Open Problems

In the list below we collect some open problems related to the topics treated in this book.

(OP-1) Let G be an amenable periodic which is not locally finite. Does there exist a finite set A and a τ : AG → AG which is surjective but not injective? (OP-2) Let G be a periodic group which is not locally finite and let A be an infinite set. Does there exist a bijective cellular automaton τ : AG → AG which is not invertible? (OP-3) Let G be a periodic group which is not locally finite and let V be an infinite-dimensional vector space over a field K. Does there exist a bijective linear cellular automaton τ : V G → V G which is not invertible? (OP-4) Is every Gromov- residually finite (resp. residually amenable, resp. sofic, resp. surjunctive)? (OP-5) (Gottschalk’s conjecture) Is every group surjunctive? (OP-6) Let G be a periodic group which is not locally finite and let A be an infinite set. Does there exist a cellular automaton τ : AG → AG whose image τ(AG) is not closed in AG with respect to the prodiscrete topology? (OP-7) Let G be a periodic group which is not locally finite and let V be an infinite-dimensional vector space over a field K. Does there exist a linear cellular automaton τ : V G → V G whose image τ(V G)isnot closed in V G with respect to the prodiscrete topology? (OP-8) Let G and H be two quasi-isometric groups. Suppose that G is surjunctive. Is it true that H is surjunctive? (OP-9) Let G be a non-. Does there exist a finite set A and a cellular automaton τ : AG → AG which is pre-injective but not surjective? (OP-10) Does there exist a non-sofic group? (OP-11) Does there exist a surjunctive group which is non-sofic?

T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, 417 Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1, © Springer-Verlag Berlin Heidelberg 2010 418 Open Problems

(OP-12) Let G and H be two quasi-isometric groups. Suppose that G is sofic. Is it true that H is sofic? (OP-13) Let G be a non-amenable group and let K be a field. Does there exist a finite-dimensional K-vector space V and a linear cellular automaton τ : V G → V G which is pre-injective but not surjective? (OP-14) Let G be a non-amenable group and let K be a field. Does there exist a finite-dimensional K-vector space V and a linear cellular automaton τ : V G → V G which is surjective but not pre-injective? (OP-15) (Kaplanski’s stable finiteness conjecture) Is the group algebra K[G] stably finite for any group G and any field K? Equivalently, is every group L-surjunctive, that is, is it true that, for any group G,any field K, and any finite-dimensional K-vector space V , every injective linear cellular a automaton τ : V G → V G is surjective? (OP-16) (Kaplanski’s zero-divisors conjecture) Is it true that the group alge- bra K[G] has no zero-divisors for any torsion- G and any field K? Equivalently, is it true that, for any torsion-free group G and any field K, every non-identically-zero linear cellular automaton τ : KG → KG is pre-injective? (OP-17) Is every unique-product group orderable?

Comments

(OP-1) The answer to this question is affirmative if G is non-periodic, i.e., it contains an element of infinite order (see Exercise 3.23), or if G is non- amenable (Theorem 5.12.1). On the other hand, if G is a locally finite group and A is a finite set, then every surjective cellular automaton τ : AG → AG is injective (see Exercise 3.21). An example of an amenable periodic group which is not locally finite is provided by the Grigorchuck group described in Sect. 6.9. (OP-2) The answer is affirmative if G is not periodic (cf. [CeC11, Corol- lary 1.2]). On the other hand, if G is locally finite and A is an arbitrary set, then every bijective cellular automaton τ : AG → AG is invertible (cf. Exercise 3.20 or [CeC11, Proposition 4.1]). (OP-3) The answer is affirmative if G is not periodic (cf. [CeC11,The- orem 1.1]). On the other hand, if G is locally finite and V is an arbitrary vector space, then every bijective linear cellular automaton τ : V G → V G is invertible (cf. [CeC11, Proposition 4.1]). (OP-5) Every sofic group is surjunctive (cf. Theorem 7.8.1). (OP-6)WhenA is a finite set and G is an arbitrary group, it follows from Lemma 3.3.2 that the image of every cellular automaton τ : AG → AG is closed in AG.WhenA is an infinite set and G is a non-periodic group, it is shown in [CeC11, Corollary 1.4] that there exists a cellular automaton τ : AG → AG whose image is not closed in AG. On the other hand, when G Comments 419 is locally finite, then, for any set A, the image of every cellular automaton τ : AG → AG is closed in AG (cf. Exercise 3.22 or [CeC11, Proposition 4.1]). (OP-6)WhenV is a finite-dimensional vector space over a field K and G is an arbitrary group, it follows from Theorem 8.8.1 that the image of every linear cellular automaton τ : V G → V G is closed in V G.WhenV is an infinite-dimensional vector space and G is a non-periodic group, it is shown in [CeC11, Theorem 1.3] that there exists a linear cellular automaton τ : V G → V G whose image is not closed in V G. On the other hand, when G is locally finite, then, for any vector space V , the image of every linear cellular automaton τ : V G → V G is closed in V G (cf. [CeC11, Proposition 4.1]). (OP-9) The answer is affirmative if G contains a nonabelian free subgroup (cf. Proposition 5.11.1). (OP-13) The answer is affirmative if G contains a nonabelian free subgroup (cf. Corollary 8.10.2). (OP-14) The answer is affirmative if G contains a nonabelian free subgroup (cf. Corollary 8.11.2). (OP-15) See the discussion in the notes at the end of Chap. 8. (OP-16) See the discussion in the notes at the end of Chap. 8. References

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Symbol Definition Page  the empty word 367  the dominance relation in the set of growth functions γ : N → [0, +∞) 162 ∼ the equivalence relation in the set of growth functions γ : N → [0, +∞) 162 p T p→p the  -norm of a linear map T : p(E) → p(E) 194 0R, 0 the zero element of the ring R 291 1G the identity element of the group G 2 1R, 1 the unity element of the ring R 291 G γS , γS the growth function of the group G relative to the finite symmetric generating subset S ⊂ G 160 G ΔS , ΔS the discrete laplacian on the group G associated with the subset S ⊂ G 9 (2) ΔS the restriction of ΔS to the Hilbert space 2(G) 201 ∂E(Ω)theE-boundary of the subset Ω ⊂ G 116 ιS(G) the isoperimetric constant of the group G with respect to the finite symmetric generating subset S ⊂ G 191 G λS , λS the growth rate of the group G with respect to the finite symmetric generating subset S ⊂ G 169 λ(e) the label of the edge e ∈ E in a labeled graph G =(Q, E) 153 λ(π) the label of the path π in a labeled graph G =(Q, E) 154, 223 π− the initial vertex of the path π in an S-labeled graph 154

T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, 429 Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1, © Springer-Verlag Berlin Heidelberg 2010 430 List of Symbols

Symbol Definition Page π+ the terminal vertex of the path π in an S-labeled graph 154 σ(T ) the real spectrum of T ∈L(X) 406 ψq,r the S-labeled graph isomorphism from BS(r)ontoB(q, r) such that ψq,r(1G)=q 265 Ω−E the E-interior of the subset Ω ⊂ G 115 Ω+E the E-closure of the subset Ω ⊂ G 115 A∗ the monoid consisting of all words on the alphabet A 367 AG the set of all configurations x: G → A 2 B(q, n) the ball of radius n in an S-labeled graph Q =(Q, E) centered at the vertex q ∈ Q 265 G BS (g, n), BS(g, n) the ball of radius n in G centered at the element g ∈ G with respect to the word metric 153 G BS (n), BS(n) the ball of radius n in G centered at the identity element 1G ∈ G with respect to the word metric 153 CA(G; A) the monoid consisting of all cellular automata τ : AG → AG 13 CA(G, H; A) the submonoid of CA(G; A) consisting of all cellular automata τ : AG → AG admitting a memory set S such that S ⊂ H 16 i (C (G))i≥0 the lower central series of the group G 93 CS(G) the Cayley graph of the group G with respect to the finite symmetric generating subset S ⊂ G 156 dF the normalized Hamming distance on Sym(F ) 251 G dS , dS the word metric on G with respect to the finite symmetric generating subset S ⊂ G 152 dQ the graph metric in the edge-symmetric S-labeled graph Q 155 D(G) the derived subgroup of the group G 92 i (D (G))i≥0 the derived series of the group G 92 G entF (X) the entropy of the subset X ⊂ A with respect to the right Følner net F 125 F (X) the free group based on the set X 371 Fn the free group of rank n 372 Fix(α) the set of fixed points of the permutation α 251 Fix(H) the set of configurations x ∈ AG fixed by H 4 G = X; R the presentation of the group G given by the generating subset X and the set of relators R 375 List of Symbols 431

Symbol Definition Page gx the configuration defined by gx(h)=x(g−1h) 2 G =(X, Y, E) the bipartite graph with left (resp. right) vertex set X (resp. Y )andsetofedgesE 391 HR the Heisenberg group with coefficients in the ring R 94 ICA(G; A) the group consisting of all invertible cellular automata τ : AG → AG 24 IdX the identity map on the set X 2 (w) the length of the word w ∈ A∗ 35 p(E) the Banach space of all p-summable functions x: E → R 193 ∞(E) the Banach space of all bounded functions x: E → R 78 G ∈ S (g), S(g) the word-length of the element g G with respect to the finite symmetric generating subset S ⊂ G 152 LCA(G; V ) the algebra of all linear cellular automata τ : V G → V G 287 LCA(G, H; V ) the subalgebra of LCA(G; V ) consisting of all linear cellular automata τ : V G → V G admitting a memory set S such that S ⊂ H 289 L(X) the language associated with the subshift X 35 Ln(X) the set of admissible words of length n of the subshift X 35 L(X) the space of all continous endomorphisms of the Banach space X 406 (p) p MS the  -Markov operator associated with the finite subset S ⊂ G 195 Matd(R) the ring consisting of all d × d matrices with entries in the ring R 305 mdimF (X) the mean dimension of the vector subspace X ⊂ V G with respect to the right Følner net F 308 M(E) the set of all means on the set E 79 PM(E) the set of all finitely additive probability measures on the set E 79 N (Γ ) the space of all normal subgroups of the group Γ or, equivalently, the space of all Γ -marked groups 61 NL(B) ⊂ X the left-neighborhood of the subset B ⊂ Y in the bipartite graph (X, Y, E) 392 NL(y) ⊂ X the left-neighborhood of the vertex y ∈ Y in the bipartite graph (X, Y, E) 391 432 List of Symbols

Symbol Definition Page NR(A) ⊂ Y the right-neighborhood of the subset A ⊂ X in the bipartite graph (X, Y, E) 392 NR(x) ⊂ Y the right-neighborhood of the vertex x ∈ X in the bipartite graph (X, Y, E) 391 P(E) the set of all subsets of the set E 77 per(B) the period of the matrix B 227 per(G) the period of the labeled graph G 227 per(X) the period of the irreducible sofic subshift X 227 Z pern(X)thenumberofn -periodic configurations in the subshift X 227 Q(r) the set of all vertices of the S-labeled graph Q =(Q, E) for which there exists an S-labeled graph isomorphism ψq,r : BS(r) → B(q, r) satisfying ψq,r(1G)=q 265 Q =(Q, E)theS-labeled graph with vertex set Q and edge set E ⊂ Q × S × Q 153 R[G] the group ring of the group G with coefficients in the ring R 292 Rop the opposite ring of the ring R 293 Sym(X) the of the set X 359 Sym0(X) the subgroup of Sym(X) consisting of all permutations with finite support 360 + Sym0 (X) the on X 364 Symn the symmetric group of degree n 366 + Symn the alternating group of degree n 366 Sym(X, ) the subgroup of Sym(X)thatpreservethe partial order  of the set X 179, 333 U(R) the multiplicative group consisting of all invertible elements in the ring R, 292 V [G] the vector subspace of V G consisting of all configurations x: V → G with finite support 288 G x|Ω the restriction of the configuration x ∈ A to the subset Ω ⊂ G 3 X∗ the topological dual of the real normed space X 384 X(A) the subshift of finite type defined by the set of admissible patterns A 32 Xf the set of all configurations in the subshift X whose G-orbit is finite 71 XP the subshift defined by the set of forbidden patterns P 32 XG the subshift defined by the labeled graph G 223 Z(G) the center of the group G 94 Index

Δ-irreducible subshift, 34 back-tracking, 156 Baire theorem, 403 action Banach-Alaoglu theorem, 385 continuous —, 3 base equivariantly approximable —, 279 — of a uniform structure, 353 expansive —, 65 free —, 368 faithful —, 50 based free group, 367 topologically mixing —, 29 bi-invariant metric, 251 topologically transitive —, 32 bi-orderable group, 341 uniformly continuous —, 64 bipartite additive cellular automaton, 335 — graph, 391 adjacency matrix of a labeled graph, 227 — subgraph, 391 admissible pattern, 32 finite — graph, 392 admissible word, 35 locally finite — graph, 392 affine Boolean ring, 340 — group, 93 boundary, 116 — map, 387 Burnside problem, 214 algebra, 286 — homomorphism, 289 Cantor-Bernstein theorem, 398 — isomorphism, 290 Cayley graph, 156 almost cellular automaton, 6 — -homomorphism, 234, 254 additive —, 335 — equal configurations, 112 induced —, 17 — perfect group, 55 invertible —, 24 — periodic configuration, 72 linear —, 284, 335 alphabet, 2 reversible —, 24 alternating group, 364 characteristic map, 79 —ofrankn, 366 closed path, 155 amenable closure, 115 — group, 87 cluster point of a net, 345 elementary — group, 215, 338 color, 2 Artinian module, 69 commensurable groups, 171 automaton commutative-transitive group, 279 additive cellular —, 335 commutator cellular —, 6 — of two group elements, 92 linear cellular —, 284 — subgroup, 92 automorphism group, 45 simple —, 176

T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, 433 Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1, © Springer-Verlag Berlin Heidelberg 2010 434 Index compact topological space, 347 entourage, 352 completion entropy proamenable —, 108 topological —, 142 pronilpotent —, 108 equipotent sets, 372 prosolvable —, 108 equivalence of growth functions, 162 complexity of a paradoxical decomposition, equivariant map, 5 106 equivariantly approximable action, 279 composition of paths, 154 even subshift, 35 concatenation, 367 expansive action, 65 configuration, 2 expansivity H-periodic —, 3 constant, 65 almost periodic —, 72 entourage, 65 Garden of Eden —, 111 exponential growth, 164 language of a —, 73 Toeplitz —, 74 Følner conjugate elements, 362 — conditions, 96 connected labeled graph, 155 —theorem,99 Connes embedding conjecture, 277 left — net, 96 context-free subshift, 225 left — sequence, 96 convergent net, 344 right — net, 96 convex subset, 383 right — sequence, 96 convolution product, 291 faithful action, 50 convolutional encoders, 335 Fibonacci sequence, 219 Curtis-Hedlund theorem, 20 field, 284 cycle, 361 filter, 409 — generated, 409 convergent —, 412 Day’s problem, 105 Fr´echet —, 409 Dedekind finite ring, 336 limit of a —, 412 degree principal —, 409 — of a graph, 155 residual —, 409 — of a symmetric group, 366 ultra—, 410 — of a vertex, 155 finite intersection property, 347 — of an alternating group, 366 finitely derived — additive probability measure, 77 —series,92 — generated group, 152, 375 — subgroup, 92 — presented group, 376 directed set, 343 bi-invariant — additive probability directly finite ring, 327 measure, 85 discrete uniform structure, 352 left-invariant — additive probability divisible group, 38 measure, 85 dominance of growth functions, 162 right-invariant — additive probability measure, 85 edge forbidden — -symmetric labeled graph, 155 — pattern, 32 — of a bipartite graph, 391 — word, 35 inverse —, 155 Fr´echet filter, 409 of a labeled graph, 153 free elementary — base, 368 — amenable group, 215, 338 — base subset, 368 — reduction, 370 — group, 368 empty — group of rank k, 372 — path, 154 — ultrafilter, 410 — word, 367 based — group, 367 Index 435

rank of a — group, 373 LEF —, 247 fully residually free group, 279 linear —, 51 locally P —, 58 Garden of Eden locally indicable —, 337 — configuration, 111 marked —, 61 — pattern, 112 metabelian —, 92 — theorem, 114, 128 nilpotent —, 93 — theorem for linear cellular automata, orderable —, 331 312 periodic —, 29, 105 generating subset, 151 polycyclic —, 106, 108 generator of a presentation, 375 presentation of a —, 375 golden mean subshift, 35 profinite —, 41 graph residually C —, 238 — metric, 155 residually P —, 62, 131 bipartite —, 391 residually amenable —, 132 Cayley —, 156 residually finite —, 37 degree of a regular labeled —, 155 simple —, 44 finite labeled —, 154 sofic —, 254 labeled —, 153 solvable —, 93 loop in a labeled —, 154 surjunctive —, 57 regular labeled —, 155 symmetric —, 359 tree, 156 symmetric — of rank n, 366 Grigorchuk group, 179 unique-product —, 331 abelianization of the —, 222 virtually P —, 41 group growth — algebra, 294 — function, 160, 162 —ofp-adic integers, 41 — rate, 169 — of intermediate growth, 190 — type of a group, 163 — ring, 292 equivalence class of — functions, 163 affine —, 93 equivalence of — functions, 162 almost perfect —, 55 exponential —, 164 alternating —, 364 intermediate —, 190 alternating — of rank n, 366 polynomial —, 164 amenable —, 87 subexponential —, 164 automorphism —, 45 bi-orderable —, 341 Cayley graph of a —, 156 Hall commutative-transitive —, 279 — k-harem conditions, 399 divisible —, 38 — condition, 394 elementary amenable —, 215, 338 — harem theorem, 399 finitely generated —, 152, 375 — marriage theorem, 399 finitely presented —, 376 Hamming metric, 252 free —, 368 Hausdorff metric, 357 free — of rank k, 372 Hausdorff-Bourbaki fully residually free —, 279 — topology, 356 Grigorchuk —, 179 uniform structure, 356 Heisenberg —, 94 Heisenberg group, 94 Hopfian —, 44 homomorphism hyperlinear —, 277 almost- —, 234, 254 Kaloujnine —, 221 labeled graph —, 154 Klein bottle —, 341 Hopfian L-surjunctive —, 324 — group, 44 lamplighter —, 108 — module, 339 LEA —, 247 hyperlinear group, 277 436 Index

ICC-property, 338 — finitely additive probability measure, idempotent, 331 85 proper —, 331 — metric, 251 induced length — cellular automaton, 17 — of a cycle, 361 — labeled subgraph, 154 — of a word, 35, 152 inductive letter, 2 — limit, 379 limit — system of groups, 379 — along an ultrafilter, 413 initial — of a filter, 412 — topology, 346 — point of a net, 344 — uniform structure, 355 inductive —, 379 interior, 115 projective —, 380 intermediate growth, 190 linear inverse — cellular automaton, 284, 335 — edge, 155 — group, 51 — path, 155 Lipschitz-equivalence, 162 invertible cellular automaton, 24 local defining map, 6 irreducible locally — matrix, 227 — P group, 58 — subshift, 32 — convex topological vector space, 383 isomorphism — embeddable, 235 labeled graph —, 154 — finite bipartite graph, 392 isoperimetric constant, 191 — finite labeled graph, 155 — indicable group, 337 loop, 154 Kaloujnine group, 221 lower central series, 93 abelianization of the —, 222 Kesten-Day theorem, 201 majority action, 10 Klein bottle group, 341 marked group, 61 Klein Ping-Pong theorem, 376 Markov operator, 195 Markov-Kakutani theorem, 387 L-surjunctive group, 324 matching, 393 labeled graph, 153 left-perfect —, 393 — homomorphism, 154 perfect —, 393 — isomorphism, 154 right-perfect—, 393 adjacency matrix of a —, 227 mean, 78 connected —, 155 — dimension, 308 edge-symmetric —, 155 bi-invariant —, 86 finite —, 154 left-invariant —, 86 locally finite —, 155 right-invariant —, 86 path in a —, 154 memory set, 6 subgraph, 154 minimal —, 15 subshift defined by a —, 223 metabelian group, 92 labelling map, 153 metric lamplighter group, 108 bi-invariant —, 251 language graph —, 155 — of a configuration, 73 Hamming —, 252 — of a subshift over Z,35 word —, 153 Laplacian, 10 metrizable uniform structure, 352 lattice, 95 Milnor problem, 215 LEA-group, 247 minimal LEF-group, 247 — memory set, 15 left-invariant —set,72 Index 437

— subshift, 72 principal module — filter, 409 Artinian —, 69 — ultrafilter, 410 Hopfian —, 339 proamenable completion, 108 Noetherian —, 339 prodiscrete projective —, 339 — topology, 3, 346 monoid, 13 — uniform structure, 22, 355 Moore neighborhood, 166 product Morse subshift, 74 — topology, 346 entropy of the —, 144 — uniform structure, 355 profinite neighbor, 155 — completion, 55 net, 343 — group, 41 nilpotency degree, 93 — kernel, 39 , 93 — topology, 53 Noetherian projective — module, 339 — limit, 380 — ring, 340 — module, 339 non-principal ultrafilter, 410 — system of groups, 380 normal closure, 375 pronilpotent completion, 108 proper Open mapping theorem, 404 — idempotent, 331 operator norm, 384 — path, 156 opposite ring, 293 prosolvable completion, 108 orderable group, 331 quasi-isometric Ore ring, 340 — embedding, 204 — groups, 206 paradoxical decomposition quasi-isometry, 204 left —, 98 right —, 98 rank of a free group, 373 partially ordered set, 343 reduced path — form, 374 — in a labeled graph, 154 — product, 244 closed —, 155 — word, 373 closed simple —, 156 regular labeled graph, 155 composition, 154 relator of a presentation, 375 empty —, 154 residual inverse —, 155 — filter, 409 label of a —, 154 — set, 409 proper —, 156 — subgroup, 39 simple —, 156 residually pattern, 2 — C group, 238 admissible —, 32 — P group, 62, 131 forbidden —, 32 — amenable group, 132 periodic group, 29, 105 — finite group, 37 permutation, 359 restriction, 17 support of a —, 360 reversible cellular automaton, 24 polycyclic group, 106, 108 right-invariant polynomial, 164 — finitely additive probability measure, pre-injective map, 112 85 presentation — metric, 251 — of a group, 375 ring generator of a —, 375 Boolean —, 340 relator of a —, 375 Dedekind finite —, 336 438 Index

directly finite —, 327 support group —, 292 — of a configuration, 288 Noetherian —, 340 — of a pattern, 2 opposite —, 293 — of a permutation, 360 Ore —, 340 surjunctive stably finite —, 328 — group, 57 unit-regular—, 340 —subshift, 71 von Neumann finite —, 336 symbol, 2 symmetric set — group, 359 directed —, 343 — subset, 152 partially ordered —, 343 syndetic subset, 72 shift, 2 simple Tarski — group, 44 — alternative, 99 — path, 156 — number of a group, 106 closed — path, 156 Tarski-Følner theorem, 99 sofic theorem — subshift, 225 Baire —, 403 — group, 254 Banach-Alaoglu —, 385 solvable group, 93 Cantor-Bernstein —, 398 spectrum Curtis-Hedlund —, 20 real —, 406 Garden of Eden —, 114, 128 stably finite ring, 328 Garden of Eden — for linear cellular state, 2 automata, 312 strong topology, 82, 384 Gromov-Weiss —, 272 strongly irreducible subshift, 34 Hall harem —, 399 subalgebra, 287 Hall marriage —, 399 subexponential growth, 164 Kesten-Day —, 201 subgraph Klein Ping-Pong —, 376 induced labeled —, 154 Markov-Kakutani —, 387 labeled —, 154 open mapping —, 404 submonoid, 17 Tarski-Følner —, 99 subnet, 344 Tychonoff —, 348 subsemigroup, 322 Thue-Morse sequence, 73 subshift, 31 tiling, 122 N-power —, 228 Toeplitz Nth higher block —, 227 — configuration, 74 Δ-irreducible —, 34 — subshift, 74 — defined by a labeled graph, 223 topological — of finite type, 32 — dual, 384 context-free —, 225 — entropy, 142 even —, 35 — manifold, 69 golden mean —, 35 — vector space, 383 irreducible —, 32 topologically mixing language of a — over Z,35 —action,29 minimal —, 72 — subshift, 33 Morse —, 74, 144 topologically transitive action, 32 sofic —, 225 topology strongly irreducible —, 34 Hausdorff-Bourbaki —, 356 surjunctive —, 71 initial —, 346 Toeplitz —, 74 prodiscrete —, 3, 346 topologically mixing —, 33 product —, 346 subword, 35 profinite —, 53 Index 439

strong —, 384 valence of a vertex, 155 weak-∗ —, 384 vertex total ordering, 331 — of a bipartite graph, 391 totally disconnected topological space, 346 — of a labeled graph, 153 transposition, 361 degree of a —, 155 tree, 156 neighbor, 155 trivial uniform structure, 352 valence of a —, 155 Tychonoff theorem, 348 virtually P group, 41 von Neumann ultrafilter, 410 — conjecture, 105 free —, 410 — finite ring, 336 limit along an — , 413 — neighborhood, 165 non-principal —, 410 principal —, 410 ultraproduct, 244 weak-∗ topology, 384 uniform word, 367 — convexity, 407 — length, 152 — embedding, 355 — metric, 153 — isomorphism, 355 admissible —, 35 — structure, 351 empty —, 367 Hausdorff-Bourbaki — structure, 356 forbidden —, 35 induced — structure, 353 length of a —, 35 prodiscrete — structure, 22, 355 reduced —, 373 uniformly continuous subword of a —, 35 —action,64 wreath product, 52 — map, 353 unique — -product group, 331 zero-divisor, 330 — rank property, 340 — conjecture, 337 unit-regular ring, 340 left —, 330 universe, 2 right —, 330