# Coverings of a graph of groups

For topological space $X$ (connected, path connected etc.), there is classification of coverings of $X$ : for fixed $x_0\in X$, consider $\pi_1(X,x_0)$. Then there is a $1-1$ correspondance between conjugacy class of subgroups of $\pi_1(X,x_0)$ and covering spaces of $X$ (upto isomorphism). We define universal cover of $X$ to be a cover $\tilde{X}$ which is a cover of every cover $Y$ of $X$. This exist and unique up to isomorphism.

Now consider a graph of groups $(X,A)$, where $X$ is a graph and $A=(A_v,A_e)$ is a family of groups attached to vertices $v\in V(X)$ and edges $e\in E(X)$ of $X$ with injections from edge groups to end-vertex groups.

In the book "Trees"-Serre, the universal cover of $(X,A)$ is defined to be a connected graph $\tilde{X}$ with:

1) a morphism $p\colon \tilde{X}\rightarrow X$ of graphs;

2) an action of $\pi_1(X,x_0)$, $(x_0\in V(X))$ on $\tilde{X}$ such that stabilizer of $\tilde{v}\in p^{-1}(v)$ is isomorphic to vertex group $A_v$, $v\in V(X)$

(In other words, it is a graph, with action of $\pi_1(X,x_0)$, $(x_0\in V(X)$, such the quotient graph of groups $X/\pi_1(X,x_0)$ is isomorphic to given graph of groups).

Question: Is there a construction of universal cover of a graph of groups analogous to the construction of universal cover of topological spaces ( i.e. a cover of every cover of given graph of groups).

As an illustration, how can we obtain all coverings of $(X,A)$ where $X$ is the graph $\circ --\circ$ with vertex groups $\mathbb{Z}/l$, and $\mathbb{Z}/n$ and edge group trivial.

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As per the construction of Serre, the universal cover of a graph of groups is (just) a graph; so are we not moving away from category of graph of groups to category of graphs? Why we do not construct a universal cover as a graph of groups again? – joseph Sep 5 '11 at 6:51
I added the alg.top. tag. This is more than just group theory :) – David Roberts Sep 5 '11 at 6:58
@David: Thanks! I was adding tag of "geometric group theory", but didn't get it. so added just group theory. Anyway!. – joseph Sep 5 '11 at 7:02

The universal cover of a graph of groups is the Bass--Serre tree. This is described in Serre's book Trees, to which you refer. I don't have a copy to hand, so I can't give you the precise reference, but it's the main object of study throughout the book.

Let me also add that, when trying to think about coverings of graphs of groups, I find it much easier to think about coverings of graphs of spaces. The initial reference for this is paper of Scott and Wall. My students wrote some things about this point of view in this blog.

That said, it's not always feasible to describe every finite-sheeted covering of a graph of groups. For instance, in the case that you ask for, the set of coverings that you ask for is in bijection with the set of transitive permutation groups generated by a pair of permutations of order $l$ and $m$. Just the set of regular coverings is in bijection with the set of groups $\Gamma$ equipped with a generating set $\{a,b\}$, where $a$ is of order $l$ and $b$ is of order $m$. This is an unimaginably complicated set!

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@HW: The universal cover (Bass-Serre tree) of graph of groups is a tree on which the fundamental group of graph of group acts, with quotient graph of groups isomorphic to given graph of groups. But can we define it as a cover of given graph of groups which is also cover of any other cover of given graph of groups (similar to "topological universal cover")? I want to get all covers of graph of groups $\circ --\circ$, with vertex groups finite cyclic, edge group trivial. – joseph Sep 5 '11 at 10:28
Also the Bass-Serre tree is just a tree; why we do not construct universal cover of graph of groups as again certain graph of groups with obvious universal property? – joseph Sep 5 '11 at 10:28
joseph - in answer to your first question, yes, the Bass--Serre tree does surject every connected cover. The trouble with defining it via the universal property is that such definitions contain almost no information. On the other hand, the fact that it is a tree is very useful - connectedness corresponds to the existence of normal forms, and simply-connectedness corresponds to uniqueness of normal forms. – HJRW Sep 5 '11 at 20:20
Here is a combinatorial description of an arbitrary covering space of the graph of groups that you're interested in. It has two classes of vertices, $u_i$ and $v_j$. Each $u_i$ is labelled by some $\mathbb{Z}/l_i$, where $l_i$ divides $l$, and each $v_j$ is labelled by $\mathbb{Z}/m_j$, where $m_j$ divides $m$; the valence of $u_i$ is $l/l_i$, and the valence of $v_j$ is $m/m_j$; finally, each edge adjoins one $u_i$ and one $v_j$. Does that make sense? – HJRW Sep 5 '11 at 20:25

There is a paper: Covering theory for graphs of groups, by Hyman Bass, Journal of Pure and Applied Algebra (1993) Volume: 89, Issue: 1-2, Pages: 3-47.

There is defined the notion of a covering of one graph of groups by another graph of groups. All such coverings of a given graph of groups are in correspondence with the subgroups of the fundamental group of the given graph of groups.

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Have any of you looked at the following?

Higgins, P.~J. The fundamental groupoid of a graph of groups. J. London Math. Soc. (2) 13~(1) (1976) 145--149.

He gives a nice normal form for this fundamental groupoid which has been further exploited in

Moore, E. Graphs of groups: word computations and free crossed resolutions. Ph.D. thesis, University of Wales, Bangor (2001). (downloadable)

One could then investigate coverings of graphs of groups in terms of covering groupoids of this fundamental groupoid. The relevant theory of covering morphisms of groupoids is given by Higgins in

Higgins, P.~J. Notes on categories and groupoids, Mathematical Studies, Volume~32. Van Nostrand Reinhold Co. London (1971); Reprints in Theory and Applications of Categories, No. 7 (2005) pp 1-195.

and also in my book Topology and groupoids Chapter 10, which makes clear in terms of an equivalence of categories the relation between covering maps of a "good" space $X$ and covering morphisms of the fundamental groupoid of $X$.

I have not had time to compare this with the paper by Bass.

For some reason, the main authors on graphs of groups insist on choosing base points and trees.

Incidentally, the notion of covering morphism of groupoids occurred under the name "regular morphism" in a 1951 Annals paper of P.A. Smith.

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