 ◽  
Finitely Presented ◽  
Solvable Word

For various proper inclusions of classes of groups [Formula: see text], we obtain a group [Formula: see text] and a first-order sentence φ such that H⊨φ but no G∈ C satisfies φ. The classes we consider include the finite, finitely presented, finitely generated with and without solvable word problem, and all countable groups. For one separation, we give an example of a f.g. group, namely ℤp ≀ ℤ for some prime p, which is the only f.g. group satisfying an appropriate first-order sentence. A further example of such a group, the free step-2 nilpotent group of rank 2, is used to show that true arithmetic Th(ℕ,+,×) can be interpreted in the theory of the class of finitely presented groups and other classes of f.g. groups.

scholarly journals Embeddings into groups with only a few defining relations

1974 ◽  
Vol 18 (1) ◽  
pp. 1-7 ◽  
Author(s):  
W. W. Boone ◽  
D. J. Collins
Keyword(s):  
Word Problem ◽  
Finitely Presented Groups ◽  
Fixed Group ◽  
Defining Relations ◽  
Finitely Presented Group ◽  
One Relator Groups ◽  
Finitely Presented ◽  
Unsolvable Word Problem ◽  
Trivial Consequence

It is a trivial consequence of Magnus' solution to the word problem for one-relator groups [9] and the existence of finitely presented groups with unsolvable word problem [4] that not every finitely presented group can be embedded in a one-relator group. We modify a construction of Aanderaa [1] to show that any finitely presented group can be embedded in a group with twenty-six defining relations. It then follows from the well-known theorem of Higman [7] that there is a fixed group with twenty-six defining relations in which every recursively presented group is embedded.

scholarly journals FUNCTIONS ON GROUPS AND COMPUTATIONAL COMPLEXITY

2004 ◽  
Vol 14 (04) ◽  
pp. 409-429 ◽  
Author(s):  
JEAN-CAMILLE BIRGET
Keyword(s):  
Computational Complexity ◽  
Symmetric Space ◽  
Word Problem ◽  
Exponential Inequality ◽  
Isoperimetric Function ◽  
Length Functions ◽  
Finitely Presented Groups ◽  
Double Exponential ◽  
Context Free ◽  
Finitely Presented

We give some connections between various functions defined on finitely presented groups (isoperimetric, isodiametric, Todd–Coxeter radius, filling length functions, etc.), and we study the relation between those functions and the computational complexity of the word problem (deterministic time, nondeterministic time, symmetric space). We show that the isoperimetric function can always be linearly decreased (unless it is the identity map). We present a new proof of the Double Exponential Inequality, based on context-free languages.

scholarly journals THE GROUPS OF RICHARD THOMPSON AND COMPLEXITY

2004 ◽  
Vol 14 (05n06) ◽  
pp. 569-626 ◽  
Author(s):  
JEAN-CAMILLE BIRGET
Keyword(s):  
Word Problem ◽  
Simple Groups ◽  
Direct Products ◽  
Turing Machines ◽  
Group V ◽  
Dehn Functions ◽  
Finitely Presented Groups ◽  
Distortion Functions ◽  
The 1960S ◽  
Finitely Presented

We prove new results about the remarkable infinite simple groups introduced by Richard Thompson in the 1960s. We give a faithful representation in the Cuntz C⋆-algebra. For the finitely presented simple group V we show that the word-length and the table size satisfy an n log n relation. We show that the word problem of V belongs to the parallel complexity class AC1 (a subclass of P), whereas the generalized word problem of V is undecidable. We study the distortion functions of V and show that V contains all finite direct products of finitely generated free groups as subgroups with linear distortion. As a consequence, up to polynomial equivalence of functions, the following three sets are the same: the set of distortions of V, the set of Dehn functions of finitely presented groups, and the set of time complexity functions of nondeterministic Turing machines.

scholarly journals Reidemeister moves and groups

2015 ◽  
Vol 24 (10) ◽  
pp. 1540006
Author(s):  
Vassily Olegovich Manturov
Keyword(s):  
Group Theory ◽  
Knot Theory ◽  
Free Groups ◽  
Equivalence Classes ◽  
Finitely Presented Groups ◽  
Combinatorial Objects ◽  
Reidemeister Moves ◽  
Interesting Class ◽  
Finitely Presented

Recently, the author discovered an interesting class of knot-like objects called free knots. These purely combinatorial objects are equivalence classes of Gauss diagrams modulo Reidemeister moves (the same notion in the language of words was introduced by Turaev [Topology of words, Proc. Lond. Math. Soc.95(3) (2007) 360–412], who thought all free knots to be trivial). As it turned out, these new objects are highly nontrivial, see [V. O. Manturov, Parity in knot theory, Mat. Sb.201(5) (2010) 65–110], and even admit nontrivial cobordism classes [V. O. Manturov, Parity and cobordisms of free knots, Mat. Sb.203(2) (2012) 45–76]. An important issue is the existence of invariants where a diagram evaluates to itself which makes such objects "similar" to free groups: An element has its minimal representative which "lives inside" any representative equivalent to it. In this paper, we consider generalizations of free knots by means of (finitely presented) groups. These new objects have lots of nontrivial properties coming from both knot theory and group theory. This connection allows one not only to apply group theory to various problems in knot theory but also to apply Reidemeister moves to the study of (finitely presented) groups. Groups appear naturally in this setting when graphs are embedded in surfaces.

scholarly journals Homotopical height

2014 ◽  
Vol 25 (13) ◽  
pp. 1450123 ◽  
Author(s):  
Indranil Biswas ◽  
Mahan Mj ◽  
Dishant Pancholi
Keyword(s):  
Fundamental Group ◽  
Group Cohomology ◽  
Point Of View ◽  
Complex Manifolds ◽  
Almost Complex Manifolds ◽  
Finitely Presented Groups ◽  
Almost Complex ◽  
Kähler Groups ◽  
Closed Contact ◽  
Finitely Presented

Given a group G and a class of manifolds 𝒞 (e.g. symplectic, contact, Kähler, etc.), it is an old problem to find a manifold MG ∈ 𝒞 whose fundamental group is G. This article refines it: for a group G and a positive integer r find MG ∈ 𝒞 such that π1(MG) = G and πi(MG) = 0 for 1 < i < r. We thus provide a unified point of view systematizing known and new results in this direction for various different classes of manifolds. The largest r for which such an MG ∈ 𝒞 can be found is called the homotopical height ht 𝒞(G). Homotopical height provides a dimensional obstruction to finding a K(G, 1) space within the given class 𝒞, leading to a hierarchy of these classes in terms of "softness" or "hardness" à la Gromov. We show that the classes of closed contact, CR, and almost complex manifolds as well as the class of (open) Stein manifolds are soft. The classes 𝒮𝒫 and 𝒞𝒜 of closed symplectic and complex manifolds exhibit intermediate "softness" in the sense that every finitely presented group G can be realized as the fundamental group of a manifold in 𝒮𝒫 and a manifold in 𝒞𝒜. For these classes, ht 𝒞(G) provides a numerical invariant for finitely presented groups. We give explicit computations of these invariants for some standard finitely presented groups. We use the notion of homotopical height within the "hard" category of Kähler groups to obtain partial answers to questions of Toledo regarding second cohomology and second group cohomology of Kähler groups. We also modify and generalize a construction due to Dimca, Papadima and Suciu to give a potentially large class of projective groups violating property FP.

scholarly journals The computability of group constructions II

1973 ◽  
Vol 8 (1) ◽  
pp. 27-60 ◽  
Author(s):  
R.W. Gatterdam
Keyword(s):  
Characteristic Function ◽  
Word Problem ◽  
Elementary Functions ◽  
Recursively Enumerable Set ◽  
Finitely Presented Groups ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Recursively Enumerable ◽  
Finitely Presented ◽  
Computational Level

Finitely presented groups having word, problem solvable by functions in the relativized Grzegorczyk hierarchy, {En(A)| n ε N, A ⊂ N (N the natural numbers)} are studied. Basically the class E3 consists of the elementary functions of Kalmar and En+1 is obtained from En by unbounded recursion. The relativization En(A) is obtained by adjoining the characteristic function of A to the class En.It is shown that the Higman construction embedding, a finitely generated group with a recursively enumerable set of relations into a finitely presented group, preserves the computational level of the word problem with respect to the relativized Grzegorczyk hierarchy. As a corollary it is shown that for every n ≥ 4 and A ⊂ N recursively enumerable there exists a finitely presented group with word problem solvable at level En(A) but not En-1(A). In particular, there exist finitely presented groups with word problem solvable at level En but not En-1 for n ≥ 4, answering a question of Cannonito.

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