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Introduction
          Definitions and Basics
                     Motivation
           The Result and Idea




A Locally Finite Maximal Cofinitary Group

                    Bart Kastermans

                Department of Mathematics
                  University of Wisconsin

                  kasterma@math.wisc.edu
                   http://www.bartk.nl/


 July 13, 2007: First European Set Theory Meeting




              Bart Kastermans     A Locally Finite Maximal Cofinitary Group
Introduction
                    Definitions and Basics
                               Motivation
                     The Result and Idea


Outline




      Definitions and Basics.
      Motivation.
      The Result and Idea.




                        Bart Kastermans     A Locally Finite Maximal Cofinitary Group
Introduction
                  Definitions and Basics   Definitions
                             Motivation   Basics
                   The Result and Idea




Definition
Sym(N): the group of bijections N → N with operation
composition.
f ∈ Sym(N) is cofinitary iff f is the identity or has only finitely
many fixed points.
G ≤ Sym(N) is a cofinitary group (sharp group) iff all g ∈ G are
cofinitary.
G ≤ Sym(N) is a maximal cofinitary group (MCG) iff G is a
cofinitary group and is not properly contained in another cofinitary
group.




                      Bart Kastermans     A Locally Finite Maximal Cofinitary Group
Introduction
                  Definitions and Basics   Definitions
                             Motivation   Basics
                   The Result and Idea



g ∈ Sym(N) is cofinitary iff it is either the identity or has only
finitely many fixed points.
G ≤ Sym(N) is cofinitary iff all of its elements are.




    (Adeleke, Truss) A maximal cofinitary group can not be
    countable.
    (P. Neumann) There is a cofinitary group of size |R|.
    Any cofinitary group is contained in a maximal cofinitary
    group.
    (Yi Zhang) If |N| < κ ≤ |R| then it is possible that there is an
    MCG G with |G | = κ.


                      Bart Kastermans     A Locally Finite Maximal Cofinitary Group
Introduction
                  Definitions and Basics   Definitions
                             Motivation   Basics
                   The Result and Idea




g ∈ Sym(N) is cofinitary iff it is either the identity or has only
finitely many fixed points.
G ≤ Sym(N) is cofinitary iff all of its elements are.



A cofinitary group is an almost disjoint (eventually different) family
of permutations that is also a group.

                   g −1 f (n) = n ⇔ f (n) = g (n).




                      Bart Kastermans     A Locally Finite Maximal Cofinitary Group
Introduction
                   Definitions and Basics
                              Motivation
                    The Result and Idea




For A ⊆ P(N) maximal almost disjoint know:
Theorem (Miller)
A can be coanalytic.

Theorem (Mathias)
A can not be analytic.
Question
What about mcg?




                       Bart Kastermans     A Locally Finite Maximal Cofinitary Group
Introduction
                  Definitions and Basics
                             Motivation
                   The Result and Idea




Theorem (Su Gao, Yi Zhang)
V = L implies there is an mcg with a coanalytic generating set.

Theorem
V = L implies there is a coanalytic mcg.
Question
[Anatoly Vershik] Does there exist a nonrecursive cofinitary group
with a recursive set of generators?
Answer: yes.




                      Bart Kastermans     A Locally Finite Maximal Cofinitary Group
Introduction
                   Definitions and Basics
                              Motivation
                    The Result and Idea




Question
Does there exist an analytic mcg?
Observation (Blass): if there is an analytic mcg, then it is Borel.
Question
Can there exist a Borel mcg?




                       Bart Kastermans     A Locally Finite Maximal Cofinitary Group
Introduction
                 Definitions and Basics
                            Motivation
                  The Result and Idea




Theorem (Otmar Spinas)
There does not exist a locally compact mcg.

Theorem
There does not exist an eventually bounded mcg.
Question
Does there exist a closed mcg?




                     Bart Kastermans     A Locally Finite Maximal Cofinitary Group
Introduction
                  Definitions and Basics
                             Motivation
                   The Result and Idea




Get better algebraic understanding of mcg.
Theorem
No mcg has infinitely many orbits.

Theorem
MA implies that for any n, m ∈ N there exists an mcg with n finite
orbits and m infinite orbits.
Question
What is the orbit structure of the diagonal action on Nk (k > 1)?




                      Bart Kastermans     A Locally Finite Maximal Cofinitary Group
Introduction
                    Definitions and Basics
                               Motivation
                     The Result and Idea


Constructing MCG

  Use CH or MA.

              Gα+1 = Gα , gα = (Gα ∗ F (x))[x := gα ]


                                 gα =             gα,s
                                            s∈N

  Study
                           w (gα,s )         w (gα,s+1 )
  good extension: no unavoidable new fixed points.

                                  w = u −1 vu
  “Always” gives free groups.
                        Bart Kastermans       A Locally Finite Maximal Cofinitary Group
Introduction
                  Definitions and Basics
                             Motivation
                   The Result and Idea




Theorem
MA implies there exists an mcg into which every countable group
embeds.
Proof is a “cheat”:
                               G = ∗α<c Ga .
with Gα of all different isomorphism types.




                      Bart Kastermans     A Locally Finite Maximal Cofinitary Group
Introduction
                  Definitions and Basics
                             Motivation
                   The Result and Idea




Question
Are the possible complexities of free mcg the same as the possible
complexities of all mcg?
Need more methods of construction.
Test question:
Question
Does there exist a locally finite mcg?
Requires: adding elements having relations with previously added
elements.
Answer: under MA yes.




                      Bart Kastermans     A Locally Finite Maximal Cofinitary Group
Introduction
                  Definitions and Basics
                             Motivation
                   The Result and Idea




Theorem
MA implies there exists a locally finite mcg.
A finite cg has action on N with all finite orbits. On all but finitely
many of these orbits it acts regularly (transitive and fixed point
free).
Any regular action of a group is isomorphic to the group acting on
itself.




                      Bart Kastermans     A Locally Finite Maximal Cofinitary Group
Introduction
                 Definitions and Basics
                            Motivation
                  The Result and Idea




Have G = {gn : n < ω} a locally finite cofinitary group.
f :N    N a finite injective map.
Make f act nice with larger and larger parts of G .
Let Gn denote gi : i ≤ n .




                                         F




                     Bart Kastermans     A Locally Finite Maximal Cofinitary Group
Introduction
                 Definitions and Basics
                            Motivation
                  The Result and Idea




Now for the maximality just need the following lemma.
Lemma
Let G be a finite cofinitary group, and h ∈ Sym(N)  G such that
 G , h is cofinitary. Then for all finite subgroups H ≤ G and all
but finitely many n ∈ N the numbers n and h(n) are not in the
same H orbit.




                     Bart Kastermans     A Locally Finite Maximal Cofinitary Group
Introduction
                  Definitions and Basics
                             Motivation
                   The Result and Idea




Theorem
MA implies there exists a locally finite mcg.




                      Bart Kastermans     A Locally Finite Maximal Cofinitary Group

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A Locally Finite Maximal Cofinitary Group

  • 1. Introduction Definitions and Basics Motivation The Result and Idea A Locally Finite Maximal Cofinitary Group Bart Kastermans Department of Mathematics University of Wisconsin kasterma@math.wisc.edu http://www.bartk.nl/ July 13, 2007: First European Set Theory Meeting Bart Kastermans A Locally Finite Maximal Cofinitary Group
  • 2. Introduction Definitions and Basics Motivation The Result and Idea Outline Definitions and Basics. Motivation. The Result and Idea. Bart Kastermans A Locally Finite Maximal Cofinitary Group
  • 3. Introduction Definitions and Basics Definitions Motivation Basics The Result and Idea Definition Sym(N): the group of bijections N → N with operation composition. f ∈ Sym(N) is cofinitary iff f is the identity or has only finitely many fixed points. G ≤ Sym(N) is a cofinitary group (sharp group) iff all g ∈ G are cofinitary. G ≤ Sym(N) is a maximal cofinitary group (MCG) iff G is a cofinitary group and is not properly contained in another cofinitary group. Bart Kastermans A Locally Finite Maximal Cofinitary Group
  • 4. Introduction Definitions and Basics Definitions Motivation Basics The Result and Idea g ∈ Sym(N) is cofinitary iff it is either the identity or has only finitely many fixed points. G ≤ Sym(N) is cofinitary iff all of its elements are. (Adeleke, Truss) A maximal cofinitary group can not be countable. (P. Neumann) There is a cofinitary group of size |R|. Any cofinitary group is contained in a maximal cofinitary group. (Yi Zhang) If |N| < κ ≤ |R| then it is possible that there is an MCG G with |G | = κ. Bart Kastermans A Locally Finite Maximal Cofinitary Group
  • 5. Introduction Definitions and Basics Definitions Motivation Basics The Result and Idea g ∈ Sym(N) is cofinitary iff it is either the identity or has only finitely many fixed points. G ≤ Sym(N) is cofinitary iff all of its elements are. A cofinitary group is an almost disjoint (eventually different) family of permutations that is also a group. g −1 f (n) = n ⇔ f (n) = g (n). Bart Kastermans A Locally Finite Maximal Cofinitary Group
  • 6. Introduction Definitions and Basics Motivation The Result and Idea For A ⊆ P(N) maximal almost disjoint know: Theorem (Miller) A can be coanalytic. Theorem (Mathias) A can not be analytic. Question What about mcg? Bart Kastermans A Locally Finite Maximal Cofinitary Group
  • 7. Introduction Definitions and Basics Motivation The Result and Idea Theorem (Su Gao, Yi Zhang) V = L implies there is an mcg with a coanalytic generating set. Theorem V = L implies there is a coanalytic mcg. Question [Anatoly Vershik] Does there exist a nonrecursive cofinitary group with a recursive set of generators? Answer: yes. Bart Kastermans A Locally Finite Maximal Cofinitary Group
  • 8. Introduction Definitions and Basics Motivation The Result and Idea Question Does there exist an analytic mcg? Observation (Blass): if there is an analytic mcg, then it is Borel. Question Can there exist a Borel mcg? Bart Kastermans A Locally Finite Maximal Cofinitary Group
  • 9. Introduction Definitions and Basics Motivation The Result and Idea Theorem (Otmar Spinas) There does not exist a locally compact mcg. Theorem There does not exist an eventually bounded mcg. Question Does there exist a closed mcg? Bart Kastermans A Locally Finite Maximal Cofinitary Group
  • 10. Introduction Definitions and Basics Motivation The Result and Idea Get better algebraic understanding of mcg. Theorem No mcg has infinitely many orbits. Theorem MA implies that for any n, m ∈ N there exists an mcg with n finite orbits and m infinite orbits. Question What is the orbit structure of the diagonal action on Nk (k > 1)? Bart Kastermans A Locally Finite Maximal Cofinitary Group
  • 11. Introduction Definitions and Basics Motivation The Result and Idea Constructing MCG Use CH or MA. Gα+1 = Gα , gα = (Gα ∗ F (x))[x := gα ] gα = gα,s s∈N Study w (gα,s ) w (gα,s+1 ) good extension: no unavoidable new fixed points. w = u −1 vu “Always” gives free groups. Bart Kastermans A Locally Finite Maximal Cofinitary Group
  • 12. Introduction Definitions and Basics Motivation The Result and Idea Theorem MA implies there exists an mcg into which every countable group embeds. Proof is a “cheat”: G = ∗α<c Ga . with Gα of all different isomorphism types. Bart Kastermans A Locally Finite Maximal Cofinitary Group
  • 13. Introduction Definitions and Basics Motivation The Result and Idea Question Are the possible complexities of free mcg the same as the possible complexities of all mcg? Need more methods of construction. Test question: Question Does there exist a locally finite mcg? Requires: adding elements having relations with previously added elements. Answer: under MA yes. Bart Kastermans A Locally Finite Maximal Cofinitary Group
  • 14. Introduction Definitions and Basics Motivation The Result and Idea Theorem MA implies there exists a locally finite mcg. A finite cg has action on N with all finite orbits. On all but finitely many of these orbits it acts regularly (transitive and fixed point free). Any regular action of a group is isomorphic to the group acting on itself. Bart Kastermans A Locally Finite Maximal Cofinitary Group
  • 15. Introduction Definitions and Basics Motivation The Result and Idea Have G = {gn : n < ω} a locally finite cofinitary group. f :N N a finite injective map. Make f act nice with larger and larger parts of G . Let Gn denote gi : i ≤ n . F Bart Kastermans A Locally Finite Maximal Cofinitary Group
  • 16. Introduction Definitions and Basics Motivation The Result and Idea Now for the maximality just need the following lemma. Lemma Let G be a finite cofinitary group, and h ∈ Sym(N) G such that G , h is cofinitary. Then for all finite subgroups H ≤ G and all but finitely many n ∈ N the numbers n and h(n) are not in the same H orbit. Bart Kastermans A Locally Finite Maximal Cofinitary Group
  • 17. Introduction Definitions and Basics Motivation The Result and Idea Theorem MA implies there exists a locally finite mcg. Bart Kastermans A Locally Finite Maximal Cofinitary Group