Embeddings of relatively free groups into finitely presented groups

Periodicity Theorem ◽

AbstractWe generalize a well-known periodicity lemma from the case of free groups to the case of acylindrically hyperbolic groups. This generalization has been used to describe solutions of certain equations in acylindrically hyperbolic groups and to characterize verbally closed finitely generated acylindrically hyperbolic subgroups of finitely presented groups.

Reidemeister moves and groups

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216515400064 ◽

2015 ◽

Vol 24 (10) ◽

pp. 1540006

Author(s):

Vassily Olegovich Manturov

Keyword(s):

Group Theory ◽

Knot Theory ◽

Free Groups ◽

Equivalence Classes ◽

Combinatorial Objects ◽

Reidemeister Moves ◽

Interesting Class ◽

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.

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.

Journal of Symbolic Logic ◽

10.2307/2269900 ◽

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 ◽

Recursively Enumerable ◽

Finitely Presented ◽

Degrees Of Unsolvability

Constructing matrix representations of finitely presented groups

Journal of Symbolic Computation ◽

10.1016/s0747-7171(08)80095-8 ◽

1991 ◽

Vol 12 (4-5) ◽

pp. 427-438 ◽

Cited By ~ 8

Author(s):

S.A. Linton

Keyword(s):

Matrix Representations ◽

VERIFYING THAT A GROUP IS VIRTUALLY FREE

International Journal of Algebra and Computation ◽

10.1142/s0218196791000237 ◽

1991 ◽

Vol 01 (03) ◽

pp. 339-351

Author(s):

ROBERT H. GILMAN

Keyword(s):

Finite Index ◽

Free Subgroup ◽

Universal Group ◽

Finite Presentation ◽

This paper is concerned with computation in finitely presented groups. We discuss a procedure for showing that a finite presentation presents a group with a free subgroup of finite index, and we give methods for solving various problems in such groups. Our procedure works by constructing a particular kind of partial groupoid whose universal group is isomorphic to the group presented. When the procedure succeeds, the partial groupoid can be used as an aid to computation in the group.