Index of Notation

Cambridge University Press 978-1-107-15235-9 — Groups, Languages and Automata Derek F. Holt , Sarah Rees , Claas E. Röver Index More Information

Index of Notation

∗ A (free monoid), 4 G ≀φ H (wreath product), 12 + A (free semigroup), 4 G ≀Rφ H (restricted wreath product), 13 ∗ H ⋉φ N (semidirect product), 12 Bd(A ), 204 BS(m, n) (Baumslag–Solitar group), 34, 113 Hn (Houghton group), 260 I, 48, 249 CF , 43, 63 C′(λ), 27 L(M), 37 C(p), 27 M∃,94 CS, 47, 85 M∀,95 m DCF , 43, 63 (M), 214 DCS, 47, 85 MonX, MonX |R,5,6 dfsa, 41, 52 N ⋊φ H (semidirect product), 12 dfst, 195 NSPACE, 46, 237 d(G) (minimal size of generating set), 11 NTIME, 47, 237 DOC, 43, 80 OC, 43, 80 DPCF , 48, 255 PCF , 48, 249, 255 DSPACE, 46, 237 pda,41 DTIME, 47, 237 $ (padding symbol), 92 ∗ e(G) (number of ends), 25 Ptm0(A ), 206 ε (empty word), 4 Ptm(A∗), 198 f g, f ≃ g, 103 Rˆ (symmetric closure), 26 fsa,38 Reg, 38, 53 fst, 194 RistG(), 208 RT , 47, 249 G ∗A H (amalgamated free product), 13 Γ(G, H, X), 21 SgpX, SgpX |R,5,6 Γ(G, X), 20

283

© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-15235-9 — Groups, Languages and Automata Derek F. Holt , Sarah Rees , Claas E. Röver Index More Information

Index of Names

Aanderaa, S., 47 Caprace, P.-E., 141, 173 Aleshin,ˇ S., 194, 211, 212 Charney, R., 31, 140, 176 Alonso, J., 103 Chiswell, I., xi, 229 Amir, G., 207 Ciobanu, L., 139, 173, 182, 183 Angel, O., 207 Collins, D., 221 Anisimov, A., 105, 187 Dahmani, F., 167 Antol´ın, Y., 139, 173, 182, 183 Davis, M., 173 Avenhaus, J., 226, 237 Day, M., 216 Bahls, P., 141 de la Harpe, P., 139 Bartholdi, L., 207, 213, 216, 217 Dehn, M., 7, 26, 97 Bass, H., 19 Dehornoy, P., 32, 140, 176 Baumslag, G., 34, 49, 125, 147 Dunwoody, M., 233 Bendix, P., 118 Edjvet, M., 139 Bestvina, M., 129 Elder, M., 144, 145, 147 Bhattacharjee, M., 212 Epstein, D., 34, 113, 125, 134, 138, 166, 191 Bleak, C., 263 Erschler, A., 216 Bondarenko, E., 208, 214 Escher, M., 30 Bondarenko, N., 214 Gersten, S., 102, 125, 134, 140, 141, 143, 184, Book, R., 48 241, 243, 244 Boone, W., 88, 103, 117, 221, 223, 227 Ghys, E.,´ 139 Bowditch, B., 25, 105, 165 Gillibert, P., 214 Brady, N., 105, 129 Gilman, R., 118, 128, 142, 146, 182, 254 Brady, T., 140 Glasner, Y., 211, 212 Bridson, M., 105, 128, 129, 141, 142, 146, Godelle, E., 31 166, 167, 243 Golod, E., 138, 215 Brieskorn, E., 31 Goodman, O., 112, 113, 252 Brieussel, J., 207 Greendlinger, M., 27, 140, 237 Brink, B., 30, 141, 173 Gregorac, R., 232 Brittenham, M., 147 Greibach, S., 48 Britton, J., 18 Grigorchuk, R., 194, 209, 216, 217, 239 Brough, T., 255, 267 Gromov, M., 23, 105, 114, 139, 150, 154, 165, Brunner, A., 207, 212 184, 217, 235, 241 Buckley, D., 167 Grushko, I., 11, 232 Burnside, W., 215 Guirardel, V., 167 Cannon, Jim, 128, 139, 145, 169, 247 Gupta, C., 213

284

Index of Names 285

Gupta, N., 194, 201, 213 Muller, D., 26, 106, 228 Haeﬂiger, A., 167 Muntyan, Y., 213 Hamenstadt,¨ U., 141 Nekrashevych, V., 194, 204, 207–210 Harkins, A., 140 Neumann, B., 16, 18 Hartmanis, J., 250 Neumann, H., 16, 18 Havas, G., 139 Neumann, W., 139, 143, 145, 169, 170, 173 Herbst, T., 43, 232 Neunho¨ﬀer, M., 263 Hermiller, S., 140, 143, 147, 182 Niblo, G., 141, 173 Higman, G., 16, 18, 35, 227 Novikov, P., 88, 103, 117, 221 Holt, D., 92, 113, 118, 125, 134, 138, 139, Ol’shanskii, A., 105, 165 141, 142, 147, 159, 166, 167, 176, 182, Oshiba, T., 262 190, 191, 208, 239, 252, 254, 258, 263, Otto, F., 103, 227 268 Owens, M., 233 Hopf, H., 26 Howie, J., 166 Papasoglu, P., 105, 145, 155, 165 Howlett, R., 30, 141, 173 Parikh, R., 74, 75, 235, 255, 263 Humphreys, J., 173 Paris, L., 31, 32, 140, 176 Hurt, D., 190, 191 Parkes, D., 108 Peifer, D., 140 Immerman, N., 86 Picantin, M., 214, 215 Incitti, R., 233 Pietrowski, A., 232 Juhasz,` A., 139 Pin, J-E.,´ 49 Kaimanovich, V., 207 Pyber, L., 217 Karass, A., 232 Rabin, M., 47 Kharlampovich, O., 147 Rebecchi, D., 140 Khoussainov, B., 147 Redfern, I., 189 Klimann, I., 214, 215 Rees, S., 92, 113, 141, 142, 182, 191, 252, Knuth, D., 118 258, 263, 268 Krammer, D., 30 Reeves, L., 141, 143, 173 Lakin, S., 236, 237 Riley, T., 240, 241, 243, 247 Lavreniuk, Y., 209 Rips, E., 166, 167, 187 Lehnert, J., 260, 263, 268 Rivin, I., 183 Leonov, Yu., 214 Rover,¨ C., 92, 208, 209, 239, 254, 258, 263, Linnell, P., 217 268 Lubotzky, A., 217 Rosenberg, A., 48, 251 Lyndon, R., 27 Rotman, J., 88, 221 Lysenok,¨ I., 216 Rozhkov, A., 214 Madlener, K., 103, 226, 227, 237 Saito, K., 31 Mann, A., 216, 217 Sapir, M., xi, 103, 227 Maslov, A., 262 Savchuk, D., 212, 215 Matucci, F., 263 Schutzenberger,¨ M., 56 McCammond, J., 140 Schick, T., 217 Meier, J., 176 Schreier, O., 9 Mennicke, J., 139 Schupp, P., 26, 28, 106, 228 Miasnikov, A., 147 Schweitzer, P., 260, 268 Miller, C., 7, 8, 227 Scott, E., 207 Milnor, J., 153, 216 Segal, D., 217 Mosher, L., 141, 142 Seifert, F., 187 Moussong, G., 30, 141 Sela, Z., 167 Mozes, S., 211, 212 Serre, J-P., 19 Muhlherr,¨ B., 141, 173 Shafarevich, I., 138, 215

286 Index of Names

Shalev, A., 217 Thurston, W., 126, 129, 146 Shapiro, M., 49, 112, 113, 125, 133, 139, 143, Tits, J., 30 145, 147, 169, 170, 173, 236, 246, 252, van den Dries, L., 233 254 van der Lek, H., 31 Short, H., 49, 125, 140, 141, 143, 184 Ventura, E., 214 Sidki, S., 194, 199, 201, 204, 206, 207, 212, Virag,´ B., 207 214 Vogtmann, K., 141 Sims, C., 118, 120, 124 Vorobets, M., 212 Solitar, D., 34, 232 Vorobets, Y., 212 Stallings, J., 26, 231, 233 Wilkie, A., 233 Stearn, R., 250 Williams, A., 125, 139, 192 Steinberg, B., 212 Wilson, J., 209, 214, 217 Strebel, R., 140 Suniˇ c,´ Z., 147, 207, 214 Yamada, H., 47 Sushchansky, V., 216 Zalesskii, P., 214 Taback, J., 147 Zapata, F., 214 Thomas, R., 108, 233, 258, 263 Zuk,˙ A., 217 Thompson, R., 35

Index of Topics and Terminology

2-tape fsa,93 strongly, 190 2-variable fsa,93 group, 28, 117, 127, 128, 170, 236, 237 k-pda,64 asynchronously, 147 accepting state, 38 Cayley graph, 147 of pda,60 C-graph, 147 accessible group, 232, 254 strongly geodesically, 145, 150, 155 active state, 198 structure, 127, 128 algorithm asynchronous, 128 Cannon, 112 geodesic, 135 Dehn, 97, 110, 140, 150 shortlex, 135 generalised, 112, 113, 252 automaton, 36, 37 almost convex group, 240, 247 n-variable, 91 alphabet, 4 bireversible, 211 ambiguous grammar, 51 circuit in, 204 amenable group, 207 dual, 210 A-quasiconvex subgroup, 187 finite state, 36, 38, 52 Artin group, 31, 140 group, 199 dihedral type, 31 index, 123 extra-large type, 31, 140, 182 language of, 37 finite type, 31 nested stack, 48, 239, 254 large type, 31, 140, 141 of tree automorphism, 204 right-angled, 13, 27, 31, 140 one-counter, 43 spherical type, 31, 32 pushdown, 41, 60 asynchronous reversible, 211 2-variable fsa,93 stack, 48 automatic structure, 128 synchronous, 198 ff fellow traveller property, 127 word-di erence, 129 asynchronously automaton semigroup, 213 automatic group, 128, 147 automorphism biautomatic group, 128 finite state, 199 bicombable group, 128 autostackable group, 148 combable group, 128, 142, 240, 243 Bass–Serre tree, 20 atomic monoid, 32 Baumslag–Solitar group, 34, 113, 141, 147, automatic 189, 246, 254, 264 coset system, 189 biautomatic group, 28, 128, 155, 189 shortlex, 190 bicombable group, 128

287

288 Index of Topics and Terminology

bigon context-free group, 258 geodesic, 154 context-free language, 42, 254 binary adding machine, 195 deterministic, 62 bireversible automaton, 211 context-sensitive grammar, 85, 236 blank symbol, 44 context-sensitive language, 47, 85, 113, 236 bounded tree automorphism, 204, 239 contracting group, 207 braid group, 31, 113 convergent rewriting system, 120 braid relation, 30 coset multiplier, 189 branch group, 208 coset word-acceptor, 189, 190 regular, 209 Coxeter diagram, 29 weakly, 208 Coxeter group, 27, 29, 141, 173 Britton’s Lemma, 18, 224, 225 Euclidean, 29 Cannon algorithm, 112 hyperbolic, 29 Cartan matrix, 29 parabolic subgroup, 30 Cartesian product, 11 spherical, 29 Cayley graph, 20, 39 standard subgroup, 30 Cayley graph automatic group, 147 word-hyperbolic, 30, 141 C-graph automatic group, 147 Coxeter matrix, 29 Chomsky hierarchy, 38, 249 C(p)-group, 27 + Chomsky normal form, 69, 229 C(p) T(q)-group, 27 Church’s Thesis, 44, 45 critical pair (in rewriting system), 121 Church–Turing Thesis, 44 cyclically reduced form (in free product), 16 circuit in automaton, 204 cyclically reduced word, 7, 26 circuitous tree automorphism, 206 defining relation, 6 co-context-free defining relator, 6 group, 257, 260 Dehn universal, 263 algorithm, 97, 110, 140, 150, 252 semi-deterministic, 260, 262 generalised, 112, 113, 252 co-indexed diagram, 99 semi-deterministic, 239, 268 function, 102, 161 co-word problem, 80, 256 linear, 161 context-free, 257, 260 presentation, 27, 110, 150 indexed, 239 derivation combable group, 128, 142 equivalent, 51 combing, 128 in grammar, 50 commutator, 3 left-most, 51 complement (in group), 12 right-most, 51 complete fsa,41 deterministic complete rewriting system, 120 fsa, 41, 52 complexity, xi, 46 pda, 42, 62 composite fsa,96 context-free language, 42, 62 computable function, 46 context-sensitive language, 47 cone type, 126 Turing machine, 44 configuration graph (of Turing machine), 45 diameter (of van Kampen diagram), 114, 243 confluent rewriting system, 120 direct factor, 11 congruence, 5 direct product, 11 syntactic, 51, 108 restricted, 12 conjugacy geodesic, 182 directed tree automorphism, 205 conjugacy problem, 7, 27, 134, 166, 214, 256 distance conjugate, 3 Hausdorff,25 context-free grammar, 69 divergence function, 155

Index of Topics and Terminology 289

δ-slim triangle, 151 fsa δ-thin triangle, 152 2-tape, 93 dual automaton, 210 2-variable, 93 elementary amenable group, 216 asynchronous 2-tape, 93 empty word, 4 asynchronous 2-variable, 93 endomorphic presentation, 217 complete, 41 endomorphism composite, 96 virtual, 207 deterministic, 41, 52 ends (of a graph), 25 equivalent, 53 ends (of a group), 25 non-deterministic, 41 ε-move, 53 partial, 40 ε-transition, 53 synchronous 2-tape, 93 equivalence synchronous 2-variable, 93 Nerode, 214 function equivalent fsa,53 computable, 46 equivalent derivations, 51 partial, 46 Euclidean Coxeter group, 29 total, 46 Euclidean group, 140 fundamental group of graph of groups, 19 factorisation (of w ∈ WP(G, X±)), 98 Garside group, 32, 176 failure state, 39 general linear group, 212 falsification by fellow traveller property, 143, generalised Dehn algorithm, 112, 113, 252 170 generalised sequential machine, 49, 195 families of languages, 36 generalised sequential mapping, 50 fellow traveller constant, 127 generalised word problem, 8, 167, 186 fellow traveller property, 127, 244 generating set, 5 asynchronous, 127 generator, 5 synchronous, 127 Schreier, 9, 24 FFTP, 143, 170, 176 geodesic filling function, 114 conjugacy, 182 filling length, 240, 242 geodesic automatic structure, 135 of van Kampen diagram, 114 geodesic bigon, 154 of word, 114 geodesic metric space, 20 final state, 38 geodesic triangle, 150 finite state automaton, 36, 38, 52 geodesic word, 4, 5, 21, 169 finite state automorphism, 199 geometrically finite hyperbolic group, 113, finite state transducer, 194 139, 173, 240, 252, 254 finitely generated (semigroup, monoid, group), grammar, 36, 37 5 ambiguous, 51 finitely presentable (semigroup, monoid, context-free, 69 group), 6 context-sensitive, 85, 236 finitely presented (semigroup, monoid, group), growing context-sensitive, 85 6 language of, 37 finiteness problem, 214 left-regular, 58 fractal group, 209 production rule, 37 free regular, 59 factors, 11 right-regular, 59 group, 6, 212 semi-Thue, 36 monoid, 6 start symbol, 37 product, 10, 26 terminal, 37 with amalgamation, 13, 20, 232 Type 0, 36 semigroup, 6 Type 1, 85

290 Index of Topics and Terminology

Type 2, 69 word-hyperbolic, 141 unrestricted, 36 Euclidean, 140 variable), 37 fractal, 209 graph free, 6 Cayley, 20 Garside, 32, 176 Schreier, 21 geometrically finite hyperbolic, 113, 139, graph group, 31 173, 252, 254 graph of groups, 19 graph, 31 fundamental group of, 19, 20 Grigorchuk, 215, 216, 239, 268 graph product, 13, 143, 182 Gupta–Sidki, 215, 216 Greendlinger’s Lemma, 237 Higman–Thompson, 34, 239, 260, 262, 268 Greibach normal form, 69 Houghton, 260 Grigorchuk group, 215, 216, 239, 268 hyperbolic, 28, 105, 111, 117, 139, 145, group, 4 150, 170, 182, 187, 191, 240, 252, 254 accessible, 232, 254 knot, 33 almost convex, 240, 247 lamplighter, 203 amenable, 207 language for, 4 elementary, 216 linear, 236 Artin, 31, 140 mapping class, 31, 141 dihedral type, 31 nilpotent, 3, 23, 140, 142, 147, 189, 240, extra-large type, 31, 140, 182 252, 254, 264 finite type, 31 non-Hopfian, 34 large type, 31, 140, 141 normal form for, 5 right-angled, 13, 27, 31, 140 parallel poly-pushdown, 147 spherical type, 31, 32 poly-context-free, 255 asynchronously automatic, 128, 147 polycyclic, 3, 140, 189, 264 asynchronously biautomatic, 128 real-time, 252 asynchronously bicombable, 128 recurrent, 209 asynchronously combable, 128, 142, 240, recursively presentable, 98 243 relatively hyperbolic, 113, 140, 170, 173 automatic, 28, 117, 127, 128, 170, 236, 237 residually finite, 216 strongly geodesically, 150, 155 self-similar, 207 automaton, 199 soluble, 3 autostackable, 148 solvable, 3 Baumslag–Solitar, 34, 141, 147, 189, 246, state-closed, 207 254, 264 strongly geodesically automatic, 145 biautomatic, 28, 128, 155, 189 surface, 28 bicombable, 128 virtually abelian, 170, 182 braid, 31, 113 word-hyperbolic, 28, 30, 105, 111, 139, branch, 208 145, 150, 170, 187, 191, 240, 252, 254 regular, 209 group extension, 12 weakly, 208 growing context-sensitive grammar, 85 Cayley graph automatic, 147 growth C-graph automatic, 147 function, 22, 183, 216 co-context-free, 257, 260 intermediate, 199, 216 universal, 263 polynomial, 23 combable, 128, 142 rate, 217 complement in, 12 series, 23, 139, 183 context-free, 258 subgroup, 217 contracting, 207 uniformly exponential, 217 Coxeter, 27, 29, 141, 173 growth series, 118

Index of Topics and Terminology 291

Grushko’s theorem, 11, 232 one-counter, 43, 80, 232 Grzegorczyk hierarchy, 226 parallel poly-pushdown, 49 Gupta–Sidki group, 215, 216 poly-context-free, 48, 249, 254 Hausdorff distance, 25 quasi-real-time, 48 Higman–Thompson group, 34, 239, 260, 262, rational, 38 268 real-time, 47, 240, 249 HNN-extension, 16, 20, 34, 142, 223, 232, 246 recursive, 45 multiple, 19, 224 recursively enumerable, 45 Houghton group, 260 regular, 38, 53 hyperbolic Coxeter group, 29 star-free, 49, 182 hyperbolic group, 28, 105, 111, 117, 139, 145, tidy real-time, 252 150, 170, 182, 187, 191, 240, 252, 254 left greedy normal form, 177 geometrically finite, 240 left-most derivation, 51 hyperbolic metric space, 153 left-regular grammar, 58 lenlex ordering, 5, 120 inactive state, 198 letter, 4 index automaton, 123 level transitive, 208 indexed co-word problem, 239 limit space, 210 indexed language, 48, 249, 254 linear group, 236 inner product, 154 linearly bounded Turing machine, 47 instantaneous description (of Turing machine), locally confluent rewriting system, 120 45 locally excluding language, 49, 182 interior proints (of triangle), 151 locally testable language, 49, 182 inverse transducer, 197 L-quasiconvex subset, 184 isodiametric function, 243 L-rational subset, 184 isodiametric inequality, 243 MAF package, 139, 192 isomorphism problem, 7, 167 mapping class group, 31, 141 isoperimetric function, 102 Markov–Post theorem, 88, 90, 91 isoperimetric inequality, 243 md-reduction, 215 KBMAG, 118, 125, 186, 192 Mealy machine, 195 Kleene closure, 56 metric space knot, 32 geodesic, 20 fundamental group of, 33 minimisation of transducer, 214 group, 33 monoid, 4 Trefoil, 33 atomic, 32 Knuth–Bendix completion, 121, 125, 136 free, 6 K-step real-time Turing machine, 249 syntactic, 52, 108 lamplighter group, 203 transition, 54 language, 4, 36 morphism n-variable, 91 syntactic, 52 context-free, 42, 254 Muller–Schupp Theorem, 43, 106, 228, 258 context-sensitive, 47, 85, 113, 236 multiplier automaton, 130 deterministic context-free, 42, 62 coset, 189 deterministic context-sensitive, 47 Myhill–Nerode theorem, 54 family, 36 Nerode equivalence, 214 for a group, 4, 126 nested stack automaton, 48, 239, 254 indexed, 48, 249, 254 Nielsen–Schreier Theorem, 10 locally excluding, 49, 182 nilpotent group, 3, 23, 140, 142, 147, 189, locally testable, 49, 182 240, 252, 254, 264 of automaton, 37 Noetherian rewriting system, 120 of grammar, 37 non-deterministic

292 Index of Topics and Terminology

fsa,41 positive word, 30 pda,42 preﬁx property (of language), 63 Turing machine, 44 presentation non-Hopﬁan group, 34 Dehn, 27, 110, 150 normal form, 5, 117, 126 endomorphic, 217 Chomsky, 69, 229 recursive, 98 Greibach, 69 recursively enumerable, 98 left greedy, 177 Wirtinger, 33 rational, 184 product shortlex, 5 Cartesian, 11 Novikov–Boone theorem, 88, 91 direct, 11 nucleus, 207 free, 10 null string, 4 graph, 13, 143 n-variable automaton, 91 semidirect, 12 n-variable language, 91 wreath, 12, 147 one-counter production rule (of grammar), 37 pda,43 pumping lemma automaton, 43 for context-free languages, 72 language, 43, 80, 232 for regular languages, 60 ordering pushdown automaton, 41, 60 lenlex, 5, 120 quasi-isometric recursive path, 120 embedding, 22, 23 reduction, 9, 119 groups, 24 shortlex, 5, 120 spaces, 23 wreath product, 120 quasi-isometry, 23 padding symbol, 92 quasi-real-time language, 48 parallel poly-pushdown group, 147 quasiconvex subgroup, 166, 187, 191 parallel poly-pushdown language, 49 quasiconvex subset, 184 Parallel Wall Theorem, 173 quasigeodesic, 160, 165 Parkih’s theorem, 75 path, 24 parse tree (of grammar), 50 word, 24 partial fsa,40 radius (of van Kampen diagram), 114 partial function, 46 rank (of free group), 6 recursive, 46 rational path expression, 57 geodesic, 20 language, 38 pda,60 normal form, 184 deterministic, 42, 62 structure, 184 non-deterministic, 42 subgroup, 166 one-counter, 43 real-time stack of, 41 K-step Turing machine, 249 permutational transducer, 197 group, 252 mapping, 198, 239, 268 language, 47, 240, 249 phrase structure, 36 tidy language, 252 piece (in group presentation), 26 Turing machine, 47, 249 pinch (in HNN-extension), 18, 224 recurrent group, 209 ping-pong lemma, 14 recursive poly-context-free language, 45 group, 255 partial function, 46 language, 48, 249, 254 path ordering, 120 polycyclic group, 3, 140, 189, 264 presentation, 98

Index of Topics and Terminology 293

recursively enumerable language, 45 Seifert–van Kampen Theorem, 14, 16 recursively enumerable presentation, 98 self-similar group, 207 recursively presentable group, 98 semi-deterministic co-context-free, 260, 262 reduced semi-deterministic co-indexed, 239, 268 form (in free product), 16 semi-Thue grammar, 36 word, 7 semidirect product, 12 word (in rewriting system), 119 semigroup, 4 reduction ordering, 9, 119 free, 6 regular shortlex branch group, 209 automatic coset system, 190 expression, 57 automatic structure, 135 grammar, 59 normal form, 5 language, 38, 53 ordering, 5, 120 Reidemeister–Schreier rewriting, 9 slim triangle, 151 Reidemeister–Schreier Theorem, 10 small cancellation, 26, 140, 182, 237 relation, 6 soluble group, 3 braid, 30 soluble word problem, 7 defining, 6 solvable group, 3 relatively hyperbolic group, 113, 140, 170, 173 spherical Coxeter group, 29 relator, 6 split extension, 12 defining, 6 square complex, 211 residually finite group, 216 stack restricted of pda,41 direct product, 12 stack automaton, 48 standard wreath product, 13 standard wreath product, 13 wreath product, 13 star-free language, 49, 182 reversible automaton, 211 start state, 38 rewrite rules, 111, 119 of pda,60 rewriting start symbol (of grammar), 37 Reidemeister–Schreier, 9 state rewriting system, 37, 111, 118 accepting, 38 complete, 120 accepting of pda,60 confluent, 120 active, 198 convergent, 120 failure, 39 locally confluent, 120 final, 38 Noetherian, 120 inactive, 198 terminating, 120 start, 38 right-angled Artin group, 13, 27, 31, 140 start of pda,60 right-most derivation, 51 state-closed group, 207 right-regular grammar, 59 stratified rigid stabiliser, 208 linear set, 76 rooted tree automorphism, 205 semilinear set, 76 roots string, 4 fundamental, 174 strongly automatic coset system, 190 negative, 174 strongly geodesically automatic group, 145, of Coxeter group, 174 150, 155 positive, 174 subgroup Schreier generator, 9 A-quasiconvex, 187 Schreier graph, 21, 184, 210 quasiconvex, 166, 187, 191 Schreier transversal, 9 rational, 166 section of tree automorphism, 204 subgroup growth, 217

294 Index of Topics and Terminology

surface group, 18, 28 Type 1 grammar, 85 symmetric closure (of group presentation), 26 Type 2 grammar, 69 symmetric closure (of relators), 110 unique word-acceptor, 118 synchronous unrestricted grammar, 36 2-variable fsa,93 useful variable, 69 automaton, 198 useless variable, 69, 229 fellow traveller property, 127 van Kampen diagram, 27, 99 syntactic congruence, 51, 108 diameter, 114, 243 syntactic monoid, 52, 108 filling length, 114 syntactic morphism, 52 radius, 114 terminal (of grammar), 37 variable terminating rewriting system, 120 of grammar, 37 thin triangle, 152 useful, 69 Thompson’s group F, 35, 113, 196 useless, 69, 229 Thurston’s Geometrisation Conjecture, 146 virtual endomorphism, 207 tidy real-time language, 252 virtually abelian group, 170, 182 total function, 46 weakly branch group, 208 transducer Whitehead problem, 167 deterministic, 195 Wirtinger presentation, 33 finite state, 194 word, 4 inverse, 197 cyclically reduced, 7, 26 minimisation of, 214 empty, 4 permutational, 197 geodesic, 4, 5, 21, 169 transition function positive, 30 of fsa,52 quasigeodesic, 24 of pda,61 reduced, 7 transition monoid, 54 reduced (in rewriting system), 119 transversal word metric, 153 Schreier, 9 word problem, 7, 39, 133, 166, 213 tree automorphism generalised, 8, 167, 186 automaton of, 204 of group, 97 bounded, 204, 239, 268 soluble, 7 circuitous, 206 word-acceptor, 117, 118, 127 directed, 205 coset, 189, 190 rooted, 205 unique, 118 section of, 204 word-difference, 127 triangle automaton, 129 δ-slim, 151 word-hyperbolic Coxeter group, 30, 141 δ-thin, 152 word-hyperbolic group, 28, 30, 105, 111, 139, geodesic, 150 145, 150, 170, 187, 191, 240, 252, 254 Turing machine, 36, 44, 82 words over X,4 K-step real-time, 249 wreath product, 12, 147 deterministic, 44 restricted, 13 linearly bounded, 47 restricted standard, 13 non-deterministic, 44 standard, 13 real-time, 47, 249 wreath product ordering, 120 Type 0 grammar, 36