[GAP Forum] The Higman-Thompson group

[Stefan Kohl]

Dear Forum,

Two weeks ago, I have posted the following example of a finitely-generated
infinite simple group to the group-pub-forum:

     Def.: Given disjoint residue classes r_1(m_1) and r_2(m_2) of the
           integers, let the class transposition (r_1(m_1),r_2(m_2)) be the
           permutation which interchanges r_1 + k * m_1 and r_2 + k * m_2
           for each integer k and which fixes all other points.

     Then our group is

             G := < (0(2),1(4)), (0(4),1(4)), (1(4),2(4)), (2(4),3(4)) >.

Having loaded the RCWA package, this group can be entered into GAP by

     gap> G := Group(List([[0,2,1,4],[0,4,1,4],[1,4,2,4],[2,4,3,4]],
     >                    ClassTransposition));
     <rcwa group over Z with 4 generators>

Last week, in an answer to my posting John P. McDermott reported that he
has found out that this group is isomorphic to the (first) Higman-Thompson
group, which is defined and investigated in

[Higman74] Graham Higman. Finitely Presented Infinite Simple Groups.
            Notes on Pure Mathematics, 1974, Department of Pure Mathematics,
            Australian National University, Canberra, ISBN 0 7081 0300 6.

The 'standard generators' kappa, lambda, mu and nu given there correspond
to (0(2),1(2)), (1(2),2(4)), (0(2),1(4)) and (1(4),2(4)), respectively.

As the Higman-Thompson group is simple, verifying the isomorphism
requires now only a (very quick and easy) computational check whether
the generators satisfy the 16 relations given on page 50 of Higman's book:

--------------------------------------------------------------------------
gap> k := ClassTransposition(0,2,1,2);; # kappa in Higman74
gap> l := ClassTransposition(1,2,2,4);; # lambda    "
gap> m := ClassTransposition(0,2,1,4);; # mu        "
gap> n := ClassTransposition(1,4,2,4);; # nu        "
gap> H := Group(k,l,m,n);
<rcwa group over Z with 4 generators>
gap> G = H;
true
gap> HigmanThompsonRels :=
 > [ k^2, l^2, m^2, n^2,                           # (1) in Higman74, p.50.
 >   l*k*m*k*l*n*k*n*m*k*l*k*m,                    # (2)         "
 >   k*n*l*k*m*n*k*l*n*m*n*l*n*m,                  # (3)         "
 >   (l*k*m*k*l*n)^3, (m*k*l*k*m*n)^3,             # (4)         "
 >   (l*n*m)^2*k*(m*n*l)^2*k,                      # (5)         "
 >   (l*n*m*n)^5,                                  # (6)         "
 >   (l*k*n*k*l*n)^3*k*n*k*(m*k*n*k*m*n)^3*k*n*k*n,# (7)         "
 >   ((l*k*m*n)^2*(m*k*l*n)^2)^3,                  # (8)         "
 >   (l*n*l*k*m*k*m*n*l*n*m*k*m*k)^4,              # (9)         "
 >   (m*n*m*k*l*k*l*n*m*n*l*k*l*k)^4,              #(10)         "
 >   (l*m*k*l*k*m*l*k*n*k)^2,                      #(11)         "
 >   (m*l*k*m*k*l*m*k*n*k)^2 ];;                   #(12)         "
gap> Set(HigmanThompsonRels);
[ IdentityMapping( Integers ) ]
--------------------------------------------------------------------------

In fact, G = H is the group which is generated by the set of all class
transpositions which interchange residue classes modulo powers of 2.

Def.: Given a set P of odd primes, let CT_P(Z) be the group which is
       generated by all class transpositions (r_1(m_1),r_2(m_2)) for which
       all odd prime factors of m_1 and m_2 lie in P.

In this notation, G is the group CT_P(Z), where P = {} (i.e. the empty set).

By Corollary 3.7 in

   http://www.cip.mathematik.uni-stuttgart.de/~kohlsn/preprints/simplegp.pdf,

the groups CT_P(Z) are all simple.

The intersection of these uncountably many infinite simple groups is our
group G, hence is isomorphic to the Higman-Thompson group.

All groups CT_P(Z) are subgroups of the group CT(Z), which is generated
by the set of all class transpositions. Thus, very roughly we can depict
the situation as follows:

                                CT(Z)

                          /       |       \
                        /         |         \
                      /           |           \
                    /             |             \

              CT_{3}(Z) ... CT_{5,7,23}(Z) ... CT_{p = 1 mod 4}(Z) ...

                    \             |             /
                      \           |           /
                        \         |         /
                          \       |       /

                       G (Higman-Thompson group)

Our group G preserves a certain tree structure. The groups CT_P(Z) for
nonempty sets P of odd primes do not do so, which apparently makes
investigating them essentially more difficult -- even if P is finite,
or just {3}, say.

Any ideas, comments, hints, questions, suggestions, ...
are greatly appreciated.

Best wishes,

     Stefan Kohl