TY - JOUR

T1 - The bounded and precise word problems for presentations of groups

AU - Ivanov, S. V.

N1 - Funding Information:
The work on this article was supported in smaller part by NSF grant DMS 09-01782.
Publisher Copyright:
© 2020 American Mathematical Society.

PY - 2020/3

Y1 - 2020/3

N2 - We introduce and study the bounded word problem and the precise word problem for groups given by means of generators and defining relations. For example, for every finitely presented group, the bounded word problem is in NP, i.e., it can be solved in nondeterministic polynomial time, and the precise word problem is in PSPACE, i.e., it can be solved in polynomial space. The main technical result of the paper states that, for certain finite presentations of groups, which include the Baumslag-Solitar one-relator groups and free products of cyclic groups, the bounded word problem and the precise word problem can be solved in polylogarithmic space. As consequences of developed techniques that can be described as calculus of brackets, we obtain polylogarithmic space bounds for the computational complexity of the diagram problem for free groups, for the width problem for elements of free groups, and for computation of the area defined by polygonal singular closed curves in the plane. We also obtain polynomial time bounds for these problems.

AB - We introduce and study the bounded word problem and the precise word problem for groups given by means of generators and defining relations. For example, for every finitely presented group, the bounded word problem is in NP, i.e., it can be solved in nondeterministic polynomial time, and the precise word problem is in PSPACE, i.e., it can be solved in polynomial space. The main technical result of the paper states that, for certain finite presentations of groups, which include the Baumslag-Solitar one-relator groups and free products of cyclic groups, the bounded word problem and the precise word problem can be solved in polylogarithmic space. As consequences of developed techniques that can be described as calculus of brackets, we obtain polylogarithmic space bounds for the computational complexity of the diagram problem for free groups, for the width problem for elements of free groups, and for computation of the area defined by polygonal singular closed curves in the plane. We also obtain polynomial time bounds for these problems.

KW - Diagrams

KW - Polygonal curves

KW - Polylogarithmic space

KW - Presentations of groups

KW - Word problems

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U2 - 10.1090/memo/1281

DO - 10.1090/memo/1281

M3 - Article

AN - SCOPUS:85085614177

SN - 0065-9266

VL - 264

SP - 1

EP - 118

JO - Memoirs of the American Mathematical Society

JF - Memoirs of the American Mathematical Society

IS - 1281

ER -