Length Functions on Subgroups in Finitely Presented Groups

2000 ◽  
Author(s):  
A. Yu. Olshanskii ◽  
M. V. Sapir
Keyword(s):  
Length Functions ◽  
Finitely Presented Groups ◽  
Finitely Presented

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.

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

VERIFYING THAT A GROUP IS VIRTUALLY FREE

1991 ◽  
Vol 01 (03) ◽  
pp. 339-351
Author(s):  
ROBERT H. GILMAN
Keyword(s):  
Finite Index ◽  
Free Subgroup ◽  
Universal Group ◽  
Finitely Presented Groups ◽  
Finite Presentation ◽  
Finitely Presented

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.

scholarly journals Finitely presented groups and the Whitehead nightmare

2017 ◽  
Vol 11 (1) ◽  
pp. 291-310
Author(s):  
Daniele Ettore Otera ◽  
Valentin Poénaru
Keyword(s):  
Finitely Presented Groups ◽  
Finitely Presented

scholarly journals A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 282 ◽  
Author(s):  
Andrea Coladangelo
Keyword(s):  
Optimal Strategy ◽  
Elementary Proof ◽  
Quantum Correlations ◽  
Entangled State ◽  
Finitely Presented Groups ◽  
Self Testing ◽  
Question And Answer ◽  
Questions And Answers ◽  
Non Local ◽  
Finitely Presented

We describe a two-player non-local game, with a fixed small number of questions and answers, such that an ϵ-close to optimal strategy requires an entangled state of dimension 2Ω(ϵ−1/8). Our non-local game is inspired by the three-player non-local game of Ji, Leung and Vidick \cite{ji2018three}. It reduces the number of players from three to two, as well as the question and answer set sizes. Moreover, it provides an (arguably) elementary proof of the non-closure of the set of quantum correlations, based on embezzlement and self-testing. In contrast, previous proofs \cite{slofstra2019set, dykema2017non, musat2018non} involved representation theoretic machinery for finitely-presented groups and C∗-algebras.

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