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Tom 154, kii. 2
УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА
Физико-математические пауки
2012
UDK 510.225
ON COFINITARY GROUPS
B. Ka.stermans, Y. Zhang
Abstract
A cofinit.ary group is a subgroup of the symmetric group on the natural numbers in which all non-identity members have finitely many fixed points. In this paper we describe some questions about these groups that interest us as well as questions on related cardinal invariants and isomorphism types.
Key words: cofinit.ary groups, cardinal invariants, isomorphism types.
Introduction
This paper is a review of some talks we have given on various occasions. We hope people reading this will become more interested in these questions and help with their resolution. We begin by defining the main notions of this paper.
Definition 1.
(i) We write Sym(N) for the symmetric group of the natural numbers; the group consisting of all Injections from the natural numbers to the natural numbers, with the operation being composition.
(ii) An element g G Sym(N) is cofinitary if and only if it either has finitely many-fixed points or is the identity.
(iii) A group G < Sym(N) is cofinitary or a cofinitary group if and only if all of its elements are cofinitary.
(iv) A group G < Sym(N) is a maximal cofinitary group (meg) if and only if it is a cofinitary group and is not properly contained in another cofinitary group.
One of the sources of interest in these groups is their connection with almost disjoint families. If we have a collection A of infinite objects, we call elements x, y G A almost disjoint if and only if x n y is finite. We call the family almost disjoint if and only if all distinct x, y G A are almost disjoint. The family is maximal almost disjoint if and only if it is almost disjoint and not properly contained in another almost disjoint family.
If we apply the definitions in the last paragraph with A = P (N), then we get the usual notion of (maximal) almost disjoint family (see. e.g.. Kurieri [1]).
Next we apply these definitions with A = Sym(N). Here we use the convention that f G Sym(N) is identified with its graph, graph(f), which is a subset of the countable set N x N. With this we get the notion of a (maximal) almost disjoint family of permutations. Requiring the group structure on top of this, one obtains the notion of maximal cofinitary group as in Definition 1. We see this by considering the equivalences:
(f◦ g)(n)= n ^ g(n) = f (n) ^ (n,g(n)) G g n f,
where f, g G Sym(N). From this equivalence you see that f-1 o g has finitely many-fixed points if and only if g n f is finite.
Note that the existence of maximal cofinitary- groups follows directly- from Zorn's Lemma: the union of an increasing sequence of cofinitary- groups is a cofinitary- group (being cofinitary- IS 3. local property).
Some other basic results on these groups.
Theorem 1 (Adeleke [2], Truss [3]). A countable cofinitary group is not maximal.
This theorem can be shown using the ideas from Appendix A2 by diagonalization.
Theorem 2 (P. Neumann). There exists a cofinitary group of size |R|.
P. Neumann showed this by studying cofinitary groups with all their orbits finite (see. Cameron [4] for the proof). The following result shows that these two theorems do not determine the cardinality of maximal cofinitary groups in the context of the negation of the continuum hypothesis.
Theorem 3 (Zhang [5]). For all k such that No < k < 2" = A there exists a c.c.c. forcing G such that in MG we have that 2" = A and, there exists a maximal cofinitary group of size k .
This reasoning so far leads to two main motivations for work on cofinitary groups:
Motivation 1. How similar/different are (maximal) cofinitary groups from (maximal) almost disjoint families?
and
Motivation 2. What algebraic properties do (maximal) cofinitary groups have?
In the remainder of this paper we will work out some of the concrete questions this leads to. In Section 1 we look at the descriptive complexity: we explain and describe what is known about the possible complexities of maximal cofinitary groups. In Section 2 we consider the related cardinal invariants: we define a conple of cardinal invariants related to these families and describe some questions about them. In Section 3 we look at isomorphism types: here we explain the very algebraic question of what the possible isomorphism types of maximal cofinitary groups are. And finally in Section 4 we gather some remaining questions that did not fit in the earlier sections: questions on orbit structures and generating sets.
1. Concrete example
We observed above that settling the existence of maximal cofinitary groups is easy. Zorn's Lemma provides a maximal cofinitary group (in fact any cofinitary group can be extended to a maximal cofinitary group with the same reasoning). An object so constructed is one that is usually extremely non-constrnctive and the result therefore usually hard to describe. The following are some well-known examples of this phenomenon:
• (Suslin [6]) No well-ordering of an uncountable set of reals is analytic.
•
measurable.
The last of these items determines the least possible complexity of maximal almost disjoint families when combined with the following theorem.
Theorem 4 (Miller [9]). The axiom of constructibility implies the existence of a coanalytic maximal almost disjoint family.
These ideas and results together with Motivation 1 immediately give rise to the following question.
Question 1. What is the least possible complexity of a maximal cofinitary group?
The result analogous to Miller's result has been obtained for maximal cofinitary groups. This was done in two steps.
Theorem 5 (Gao and Zhang [10]). The axiom of constructibUity implies the existence of a maximal cofinitary group with a coanalytic generating set.
They used the method developed by Miller and an interesting and ingenious coding: the key to applying this method is in proving a lemma of the following general form.
Lemma 1 (Form of Key Lemma in Miller's method). Given
• a countable family of the right type A, and,
• a countable object f.
We can construct a new element g such that • A U g is a family of the right type,
• f <t g uniformly, and,
• if we iterate this construction (with some bookkeeping) N many times, we get a maximal family of the right type (for this item note thai we are in the context of the axiom of constructibUity).
Part of this requires being able to construct a maximal family of the right type under the continuum hypothesis. This is done here by the method of good, extensions as described in [10] and here summarized in Appendix A.
The family is then constructed by iterating the Key Lemma for u1 many steps at every step encoding the construction performed so far into the next element. By decoding this information, we can then from an element decide if it belongs to the family.
Doing this for maximal cofinitary groups, yon do the encoding into the generators yon construct. In [10]. we performed this construction with a very nice encoding, obtaining the above result.
The difficulty with cofinitary groups is that adding a generator (which yon construct) also forces lots of other elements to be added. These yon have less control over: however, these would also need to encode the construction up to that point.
It can be shown that the Key Lemma fails for cofinitary groups (see Kastermans
g
construction up to this point, but also other new elements do. This means that Miller's method as it stands does not work. However, the coding requirement can be relaxed, to have non-uniform encoding. Then by using a simple coding, we can perform the construction and obtain the following theorem.
Theorem 6 (Kastermans [11]). The axiom of constructibUity implies the existence of a coana,lytic maximal cofinitary group.
From the ideas that made us nse Motivation 1 we believe that this is the best possible result in this direction. That is. we believe that the result analogous to Mathias result mentioned above should hold for meg. There are some weak results in this direction, but really the following question (also on Velickovic problem list) is very open.
Question 2. Can there exist Borel maximal cofinitary groups?
This is the right question since Blass (see [10]) has observed that any analytic maximal cofinitary group is already Borel.
2. Cardinal invariants
In this section we describe a question on cardinal invariants related to maximal almost disjoint families. For a good general overview of results and ideas around cardinal
invariants see [12]. Here we focus on cardinal invariants related to different types of maximal almost disjoint families. These invariants are usually written as a with some subscript. We give the definitions next.
Definition 2. a
(ii) ap is the least cardinality of a maximal almost disjoint family of permutations.
(iii) ag is the least cardinality of a maximal cofinitary group.
We think the most interesting question about these cardinal invariants is the following.
Question 3. [14] What is the relationship between ap and ag ?
Other than the obvious (they can be equal), nothing is known. We mention some results related to this that are known.
Related to this Zhang in [15, 16] has shown that it is consistent that there exists a maximal cofinitary group G contained in a maximal almost disjoint family of permutations P where |G| < |Pbut that in the model for Theorem 5 ap and ag do not differ.
The consistency of a < ag was established in [17] and in [18], and the consistency of a < ap in [19]. There is an obvious question to be answered.
Question 4. Can we prove the consistency of ap, ag < a?
J. Brendle once made a conjecture that it should be provable from ZFC that a < ap, ag.
A different question on these invariants is whether they can be singular. Brendle
a
but have not yet worked through the details, that the same result can be obtained for
ag
3. Isomorphism types
This is the most immediate question following from Motivation 2. Say two groups have the same isomorphism type if and only if they are isomorphic. Write T(G) for the
G
Question 5. What is the collection {T(G) | G is a maximal cofinitary group}?
One restriction we know follows from the fact that a cofinitary group with all orbits finite is not maximal (this follows from the fact that a maximal cofinitary group cannot have infinitely many orbits). From this we see that any maximal cofinitary group has an infinite orbit, which (as Andreas Blass observed) quickly implies that a maximal
G
an infinite orbit O, k G ^^d gn G G, n G N such that O = {gn(k) | n G N}. Then g G G has g \ O completely determined by where it maps k, since if l G O, then l = gn(k) for some n, therefore g(k) = g(gn(k)) = gn(g(k)). From this you see that if g, h G G have g(k) = h(k), then g \ O = h \ O which (since G is cofinitary) means g = h. That is, we have shown G = {gn | n G N}. Then G is not maximal since no maximal cofinitary group is countable.
This, together with the obvious restrictions on cardinality, is the only restriction known.
Using the method of good extensions, the resulting groups have a lot of freeness in them. At every step in the construction, all the newly constructed elements are free
over the earlier part of the group. In the notation of the appendix (starting on page 163) we have that Ga+1 — Ga * H for some group H.
On the positive side we know that Martin's axiom implies that there exists a locally finite maximal cofinitary gronp (locally finite means that any finite subset generates a finite subgroup). In this proof we do not extend finite partial functions, but finite group actions. The locally finite isomorphism type is not determined a priori, but is determined, mostly outside of our control, during the construction (see [13]).
4. Miscellany
Following from Motivation 2. we are also interested in the orbit structure of maximal cofinitary groups. Since a cofinitary group is a subset of Sym(N), any such group has a natural action on the natural numbers: (f,n) ^ f (n). The question then becomes the following.
Question 6. What are the possible orbit structures of maximal cofinitary groups?
Above we already mentioned part of the answer: a maximal cofinitary group cannot have infinitely many orbits. We have shown though that from Martin's Axiom a maximal cofinitary group can be constructed with any finite number of finite orbits and any nonzero finite number of infinite orbits. The orbit structure of the diagonal actions has not yet been determined (here by a diagonal action we mean an action on Nk for some k defined by (f, (ni,..., nk)) ^ (fni,..., fnk).
Note that this relates to the question of isomorphism types, and the descriptive complexity of maximal cofinitary groups, since if the answer is that all diagonal actions have only finitely many orbits, then these groups are oligomorphic. This in turn means that if they are closed they are the automorphism group of a No-categorical structure.
Above we mentioned the result of Gao and Zhang that under the axiom of con-strnctibility there exists a maximal cofinitary group with a coanalytic generating set. and our result that then there exists a coanalytic maximal cofinitary group. We do not know that these results are in fact different results: it is conceivable that every maximal cofinitary group with a coanalytic generating set is already coanalytic. The only approximation to showing that they are different is our positive answer to the following question by Verhik: does there exist a computable set of generators that generate a cofinitary group whose isomorphism type is not computable? See [13] for this result. This is still far removed from the question about coanalytic generating sets and groups.
A Construction from CH and MA
In this section we describe some of the combinatorics involved in establishing the results mentioned above: this is just to give some of the flavor. We first establish some notation.
If G < H and g € H, we write (G, g) for the subgroup of H generated by the set {G,g}. F(x) denotes the free group on the generator x. If G and H are groups, we write G * H for their free product. We write WG for G * F(x), which can be identified
xG
goxkogixkl ■ ■■ xklgl+1,
where gi € G for i < l + 1, gi = id for 1 < i < l, and ki € Z \ {0}.
p : A ^ B is the notation for a partial function from A to B (as usual, p : A ^ B is the notation for a total function).
Al. Really easy from CH. First observe that with CH we do not need to do a complicated construction. Enumerate Sym(N) by (/a+i | a e w^, and do the following inductive construction of a sequence of cofinitary groups (Ga | a e w1 U{w1}):
• Go = {id},
„ J(Ga,/a+i) if (Ga,/a+i) is cofinitary;
• Ga+i — <
I Ga otherwise.
• G\ — |J Ga, if A is a limit ordinal.
a< \
Then the group G — GW1 is a maximal cofinitary group: (1) it is cofinitary since it is the union of an increasing sequence of cofinitary groups. (2) it is maximal: suppose, towards a contradiction, that there is a cofinitary group H of which G is a proper subgroup. Choose g e H \ G, note that g — /a+i for some a e w^ Note that (G, g) is cofinitary. From this we see that (Ga, g) — (Ga, /a+i) is also cofinitary. But then by the inductive construction /a+i e Ga+i < G which is a contradiction with g e H \ G. This construction is of very little use in answering questions like the ones in this
CH
arbitrary enumeration determines which elements are in and out. The same method clearly works using AC, then, however, it is easier to use Zorn's Lemma.
This is why in the next paragraph we describe a more complicated construction. This construction and the ideas therein can be tweaked to be useful to many of the above (that is, many of the obtained results above use these ideas).
CH
help us to prove a lemma of the form of Lemma 1. The enumeration that is axiomatically CH
AC
Gao and Zhang [10] describe a more concrete construction fitting with Lemma 1. Given a countable cofinitary group G and an element / e Sym(N), we want to construct an element g e Sym(N) such th at (G, g) is cofinitary, and (G, g, /) is either equal to (G, g) / e G
case. Then it suffices to construct the element g such th at (G, g) is cofinitary and / n g
g
Definition 3. Let p, q : N ^ N be finite partial injective functions, and w e WG. Then q is a good extension of p with respect to w if and only if p C q, and for every n e N such th at w(q)(n) — n there exist l e N, and u, z e WG such that
• w — u-izu without cancellation,
• z(p)(1) — I, and
• u(q)(n) — l.
Note that if w(p)(n) — n we can choose z — w and u — id.
With these definitions the following lemmas can be proved (see [10]).
Lemma 2. Let G < Sym(N), p : N ^ N finite and injective, / e Sym(N) \ G with (G, /) cofinitary, and w e WG. Then
• (Domain Extension Lemma) For each n e N \ dom(p), for all but finitely many k e N the extern ion p U {(n, k)} is a good extensi on of p with respect to w.
• (Range Extension Lemma) For each k e N \ ran(p), for all but finitely many n e N the extern ion p U {(n, k)} is a good extensi on of p with respect to w.
• (Hitting / Lemma) For all but finitely many n e N extension p U{(n, /(n))}
pw
With these lemmas we can do the construction by iterating the following lemma.
Lemma 3. Let G < Sym(N) be countable, f G Sym(N) \ G such that (G, f) cofinitary. Then vie can construct a g G Sym(N) such th at (G, g) is cofinitary, (G, g) = = G * F (x) and, f П g is infinite.
We enumerate WG as E = (wn | n G N). This lemma is then proved by iterating the
f
f
with reference to larger initial segments of the enumeration E. Since Hitting f is used infinitely often, the resulting g satisfying f П g is infinite. Because of this either g = f or (G, g, f) is not cofinitary (since f-1g has infinitely many fixed points, but is not the identity). We see that (G, g) = G * F(x) since for every w G WG from some point we are only taking good extensions with reference to it and all of its snbwords. Then all
w(g)
We can use these same ideas on constructions from MA. Given a group G we define the partial order PG:
• (p, F) G PG if p : N ^ N finite, and G С WG finite.
• (p1; F1) < (po, F0) if and only if p0 С p1, F0 С F1, and p1 is a good extension of po with reference tо all w G F0.
This partial order is c.c.c. since all elements with the same first element are corii-
f
{(p, F) | n G dom(p)}, {(p, F) | n G ran(p)}, and {(p, F | |p П f | > n}. Finally the obvious denseness of {(p, F) | w G F} for w G WG replaces the initial segments of the enumeration above.
The second author's research is partially supported by NSFC grant No. 10971237.
Резюме
Б. Кастермаис, И. Чж.аи. К вопросу о кофипитарпых группах.
Кофипитарпая группа представляет собой подгруппу симметрической группы па множестве натуральных чисел, в которой все пеедипичпые элементы имеют бесконечно много неподвижных точек. В данной работе мы рассматриваем некоторые интересующие пас вопросы, касающиеся таких групп, а также связанных с ними кардинальных инвариантов и тгшов изоморфизма.
Ключевые слова: кофипитарпые группы, кардинальные инварианты, типы изоморфизма.
References
1. Kunen K. Set theory: An Introduction to Independence Proofs. - Amsterdam; N. Y.; Oxford: North-Holland Pub. Co., 1980. - 313 p.
2. Adeleke S.A. Embeddings of infinite permutation groups in sharp, highly transitive, and homogeneous groups // Proc. Edinb. Math. Soc. - 1988. - V. 31, No 2. - P. 169-178.
3. Truss J.K. Embeddings of infinite permutation groups // Proceedings of Groups - St. Andrews 1985 / Ed. by E.F. Robertson, C.N. Campbell. - Cambridge: Cambridge Univ. Press, 1986. - P. 335-351.
4. Cameron P.J. Cofinitary permutation groups / / Bull. London Math. Soc. - 1996. - V. 28, No 2. - P. 113-140.
5. Zhang Y. Maximal cofinitary groups // Arch. Math. Logic. - 2000. - V. 39, No 1. -P. 41-52.
6. Kechris A.S. Classical Descriptive Set Theory. - N. Y.: Springer-Verlag, 1995. - 402 p.
7. Talagrand M. Compacts de Fonctions Mesurables et Filtres Non Mesurables // Studia Math. -1980. - V. 67, No 1. - P. 13-43.
8. Mathias A.R.D. Happy Families // Ann. Math. Logic. - 1977. - V. 12, No 1. - P. 59-111.
9. Miller A. W. Infinite Combinatorics and Definability // Ann. Pure Appl. Logic. - 1989. -V. 41, No 2. - P. 179-203.
10. Gao S., Zhang Y. Definable sets of generators in maximal cofinitary groups // Adv. Math. - 2008. - V. 217, No 2. - P. 814-832.
11. Kastermans B. The Complexity of Maximal Cofinitary Groups // Proc. Amer. Math. Soc. - 2009. - V. 137. - P. 307-316.
12. Blass A. Combinatorial Cardinal Characteristics of the Continuum // M. Foreman, A. Kanamori (Eds.). Handbook of Set Theory. - Dordrecht; Heidelberg; London; N. Y.: Springer, 2012. - P. 395-489.
13. Kastermans B. Isomorphism Types of Maximal Cofinitary Groups // Bull. Symbolic Logic. - 2009. - V. 15, No 3. - P. 300-319.
14. Zhang Y. Permutation groups and covering properties //J. London Math. Soc. - 2001. -V. 63, No 1. - P. 1-15.
15. Zhang Y. Adjoining cofinitary permutations //J. Symbolic Logic. - 1999. - V. 64, No 4. -P. 1803-1810.
16. Zhang Y. Adjoining cofinitary permutations (II) // Arch. Math. Logic. - 2003. - V. 42, No 2. - P. 153-163.
17. Zhang Y. Cofinitary Groups and Almost Disjoint Families: PhD Thesis. - New Brunswick: Rutgers University, 1997.
18. Michael H., Steprans J., Zhang Y. Cofinitary groups, almost disjoint and dominating families // J. Symbolic Logic. - 2001. - V. 66, No 3. - P. 1259-1276.
19. Brendle J., Spinas O., Zhang Y. Uniformity of the meager ideal and maximal cofinitary groups // J. Algebra. - 2000. - V. 232, No 1. - P. 209-225.
20. Brendle J. The almost-disjointness number may have countable cofinality // Trans. Amer. Math. Soc. - 2003. - V. 355, No 7. - P. 2633-2649.
Поступила в редакцию 16.01.12
Kastermans, Bart PliD. Assistant. Professor. Department of Mathematics, University of Colorado, Boulder, Colorado, USA.
Кастерманс, Варт доктор паук, доцепт факультета математики Университета Колорадо, г. Боулдер, шт. Колорадо, США.
E-mail: bari.kastermanseculurado.edu
Zhang, Yi PhD, Professor of Mathematics, Department of Mathematics, Sun Yat-sen University, Guangzhou, China.
Чжан, Ии доктор паук, профессор математики факультета математики Университета им. Супь Ятсепа, г. Гуанчжоу, Китай.
