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. 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