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Dear Gap-Forum,

I would be very glad if I could get help on these topics:

I am trying to write a program to compute the Reidemeister number R(f) of

a group endomorphism f:G --> G (that is, the number of the orbits in G

under the G-action given by

g . x = g x f(g^(-1))

for all g,x in G), where G is not necessarily finite (but it is finitiely

presented). The problem is equivalent to the enumeration of some double

cosets in GxG. The number R(f) can be infinite, if G is infinite (and this

is often the case, because G is usually the fundamental group of a

manifold).

1. Is there a way to use the function DoubleCosets in a smart way (that

is, check first that R(f) is finite)? I argue that a good starting point

should be to consider polycyclic groups: How can I get more details on the

(undocumented) function DoubleCosetsPcGroup? In any case, I found out that

DirectProduct() does not seem to work for PcGroups (or maybe I am doing

something wrong).

2. When using Bettina Eick's and Werner Nickel's Polycyclic share package

(I have an algorithm that is supposed to work for polycyclic groups) I

couldn't figure out how to use the function PreImagesRepresentative(hom,g)

that is documented in the manual. Here is an example of what happens:

============================================================================== gap> RequirePackage("polycyclic"); true gap> ftl:= FromTheLeftCollector(2); <<from the left collector with 2 generators>> gap> SetRelativeOrder(ftl, 1 ,2); gap> SetConjugate(ftl,2,1,[2,-1]); gap> UpdatePolycyclicCollector(ftl); gap> IsConfluent(ftl); true gap> G:=PcpGroupByCollector(ftl); Pcp-group with orders [ 2, 0 ] gap> gens:=GeneratorsOfGroup(G); [ g1, g2 ] gap> H:=RefinedDerivedSeries(G)[2]; Pcp-group with orders [ 0 ] gap> Gbar:=G/H; Pcp-group with orders [ 2, 2 ] gap> p:=NaturalHomomorphism(G,H); [ g1, g2 ] -> [ g1, g2 ] gap> gensbar:=GeneratorsOfGroup(Gbar); [ g1, g2 ] gap> gg1:=PreImagesRepresentative(p,gensbar[1]); Error, no method found! For debugging hints type ?Recovery from NoMethodFound Error no 1st choice method found for `PreImagesRepresentative' on 2 arguments \ at Error( no_method_found ); <function>( <arguments> ) called from read-eval-loop Entering break read-eval-print loop, you can 'quit;' to quit to outer loop, or you can return to continue brk> ============================================================================

Am I doing something wrong?

3. Assume H is a fully invariant subgroup of G. What is the best way to

define the restricted endomorphism f_H:H --> H and the endomorphism

induced on the quotient F:G/H --> G/H ?

4. In general, do you have hints on how to deal with actions of infinite

groups on infinite sets or groups? In case G is free, do you know how to

compute in a finite number of steps if two elements are in the same

Reidemeister orbit or not?

Thank you very much for the help,

Davide L. Ferrario

+----------------------------------+ | Davide Luigi Ferrario | | Dipartimento di Matematica | | del Politecnico di Milano | | Piazza Leonardo da Vinci, 32 | | 20133 Milano | | Italy | | ferrario@matapp.unimib.it | | ferrario@mate.polimi.it | | fax: ++39 02 2399 4568 | +----------------------------------+

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