# The Markov-Zariski topology of an infinite group

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1 Mimar Sinan Güzel Sanatlar Üniversitesi Istanbul January 23, 2014

2 joint work with Daniele Toller and Dmitri Shakhmatov 1. Markov s problem 1 and 2 2. The three topologies on an infinite group 3. Problem 1 and 2 in topological terms 4. The Markov-Zariski topology of an abelian group 5. Markov s problem 3.

3 Markov s problem 1 Definition A group G is topologizable if G admits a non-discrete Hausdorff group topology. Problem 1. [Markov Dokl. AN SSSR 1944] Does there exist a (countably) infinite non-topologizable group? Yes (under CH): Shelah, On a problem of Kurosh, Jonsson groups, and applications. In Word Problems II. (S. I. Adian, W. W. Boone, and G. Higman, Eds.) (North-Holland, Amsterdam, 1980), pp Yes (in ZFC): Ol shanskij, A note on countable non-topologizable groups. Vestnik Mosk. Gos. Univ. Mat. Mekh. (1980), no. 3, 103.

4 Definition (Markov) Markov s problem 2 A subset S of a group G is called: (a) elementary algebraic if S = {x G : a 1 x n 1 a 2 x n 2 a 3... a m x nm = 1} for some natural m, integers n 1,..., n m and elements a 1, a 2,..., a m G. (b) algebraic, if S is an intersection of finite unions of elementary algebraic subsets. (c) unconditionally closed, if S is closed in every Hausdorff group topology of G. Every centralizer c G (a) = {x G : axa 1 x 1 = 1} is an elementary algebraic set, so Z(G) is an algebraic set. (a) (b) (c) Problem 2. [Markov 1944] Is (c) (b) always true?

5 The Zariski topology E G the family of elementary algebraic sets of G. A a G the family of all finite unions of elementary algebraic sets of G. A G the family of all algebraic sets of G. The Zariski topology Z G of G has A G as family of all closed sets. It is a T 1 -topology as E G contains al singletons. Example (a) E Z = {Z, } {{n} : n Z}, so A G = A a G = {Z} [Z]<ω. Hence, Z Z is the cofinite topology of Z. (b) Analogously, if G is a torsion-free abelian group and S = {x G : nx + g = 0} E G, then either S = G or S 1, so again Z G is the cofinite topology of G. (c) [Banakh, Guran, Protasov, Top. Appl. 2012] Z Sym(X ) coincides with the point-wise convergence topology of the permutation group Sym(X ) of an infinite set X. (a) and (b) show that Z G need not be a group topology.

6 Bryant, Roger M. The verbal topology of a group. J. Algebra 48 (1977), no. 2, Wehrfritz s MR-review to Bryant s paper: This paper is beautiful, short, elementary and startling. It should be read by every infinite group theorist. The author defines on any group (by analogy with the Zariski topology) a topology which he calls the verbal topology. He is mainly interested in groups whose verbal topology satisfies the minimal condition on closed sets; for the purposes of this review call such a group a VZ-group. The author proves that various groups are VZ-groups. By far the most surprising result is that every finitely generated abelian-by-nilpotent-by-finite group is a VZ-group. Less surprisingly, every abelian-by-finite group is a VZ-group. So is every linear group. Also, the class of VZ-groups is closed under taking subgroups and finite direct products.

7 The Markov topology and the P-Markov topology The Markov topology M G of G has as closed sets all unconditionally closed subsets of G, in other words M G = inf{all Hausdorff group topologies on G}, where inf taken in the lattice of all topologies on G. P G = inf{all precompact group topologies on G} - precompact Markov topology (a group is precompact if its completion is compact). Clearly, Z G M G P G are T 1 topologies. Problem 2. [topological form] Is Z G = M G always true? Perel man (unpublished): Yes, for abelian groups Markov [1944]: Yes, for countable groups. Hesse [1979]: No in ZFC (Sipacheva [2006]: under CH Shelah s example works as well).

8 Markov s first problem through the looking glass of M G A group G Z-discrete (resp., M-discrete, P-discrete), if Z G (resp., M G, resp., P G ) is discrete. Analogously, define Z-compact, etc. G is Z-discrete if and only if there exist E 1,..., E n E G such that E 1... E n = G \ {e G }; G is M-discrete iff G is non-topologizable. So, G is non-topologizable whenever G is Z-discrete. Ol shanskij proved that for Adian group G = A(n, m) the quotient G/Z(G) m is a countable Z-discrete group, answering positively Porblem 1. Example (a) Klyachko and Trofimov [2005] constructed a finitely generated torsion-free Z-discrete group G. (b) Trofimov [2005] proved that every group H admits an embedding into a Z-discrete group.

9 Example (negative answer to Problem 2) (Hesse [1979]) There exists a M-discrete group G that is not Z-discrete. Criterion [Shelah] An uncountable group G is M G -discrete whenever the following two conditions hold: (a) there exists m N such that A m = G for every subset A of G with A = G ; (b) for every subgroup H of G with H < G there exist n N and x 1,..., x n G such that the intersection n i=1 x 1 i Hx i is finite. (i) The number n in (b) may depend of H, while in (a) the number m is the same for all A [G] G. (ii) Even the weaker form of (a) (with m depending on A), yields that every proper subgroup of G has size < G (if G = ω 1, groups with this property are known as Kurosh groups).

10 (iii) Using the above criterion, Shelah produced an example of an M-discrete group under the assumption of CH. Namely, a torsion-free group G of size ω 1 satisfying (a) with m = and (b) with n = 2. So every proper subgroup H of G is malnormal (i.e., H x 1 Hx = {1}), so G is also simple. Proof. Let T be a Hausdorff group topology on G. There exists a T -neighbourhood V of e G with V G. Choose a T -neighbourhood W of e G with W m V. Now V G and (a) yield W < G. Let H = W. Then H = W ω < G. By (b) the intersection O = n Hx i is finite for some n N and i=1 x 1 i elements x 1,..., x n G. Since each x 1 i Hx i is a T -neighbourhood of 1, this proves that 1 O T. Since T is Hausdorff, it follows that {1} is T -open, and therefore T is discrete.

11 Z-Noetherian groups A topological space X is Noetherian, if X satisfies the ascending chain condition on open sets (or, equivalently, the minimal condition on closed sets). Obviously, a Noetherian space is compact, and a subspace of a Noetherian space is Noetherian itself. Actually, a space is Noetherian iff all its subspaces are compact (so an infinite Noetherian spaces are never Hausdorff). Theorem (Bryant) A subgroup of a Z-Noetherian group is Z-Noetherian, (D.D. - D. Toller) A group G is Z-Noetherian iff every countable subgroup of G is Z-Noetherian. Using the fact that linear groups are Z-Noetherian, and the fact that countable free groups are isomorphic to subgroups of linear groups, one gets Theorem (Guba Mat. Zam.1986, indep., D. Toller - DD, 2012) Every free group is Z-Noetherian.

12 The Zariski topology of a direct product The Zariski topology Z G of the direct product G = i I G i is coarser than the product topology i I Z G i. These two topologies need not coincide (for example Z Z Z is the co-finite topology of Z Z, so neither Z {0} nor {0} Z are Zariski closed in Z Z, whereas they are closed in Z Z Z Z ). Item (B) of the next theorem generalizes Bryant s result. Theorem (DD - D. Toller, Proc. Ischia 2010) (A) Direct products of Z-compact groups are Z-compact. (B) G = i I G i is Z-Noetherian iff every G i is Z-Noetherian and all but finitely many of the groups G i are abelian. According to Bryant s theorem, abelian groups are Z-Noetherian. Corollary A nilpotent group of nilpotency class 2 need not be Z-Noetherian. Take an infinite power of finite nilpotent group, e.g., Q 8.

13 Z-Hausdorff groups and M-Hausdroff groups If {F i i I } is a family of finite groups, and G = i I F i, then the product i I Z F i is a compact Hausdorff group topology, so Z G M G P G i I Z F i. (1) G is Z-Hausdorff if and only if Z G = M G = P G = i I Z F i. (2) G is M-Hausdorff if and only if M G = P G = i I Z F i. Theorem (DD - D. Toller, Proc. Ischia 2010) If {F i i I } is a non-empty family of finite center-free groups, and G = i I F i, then Z G = M G = P G = i I Z F i is a Hausdorff group topology on G. Theorem (Gaughan Proc. Nat. Acad. USA 1966) The permutation group Sym(X ) of an infinite set X is M-Hausdorff. Since Z-Hausdorff M-Hausdorff, this follows also from Banakh-Guran-Protasov theorem. In particular, M Sym(X ) =Z Sym(X ) coincides with the point-wise convergence topology of Sym(X ).

14 P-discrete groups A group G is P-discrete iff G admits no precompact group topologies (i.e., G is not maximally almost periodic, in terms of von Neumann). In parfticular, examples of P-discrete groups are provided by all minimally almost periodic (again in terms of von Neumann, these are the groups G such that every homomorphism to a compact group K is trivial). Example (a) (von Neumann and Wiener) SL 2 (R); (b) The permutation group Sym(X ) of an infinite set X (as M Sym(X ) is not precompact). Theorem (DD - D. Toller, Topology Appl. 2012) Every divisible solvable non-abelian group is P-discrete.

15 Proof. Let G be a divisible solvable non-abelian group. It suffices to see that G admits no precompact group topology. To this end we show that every divisible precompact solvable group must be abelian. Let G be a divisible precompact solvable group. Then its completion K is a connected group. On the other hand, K is also solvable. It is enough to prove that K is abelian. Arguing for a contradiction, assume that K Z(K), is not abelian. By a theorem of Varopoulos, K/Z(K) is isomorphic to a direct product of simple connected compact Lie groups, in particular, K/Z(K) cannot be solvable. On the other hand, K/Z(K) has to be solvable as a quotient of a solvable group, a contradiction. Corollary For every field K with chark = 0 the Heisenberg group 1 K K H K = 1 K is P-discrete. 1

16 The Zariski topology of an abelian group: Markov s problem 3 Definition (Markov, Izv. AN SSSR 1945) A subset A of a group G is potentially dense in G if there exists a Hausdorff group topology T on G such that A is T -dense in G. Example (Markov) Every infinite subset of Z is potentially dense in Z. By Weyl s uniform disitribution theorem for every infinite A = (a n ) in Z there exists α R such that (a n α) is uniformly distributed modulo 1, so the subset (a n α) of R/Z is dense in R/Z (so in α as well). Now the topology T on Z induced by Z = α R/Z works. Problem 3 [Markov] Characterize the potentially dense subsets of an abelian group. A hint. [two necessary conditions] a potentially dense set is Zarisky-dense; if G has a countable potentially dense set, then G 2 c.

17 Theorem (Tkachenko-Yaschenko, Topology Appl. 2002) If an Abelian group with G c is either torsion-free or has exponent p, then every infinite set of G is potentially dense. Question [Tkachenko-Yaschenko] Can this be extended to groups with G 2 c? The answer is (more than) positive: Theorem (DD - D. Shakhmatov, Adv. Math. 2011) For a countably infinite subset A of an Abelian group G TFAE: (i) A is potentially dense in G, (ii) there exists a precompact Hausdorff group topology on G such that A becomes T -dense in G, (iii) G 2 c and A is Zarisky dense in G. The proof if based on a realization theorem for the Zariski closure by means of (metrizable) precompact group topologies.

18 For n ω and E G let G[n] = {x G : nx = 0} and ne = {nx : x E}. E E G, a G, n ω such that E = a + G[n] = {x G : nx = na}. So E G is stable under finite intersections: (a+g[n]) (b+g[m])=c +G[d], with d =GCD(m, n) (if ) Lemma If G is abelian, then A G consists of finite unions of elementary algebraic sets E G, i.e., A G = A a G. Moreover: (a) (G, Z G ) is Noetherian (hence, compact). (b) Z G H = Z H and M G H = M H or every subgroup H of G. All these propertirs are false in the non-abelian case (e.g., when G is a countable Z-discrete group). Example Z G coincides with the cofinite topology of an abelian group G iff either r p (G) < for all primes p or G has a prime exponent p.

19 An algebraic description of the Z-irredducible sets Definition A topological space X is irreducible, if X = F 1 F 2 with closed F 1, F 2 yields X = F 1 or X 2. Lemma For a countably infinite subset A of G TFAE: (a) A is irreducible; (b) A carries the cofinite tiopology; (c) there exists n N such that for every a A ( ) E = A a satisfies ne = 0 and {x E : dx = h} is finite for each h G and every divisor d of n with d n. Let T(G) = {E P(G) : E is irreducible and 0 cl ZG (E)}. For every E T(G) the set E 0 = E {0} is still irreducible. Let o(e) = o(e 0 ) be the number n determined by ( ) and let T n (G) = {E T(G) : o(e) = n}. Then T(G) = n T n(g), T 1 (G) = and T m (G) T n (G) = whenever n m.

20 E T n (G) iff every infinite subset of E is Z G -dense in G[n]. Example Let G be an infinite abelian group. (a) Every countably infinite subset of G is irreducible if G is torsion-free. (b) T 0 (G) = iff G is bounded. (c) T n (G) for some n > 1 iff there exists a monomorphism ω Z(n) G. Theorem Let S be an infinite subset of an abelian group G. Then there exist a finite F S, infinite subsets {S i : i = 1, 2,..., k} of S and a finite set {a 1, a 2,..., a k } of G such that (a) S i a i T ni (G) for some n i ω \ {1}; (b) S = F k i=1 S i; (c) cl ZG (S) = F i cl Z G (S i ) and each S i is Z G -dense in G[n i ].

21 The realzation theorem Theorem (DD - D. Shakhmatov, J. Algebra 2010) Let G be an Abelian group with G c and E be a countable family in T(G). Then there exists a metrizable precompact group topology T on G such that cl ZG (S) = cl T (S) for all S E. The realization of the Zariski closure of uncountably many sets is impossible in general. Corollary For an abelian group G with G 2 c the following are equivalent: (a) every infinite subset of G is potentially dense in G; (b) G is either almost torsion-free or has exponent p for some prime p; (c) every Zariski-closed subset of G is finite. This corollary resolves Tkachenko-Yaschenko s problem. Corollary Z G = M G = P G for every abelian group G.

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