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A theorem of Serre states that any finite subgroup $F$ of an (injective) amalgam $G_1 *_H G_2$ is conjugate into one of the factors $G_1$ or $G_2$, that is, $F$ is a subgroup of a vertex stabilizer of the Bass--Serre tree.

Is the following generalization true?: Any finite subgroup $F$ of the colimit $G$ of a developable complex of groups (in the sense of Bridson--Haefliger) is conjugate into one of the vertex groups $G_v$.

I am away on a trip and do not have my usual resources! A specific reference would be appreciated :)

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3 Answers 3

This is utterly false. If a group $G$ acts nicely on a simply-connected simplicial complex $X$, then $G$ can be expressed as a complex of groups on the quotient space $X/G$. What you want is equivalent to having all torsion elements of $G$ fix points in $X$. It is trivial to come up with group actions on simply-connected simplicial complexes in which torsion elements act freely (though it may be worth remarking that this is impossible if the complex is either contractible or nonpositively curved).

EDIT : In response to your further question, here are some references. For the fact that a finite group cannot act freely on a CAT(0) space, it is an easy center of mass argument and can be found in Bridson-Haefliger. For the fact that a finite group $G$ cannot act freely on a contractible finite-dimensional space $X$, if it did then $X/G$ would be a finite-dimensional $K(G,1)$, which is impossible since $G$ has infinitely many nonzero homology groups (see, for instance, Brown's book on group cohomology).

It is maybe worth noting that a tree is both CAT(0) and contractible, so both of these facts generalize the observation of Serre you made in your question.

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@J Newman: In general, a finite group can act on a contractible complex without having a global fixed point (even though, prime order elements will have fixed points). The right condition for the generalization is that the complex of groups is nonpositively curved. Then everything is fine as you observed. –  Misha Apr 4 '12 at 1:06

Okay, thanks, I suppose... my naive generalization might not work out over the 2-sphere.

Let me now specify that the complexes I am interested in are single n-cells with their boundary divided up, e.g. the fundamental domain of a crystallographic group of rank n. May I politely ask: What is a specific reference for the contractible case?

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Please add modifications to the question by editing the question itself. –  Mariano Suárez-Alvarez Jun 13 '11 at 12:33

This might be helpful (theorem 3)

http://arxiv.org/abs/math/0605303

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