From: Paul Robert Brown 
Date: Fri, 25 Oct 1996 10:36:36 -0700 (PDT)
Subject: Re: assigning orders to permutations
 
Hello, Forum and Derek. When 1/l + 1/m + 1/n is less than or equal to
one, this can be accomplished by taking quotients of triangle
groups. Let T(l,m,n) be the triangle group which acts by reflections
on the Euclidean or hyperbolic plane with fundamental domain a
triangle with angles pi/l, pi/m, and pi/n. T(l,m,n) contains normal
surface subgroups of arbitrarily large index, so the groups we desire
are automorphism groups of suitably triangulated surfaces; the finite
set Omega could be taken to be some set of simplices of the surface,
eg., the set of 2-cells, in which case the automorphism group acts
transitively. (Note that I've chosen the surface group to be normal in
the triangle group, so this is the case.)
 
When 1/l + 1/m + 1/n > 1, the triangle group acts on a 2-sphere, so I
would be interested to know what additional requirements you would
like to place on the set Omega (else I'll just make Omega large by
taking the disjoint union of lots of 2-spheres and letting the
triangle group act diagonally). Do you want the action to be
transitive on the set?
 
        Best,                  
 
                Paul

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