# rt.representation theory – Are there “wild-type” finitely presented discrete groups G for which it is not possible to explicitly determine all indecomposable finite G-sets?

I would like to know if there are finitely presented discrete groups $$G$$ for which the task of explicitly determining all indecomposable finite $$G$$-sets is known to be essentially impossible. Perhaps this might follow from considerations about wild representation type, or considerations about the word problem for groups. I have not been able to locate any discussion of either of these possibilities, specifically for $$G$$-sets (not linear representations), anywhere in the literature. Since finite-dimensional permutation representations of $$G$$ are very closely related to finite $$G$$-sets, I would also be happy to learn of answers to my question but with “$$G$$-sets” replaced by “permutation representations of $$G$$.” I am not an algebraist and I apologize if any part of this question is foolish or naive.

The previous paragraph is a complete statement of my question.
The rest of what I write, below, consists only of background information about my question.

1. Tame vs. wild type. Given a finite-dimensional algebra $$A$$ over an algebraically closed field $$k$$, a 1977 theorem of Drozd establishes that $$A$$ is either

• of tame type, meaning roughly that the isomorphism classes of finitely generated $$A$$-modules of a fixed degree fit into finitely many one-parameter continuous families with at most finitely many exceptions (this is supposed to be analogous to the Jordan form for square matrices, in which the Jordan block size is a discrete parameter, and the eigenvalue is a continuous parameter), or

• of wild type, in which case the category of finitely generated representations over every finite-dimensional $$k$$-algebra embeds into the category of finitely generated representations of $$A$$.

One consequence of Drozd’s theorem is that, for some choices of $$A$$ (namely, the algebras of wild type), the task of explicitly determining all the isomorphism classes of finitely generated $$A$$-modules seems to be more or less impossible. For more precise statements and very helpful discussion, I think this document by G. Leuschke is great: http://www.leuschke.org/uploads/Research/SU-wildhyps.pdf

It is natural to ask, for $$A$$ not necessarily finite-dimensional, whether some analogue of Drozd’s result may still hold. Some positive results in that direction are described in Leuschke’s paper I linked to, above.

If $$G$$ is an infinite discrete group, the category of finitely generated modules over the group algebra $$k(G)$$ is not quite the same thing as the category of finite-dimensional $$k$$-representations of $$G$$: the latter category is smaller. My naive and uneducated guess would be that, for appropriate infinite discrete groups $$G$$, the category of finite-dimensional $$k$$-representations of $$G$$ is of wild type. In fact, I suspect that this already happens when $$G$$ is a free group on two generators, and that perhaps this is not hard to derive from the fact that the category of finite-dimensional $$klangle x,yrangle$$-modules has wild type (see the example on page 2 in Leuschke’s paper, linked to above).

However, if we restrict to only the permutation representations of $$G$$, then I do not know what to expect. Perhaps there are few enough permutation representations that these kinds of “wild type” difficulties do not occur.

2. Undecidability. Perhaps undecidability issues, like the undecidability of the word problem for groups, can also have consequences for the explicit description of all finite $$G$$-sets. This is suggested by the 1991 article “Wild representation type and undecidability” by M. Prest, which makes some progress toward showing that explicit description of all finite-dimensional representations of an algebra of wild type is undecidable in the sense that word problem is undecidable. Some specific cases are proven in the 1987 paper “Decidability for theories of modules” by F. Point and M. Prest. The case of modules over $$mathbb{Z}(G)$$, with $$G$$ finite, is specifically considered in the three-part series “Decidability for modules over a group ring” by C. Toffalori, whose parts were published in 1993, 1994, and 1996. Evidently the conjecture, proven by Toffalori in many cases, is that the theory of $$mathbb{Z}(G)$$-modules is decidable if and only if the order of $$G$$ is finite.

Again, permutation representations are not discussed in those papers, so I do not know how much of the analysis given there applies to permutation representations, or to $$G$$-sets.

3. Thoma’s theorem. If I understand correctly, the 1968 paper “Eine Charakterisierung diskreter Gruppen vom Typ I” by E. Thoma shows that a discrete group $$G$$ is “tame” if and only if $$G$$ is virtually abelian (i.e., $$G$$ contains an abelian subgroup of finite index). In this context, “tame” means that every unitary representation of $$G$$ is given by a sum of representations $$T$$ such that the algebra $$mathcal{C}(T)$$ is commutative; see II.8 of Kirillov’s book “Elements of the Theory of Representations” for some discussion. I do not know how closely “non-tame” in this context is related to “of wild type.” If the two are sufficiently closely related, then presumably Thoma’s result shows that the complex representations of a discrete group $$G$$ are of wild type whenever $$G$$ is not virtually abelian. But still I do not how to conclude anything about tameness or wildness of permutation representations or $$G$$-sets from this.

4. Burnside semirings. Given a discrete group $$G$$, the “Burnside semiring of $$G$$” is the semiring of isomorphism classes of finite $$G$$-sets, with addition given by disjoint union, and multiplication given by Cartesian product. I will write $$A(G)$$ for the Burnside semiring of $$G$$. (This is somewhat an abuse of notation, since some authors instead write $$A(G)$$ for the Burnside ring of $$G$$, which is the Grothendieck group completion of the Burnside semiring of $$G$$.) Of course my question is really simply about the computability of $$A(G)$$. All the literature on Burnside semirings and Burnside rings that I have been to find, however, only has concrete results when $$G$$ is assumed finite, and I suspect that the really problematic cases are when $$G$$ is infinite, e.g. a free group on two generators. Issues of wild type and/or undecidability do not seem to be discussed in the literature on Burnside (semi)rings.

I am aware of the (proven) Segal conjecture, that is, given a finite group $$G$$, the degree $$0$$ stable cohomotopy ring of $$BG$$ is isomorphic to the completion of the Burnside ring of $$G$$ at its augmentation ideal. But again, this does not seem to help in answering my question.

Finally, I ought to explain why I care about this question. A colleague has asked me about a problem which reduces to understanding certain finite but disconnected covering spaces of certain topological spaces. When $$X$$ itself is path-connected, such a covering space of $$X$$ is of course described by its monodromy representation, a $$G$$-set where $$G = pi_1(X)$$. By calculating the Burnside semiring $$A(G)$$, we get complete control over all the possibilities that may arise in the problem my colleague is trying to solve. It was a nice exercise to calculate $$A(G)$$ for all cyclic groups $$G$$. I would like to be able to tell my colleague–and our readers, as the paper is written up–that these Burnside ring calculations either can or cannot in principle be carried out for each given group $$G$$. At least I would like to be able to tell my colleague whether there are any known undecidability or wild type results that would obstruct any possible explcit calculation of $$A(G)$$ for certain $$G$$.

Thank you for taking the time to read this question. I apologize again if my question is naive or has an answer which is obvious to some readers. To me the answer is not obvious.