LEADER 04365nam 2200565 450 001 9910820875003321 005 20200810212242.0 010 $a1-4704-5804-7 035 $a(CKB)4100000011244159 035 $a(MiAaPQ)EBC6195968 035 $a(RPAM)21609889 035 $a(PPN)250656353 035 $a(EXLCZ)994100000011244159 100 $a20200810d2020 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 14$aThe bounded and precise word problems for presentations of groups /$fS.V. Ivanov 210 1$aProvidence, RI :$cAmerican Mathematical Society,$d2020. 215 $a1 online resource (118 pages) 225 1 $aMemoirs of the American Mathematical Society,$x0065-9266 ;$vnumber 1281 311 $a1-4704-4143-8 320 $aIncludes bibliographical references. 327 $aPreliminaries -- Proof of proposition 1.1 -- Calculus of brackets for group presentation (1.2) -- Proofs of theorem 1.2 and corollary 1.3 -- Calculus of brackets for group presentation (1.4) -- Proof of theorem 1.4 -- Minimizing diagrams over (1.2) and proofs of theorem 1.5 and corollary 1.6 -- Construction of minimal diagrams over (1.4) and proof of theorem 1.7 -- Polygonal curves in the plane and proofs of theorems 1.8, 1.9 and corollary 1.10. 330 $a"We introduce and study the bounded word problem and the precise word problem for groups given by means of generators and defining relations. For example, for every finitely presented group, the bounded word problem is in NP, i.e., it can be solved in nondeterministic polynomial time, and the precise word problem is in PSPACE, i.e., it can be solved in polynomial space. The main technical result of the paper states that, for certain finite presentations of groups, which include the Baumslag-Solitar one-relator groups and free products of cyclic groups, the bounded word problem and the precise word problem can be solved in polylogarithmic space. As consequences of developed techniques that can be described as calculus of brackets, we obtain polylogarithmic space bounds for the computational complexity of the diagram problem for free groups, for the width problem for elements of free groups, and for computation of the area defined by polygonal singular closed curves in the plane. We also obtain polynomial time bounds for these problems"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society ;$vnumber 1281. 606 $aGroup theory and generalizations -- Special aspects of infinite or finite groups -- Geometric group theory [See also 05C25, 20E08, 57Mxx]$2msc 606 $aGroup theory and generalizations -- Special aspects of infinite or finite groups -- Generators, relations, and presentations$2msc 606 $aGroup theory and generalizations -- Special aspects of infinite or finite groups -- Cancellation theory; application of van Kampen diagrams [See also 57M05]$2msc 606 $aConvex and discrete geometry -- Polytopes and polyhedra -- Combinatorial properties (number of faces, shortest paths, etc.) [See also 05Cxx]$2msc 606 $aWord problems (Mathematics) 606 $aPresentations of groups (Mathematics) 615 7$aGroup theory and generalizations -- Special aspects of infinite or finite groups -- Geometric group theory [See also 05C25, 20E08, 57Mxx]. 615 7$aGroup theory and generalizations -- Special aspects of infinite or finite groups -- Generators, relations, and presentations. 615 7$aGroup theory and generalizations -- Special aspects of infinite or finite groups -- Cancellation theory; application of van Kampen diagrams [See also 57M05]. 615 7$aConvex and discrete geometry -- Polytopes and polyhedra -- Combinatorial properties (number of faces, shortest paths, etc.) [See also 05Cxx]. 615 0$aWord problems (Mathematics) 615 0$aPresentations of groups (Mathematics) 676 $a512/.2 686 $a20F05$a20F06$a20F10$a68Q25$a68U05$a52B05$a20F65$a68W30$2msc 700 $aIvanov$b S. V$g(Sergei V.),$01715714 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910820875003321 996 $aThe bounded and precise word problems for presentations of groups$94110586 997 $aUNINA