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Some notation

Sym(n) 3 GL(n,K) 3 D(n) 4 D(oo) 4 Z 4; 351 Z(n) 4 T(n,K) 6 UT(n,K) 7 (8), (X, Y) 12-13 .A 24 IXI 24 F(.A) 24 = 26 !It 28 [x, y] 49 [G,G] 49

m" 56 m(.A, n) 57; 197 B(a, r) 74 St(a) 89 all 115 aD.. 115 .AI 116; 367 ¢(p) 116

Ipi 117; 371 laill 117 I aD.. I 117 D..(2) 125 1D..(2)1125 :lj 139; 271 r(Il) 140 :1:= 140 !ltj 140; 272 G(i) 140; 272; 197 r(D..) 140 r(D..) 141

(Il,r,q) 155 (ql, r, q2) 155; 220 (III, r, Il2) 158 a(IlI, r, Il2) 158 a(ql, r, q2) 158; 220 r 1\ II 158; 220 r 1\ q 158; 220 r(.A,{,) 160 -< 165 a, (3",8, €, (, 'f}, £ 165; 218 a,~,1,d,h,n 166; 218 r(q) 167 Xi 197; 271 G(oo) 197; 272; 378 ~A 271 O"±(R) 332 O"Ll.(R) 332 B(m,n) 336 ~ 378

Author Index

Adian S.I. xxii, xxiv, 195,311,314,337 Aleshin S.V. 194 Anosov D.V. xxv, 471 Appel K.I. xxi Atabekyan V.S. xxvi, 305, 429, 432, 446 Ashmanov 1S. xxvi

Bear R. 296, 304, 393 Banach S. 413 Bass H. 411 Birkhoff G. 59,438 Bonnet D. xxi Boone W.W. 133 Borevich Z.l 446 Bovdi A.A. 423 Britton J. xx, 135 Bryant R.M. 435 Burnside W. xxi, 56, 57, 70,104,

194-196

CammR. 409 Cauchy A. 41, 79 Chandler B. xxii Chernikov S.N. 67, 304, 389 Collins D.J. 355

Dehn M. xx, 131 Deryabina G.S. xxvi, 305, 383,445 Descartes R. xix

Euler L. xxi, 97, 100

Fermat P. 196 F~lner E. 414-415 Frattini G. 392

Galois E. 54 Gauss C. xxi Gol'berg A.1 xxi, 133 Golod E.S. xxi,194 Golovin O.N. 398

499

Greendlinger M.D. xx, 135 Grigorchuk R.I. xxv, 194,411,412,

414-415,471 Gromov M. xxii Gruenberg K.W. 397,404 Guba V.S. xxv-xxvi, 444, 447, 471 Gupta N.D. 194

Hall M., Jr. 194 Hall P. xxv, 194,296,345,422-423,

439,443 HausdorffF. 413 Higman G. 194,295,393 Hilbert 423 Hiibschmann J. 355

Ivanov S.V. xviii, xxvi, 299, 306, 391, 423

Iwasawa K. 305

Jordan C. xxii, 73, 85, 269

van Kampen E.R. xix, xx, xxiii, 112, 117,364

Kargapolov M.I. 296 Kegel O.H. 67 Kleiman Yu.G. 315 Klein F. xix, 79, 103 Kostrikin A.I. xxi, 195,315,446 Kulatilaka C.R. 296 Kurosh A.G. 304,367,389

Lagrange J. 15 Levi F. 194 Levchuk V.M. 438 Lipschutz S. xxi Lobachevskii N.1 132 Lossov K.1 409 Lyndon R.C. xx-xxiv, 126, 134, 135,

354,364

500 AUTHOR INDEX

Magnus W. xxii Makhnyov A.A. 446 Mal'tsev A.I. 396-398,408 Markov A.A. xxiv, 339 Merzlyakov Yu.1. 439 Meskin S. 438 Miller O.A. 54 Milnor J. 411 Mobius A. 79,103 MoivreA. 8 Moreno, H.C. 54 Mycielski J. 340

Neumann B.H. 340, 393, 409, 435 Neumann H. 315, 393 Neumann P.M. 429 von Neumann J. xxv, 412-414 Nielsen J. 27, 429 Novikov P.S. xxii, xxiv, 133,311,315,

398,446

Obraztsov V.N. xxvi, 217, 383, 391

Peiffer R. 346, 355 Platonov V.P. 340 Poincare H. xix Pride S.J. 438

Razmyslov Yu.P. 194,315 Reznichenko E.A. 340 Rickert N.W. 415 Rips E. xxiii Robinson D.J.S. 392 Roseblade J. 423

Sadovskii L.E. 307 Sanoy I.N. 194 Schiek H. xx, 135

Schmidt O.Yu xiv, 60, 67, 296, 304 Schmidt R. 305 Schoenflies A. 85 Schupp P.E. xxi, 119, 135 Schreier O. 27,429 Schur I. 42, 194, 336,439 Semyonov Yu.S. 438 Shelah S. 339,391,392 Shevrin L.N. 445 Shirvanyan V.L. 394 Shmel 'kin A.L. 397, 404, 446 Shunkov V.P. 67,70,389 Sidki S. 194 Soldatova V.V. 135 Stallings J.R. 354 Stirling J. 419 Storozhev A.M. 307, 446 Strunkov S.P. 409 Sushchanskii V.I. 194 Suzuki M. 305 Swan R.O. 354 Sylow L. 41

Tarski A. xxiv, 296, 304, 413 Tartakovskii V.A. xx, xxii ThueA. 35 Trofimov V.l. 412 Turner-Smith R.F. 439

van der Waerden B.L. 194 Weinbaum C.M. xx-xxi, 126 Wehrfritz B.A.F. 67 WiegoldJ. 429,444

Yakovlev B.V. 307

Zaitsev OJ. 392, 393 Zeno (paradox) 58 Zyabrev I.N. 340,412

SUbject Index

A-compatibility 143 A-compatible decompositions of

words 34,377 A-edge 152 A-map 166 AO-map 175 (A,j)-triple 318 abelian extension 331 abelian group 1 adjacent edges 127,153 algebraic dependence among rela-

tions 355 algebraic number of R-cells 332 algebraic operation 1 almost polycyclic group 63 amenability criterion of Grigorchuk 414 amenability criterion of FlIllner 414 amenable group 413 annular map 115 Artinian group 62 associativity 396 aspherical 146 automorphism 20, 37 auxiliary numerical parameters 164 atoroidal 148

B-map 226 BO-map 233 barycentric subdecomposition 88, 92 basis of a free group 25 basis of the free group of a variety 57 bifurcation 123 bond 371 bound 372 boundary 84,87 boundary arc 128 bouquet of algebraic dependence 356 Burnside variety 56

C-map 244 cancellable pair of cells 124, 140,461

501

Cartesian product of groups 5 Cartesian subgroup 397 cell 92 cell of rank. 0 (O-cell) 140 cell of type 1 218,448 cell of type 2 218,448 cell with period A 197 cell decomposition 92 central extension 331 centralizer 38 centre of a group 39 characteristic subgroup 435 Chernikov group 67 circular dependence 346 circular map 115 circular star 85 closed curve 85 closed set 73 closed surface 87 closure 84 combinatorial deformation 105 combinatorially contractible cycle 107 combinatorially homotopic 105 commutator 49,331 commutator subgroup 49 compact space 79 compatible sections 460 complete system of contiguity sub-

maps 159,221,246 concealed cell 160,222,247 Condition A 271 Condition B 224 Condition C(k) 127 Condition C'(/..) 126 condition max-oo 392 condition min-oo 392 conditionR 271,378 conjugacy class 37 conjugacy in rank i 141,369 conjugacy problem 132 conjugate elements 18

502 SUBJECT INDEX

conjugate subgroups 37 connected component 84 connected space 82 connecting line 153, 155, 158,219,220,

371 consequence of a system of laws 58 consequence of a system of relations 29,

368 contiguity arc 154, 155, 157, 158,219,

220,221 contiguity degree 155, 158,219,220,

221,371 contiguity submap 221,371 continuous mapping 75 contour of a cell 105 contour of a map 115 copy of a cell 123 corrected estimating graph 224 coset 14 cycle 105 cyclically reduced word 26 cyclic group 8 cyclic section 143 cyclic shift of a word 33 cyclic subgroup 8 cyclic word 33

D-map 257, 442 defining relation 271,297 defining relation of a group 30 defining relation of a module 351 degree of vertex 93 derived group 49 diagrammatically aspherical presenta­

tion 146,369 diagrammatically atoroidal presenta­

tion 148,369 diagram over A 116,117,368 diagram on a surface X over a presenta-

tion 117 diagram of rank i 140 diagram over a presentation 368 dihedral group 4 direct product 5, 44 discrete topology 74

disjoint contiguity submaps 156 distance (metric) 74 distinguished contiguity submap 160,

222,247 distinguished system of a contiguity

submaps 160,222,247 divisible group 64 double coset 17

E-map 261 edge of a cell decomposition 105, 115 edge of a graph 95 edge on a contour 115 elementary transformations of a cell

decomposition 93 elementary transformation of words 28 embedding of a amalgam into a

group 386 embedding of spaces 77 equality in rank i 141,369 equivalence of cells of a diagram 348 equivalence of cell decompositions 92 equivalent systems of laws 59,348 equivalent triangulations 89 estimating graph 162 Euler characteristic 96 Euler number of a cell decomposition 96 exact operation 396 exponent of a group 47 exponential growth 411 extension of a graph on a surface 99 extension of a group 331 exterior arc 128

F-map 264 factor group 20 finite group 3 finitely presented group 32 finiteness condition 60 Frattini subgroup 392 free group on .>t in '13 57 free abelian group 57 free amalgam of groups 367 free Burnside group 57 free central extension 332

SUBJECT INDEX 503

free element in rank i 378 free element of G( 00) 380 free factor 364 free group 25 free group in a variety 56 free module 350 free product of groups 364 free product of groups in a variety 397 free product in rank i 378 fully residually group 432 functoriality 397 fundamental group 106, 109

G-map 265 G-module 351 generalized (A,j)-triple 441 generating elements 8 generator of a group 8, 13 geodesic path 166 graded diagram 141, 369 graded map 140

graded presentation 140, 369 graph on a surface 95 graph without I-gons and 2-gons 97 graph without loops 99 graphic equality of words 26, 142 group 1 group alphabet 24 group of type poo 7 group ring 351 group word 24 group word minimal in rank i 270 growth function 410 Gruenberg-5hmel 'kin product 397

H-map449 NO-map 452 Hausdorff space 81 heredity 396 homeomorphism 76 homomorphism 18 homotopic paths 104 homotopy 104 homotopy class 106

i-pair 461 identical relation 55 improper subgroup 8 index of a subgroup 15 infinite group 3, 4 infinite order 8 initial point of a path 105 initial segment of a word 33 inner automorphism 37 inner contour of a map 115 inner edge 175 interior arc 128 interior point 85 inverse path 83 involution 67 isomorphism 5

j-pair 141,369 Jordan arc 85

k-bond 157,220,448 k-contiguity submap 157,220 kernel of a homomorphism 18 Klein bottle 79

I-aperiodic word 35, 377 label of a path 116 law in a group 55 length of a element in a free product 365 length of a path 117,153,371 length of a word 24, 367 length of a word in a free product 367 letter 24 Lowest Parameter Principle (LPP) 165 line 153 locally finite group 60 long section of a cell 218, 257 long section of a contour 244, 273 loop 93 loose 372

M-group 305 Mal'tsev postulate 396 map 115, 140 marginal subgroup 345

504 SUBJECT INDEX

maximal set of periods 293 maximal subgroup 11 maximum condition (max) 61 measure on a group 413 metabelian group 51 metric space 74 minimal word of rank i 270 minimum condition (min) 62 Mobius band 79 multiple edge 99 multiplicity of a edge of a graph 99

n-cell92 natural homomorphism 20 neighbourhood 75 Noetherian group 62 Noetherian ring 422 normal closure 28 normal form of a free product 365 normal subgroup 17 normalizer 38

open ball 74 open set 73 open star 89 operation on 399 operation rrn 404 order of an element 9 order of a group 3 ordinary cell 160, 222, 247 orientable surface 102, 103 orientation of an arc 101 orientation of a cell decomposition 102 outer arc 175 outer contour of a map 115 outer edge 175, 233, 248

p-component of abelian group 47 p-group 39, 47 parameters 218 path 80, 105, 173 path connected 83 Peiffer transformation 355 perimeter 117 period of a rank i 143, 197,271

period oftype I 271 period of type II 271 periodic group 46 periodic word 34, 377 phase vertex 463 piece 126 polycyclic group 53 polynomial power growth 411 presentation of a group 28, 30, 368 principal cell of a bond 220 product of paths 105 product of words 24 projective plane 79 projection 78 proper power 33 proper subgroup 9

quasi-cyclic group 12 quasi-finite group 70 quotient group 20 quotient set 77 quotient space 78

9{;cell 116, 153 9{;equivalent words 116, 153 R-deducible 29 rank of a cell 140 rank of a diagram 140 rank of a long section of a contour of a

cell 219 rank of a map 140 rate of growth of a group 410 real element 68 reduced graded diagram 125, 141,369 reduced dependence 346 reduced word 24 reducible word 24 refinement of a cell decomposition 93 regular map 159,222 regular system of contiguity submaps

222,246,448 regularity 397 relation module 352 relator of first type 30, 271, 297 relator of rank i 140

SUBJECT INDEX 505

relator of second type 271, 298 representative of a coset 14 residually a group in a class 432 root of a word 348

Schur multiplier 336 section of a contour 143, 153 segment of a path 371 self-compatible cell 462 semi-circular star 87 short section of a cell 218 short section of a contour 244, 257, 273 side 86 side arc 155, 157,158,220 simple group 53 simple path 105 simple word 33, 377 simple word in rank. i 195,270,378 simply connected surface 109 singular map 159,222,246 smooth section of rank. k 167, 226 smooth section of type 1 449 smooth section of type 2 449 soluble group 51 special cell 160, 222, 247 sphere with n handles 110 spherical dependence 346 spherical map 115 star 87 subdiagram 121 subgroup 6 submap 121 subpath 116 subspace 74 subword377 subword of a cyclic word 33 surface (with boundary) 87 syllable 367 Sylow p-subgroup 343 symmetric group 3 symmetrized system of defining relations

126

Tl topology 339 terminal segments of a word 33 terminal point of a path 105

topological group 339 topological n-polygon 86 topological space 73 topological triangle 86 topology 73 toroidal map 115 torsion-free group 9 torus 78 transformation of I kind 355 transformation of II kind 356 transformation of m kind 356 triangulation 86 trivial algebraic dependence 357 trivial topology 74 type of a diagram 141,369,460 type of a sequence 355 type of a system of a submaps 222

value of a word 28 variety of groups 56 verbal product 397 verbal subgroup 56 vertex 86 vertex of a graph 95 visual equality 26, 142

weight (function) 173, 232 word equal to 1 modulo n 436 word problem 131

O-bond 154,219 O-bordering 122 0-ceIl116, 152 O-complete system of distinguished

contiguity submaps 415 O-contiguity submap 155,219 O-distinguished system of submaps 415 O-edge 116, 152 O-refinement 123

~-cellI89 'Y-cell 179, 239 ll-ce1l368

1J-.free group 56