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Open Problems 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 group which is not locally finite. Does there exist a finite set A and a cellular automaton 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-hyperbolic group 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-amenable group. 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-free group 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 [Abe] Abels, H.: An example of a finitely presented solvable group. In: Homological Group Theory Proc. Sympos., Durham, 1977. London Math. Soc. Lecture Note Ser., vol. 36, pp. 205–211. Cambridge University Press, Cambridge (1979) [Ady] Adyan, S.I.: Random walks on free periodic groups. Math. USSR, Izv. 21, 425–434 (1983) [AlS] Allouche, J.-P., Shallit, J.: Automatic Sequences: Theory, Applications, Gen- eralizations. Cambridge University Press, Cambridge (2003) [Amo] Amoroso, S., Patt, Y.N.: Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures. J. Comput. Syst. Sci. 6, 448–464 (1972) [AOP] Ara, P., O’Meara, K.C., Perera, F.: Stable finiteness of group rings in arbitrary characteristic. Adv. Math. 170, 224–238 (2002) [Aus] Auslander, L.: On a problem of Philip Hall. Ann. Math. 86, 112–116 (1967) [BanMS] Bandini, S., Mauri, G., Serra, R.: Cellular automata: from a theoreticel paral- lel computational model to its application to complex systems. Parallel Com- put. 27(5), 539–553 (2001). Cellular Automata: From Modeling to Applications (Trieste, 1998) [Bar] Bartholdi, L.: Gardens of Eden and amenability on cellular automata. J. Eur. Math. Soc. 12, 241–248 (2010) [Bas] Bass, H.: The degree of polynomial growth of finitely generated nilpotent groups. Proc. Lond. Math. Soc. 25, 603–614 (1972) [Bau] Baumslag, G.: Automorphism groups of residually finite groups. J. Lond. Math. Soc. 38, 117–118 (1963) [BaS] Baumslag, G., Solitar, D.: Some two-generator one-relator non-Hopfian groups. Bull. Am. Math. Soc. 68, 199–201 (1962) [BDV] Bekka, B., de la Harpe, P., Valette, A.: Kazhdan’s Property (T ). New Mathe- matical Monographs, vol. 11. Cambridge University Press, Cambridge (2008) [Ber] Bergman, G.M.: Right orderable groups that are not locally indicable. Pac. J. Math. 147, 243–248 (1991) [BCG] Berlekamp, E.R., Conway, J.H., Guy, R.K.: Winning Ways for Your Mathe- matical Plays, vol. 2, 2nd edn. AK Peters, Natick (2003) [BKM] Blanchard, F., K˚urka, P., Maass, A.: Topological and measure-theoretic prop- erties of one-dimensional cellular automata. Physica D 103(1–4), 86–99 (1997). Lattice Dynamics (Paris, 1995) [BoR] Botto Mura, R., Rhemtulla, A.: Orderable Groups. Lecture Notes in Pure and Applied Mathematics, vol. 27. Dekker, New York (1977) [Bou] Bourbaki, N.: Topologie G´en´erale, Chapitres 1 `a 4. Hermann, Paris (1971) T. Ceccherini-Silberstein, M. Coornaert, Cellular Automata and Groups, 421 Springer Monographs in Mathematics, DOI 10.1007/978-3-642-14034-1, © Springer-Verlag Berlin Heidelberg 2010 422 References [Bow] Bowen, L.: Measure conjugacy invariants for actions of countable sofic groups. J. Amer. Math. Soc. 23, 217–245 (2010) [BuM] Burger, M., Mozes, S.: Lattices in product of trees. Publ. Math. IHES´ 92, 151–194 (2000) [Bur] Burks, A.W.: von Neumann’s self-reproducing automata. In: Burks, A.W. (ed.) Essays on Cellular Automata, pp. 3–64. University of Illinois Press, Champaign (1971) [CFP] Cannon, J.W., Floyd, W.J., Parry, W.R.: Introductory notes on Richard Thompsons groups.
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