We still need to convince ourselves of some basic facts about the previous lecture, for example is the map injective?

**Example:** Cut the sphere along the equator. Then the diagram we have is

**Definition:**If or we say that

*G splits*over

*C*, and we call

*C*the

*edge group.*If or and is not or in the latter case, then we say that

*splits non-trivially.*

**Definition: **Let* * be a connected graph (i.e. a 1-dimensional CW-complex). For each vertex (resp. edge ) let (resp. ) be a group. If are vertices adjoining an edge *e* then let be an *injective* homomorphism. This data determines a *graph of groups* .

We say that has:

*underlying graph**vertex groups**edge groups**edge maps*

Similarly, we have:

**Definition:** Let be a connected graph. For each vertex (resp. ) let (resp. ) be a connected CW-complex. If adjoin let be -injective continuous maps. This data determines a *graph of spaces* . It has underlying graph , vertex spaces , edge spaces , etc. The graph of spaces determines a space as follows: define

where for . We say that is a *graph-of-spaces* *structure* (or *decomposition*) for .

**Remark: **There is a natural map (by collapsing all the edge and vertex spaces).

Given any graph of groups we can construct a graph of spaces with underlying graph by assigning and realizing the edge maps as continuous maps . We write for . This is well-defined up to homotopy equivalence.

**Definition: **The *fundamental group* of is just

**Examples:**

- If then
- If then
- Let be an embedded multicurve (disjoint union of circles) inside a surface. Cutting along decomposes into a graph of spaces and into a graph of groups.

**Note:** The edge maps of are only defined up to *free* (i.e. unbased) homotopy. Translated to , this means that only the conjugacy class of in matters.

**Remark:** The map induces a surjection

Here’s a way to construct a graph of groups. Let’s suppose acts on a tree without edge inversions (we can do this by subdividing edges if necessary). Let . The group acts diagonally on The quotient has a structure of a graph of spaces. The underlying graph is and there is a natural map .

Let be a vertex below . The preimage of is just where is the stabilizer of . Similarly, for below , the preimage of is

If adjoins then so the edge map is a covering map and therefore -injective. We have defined a graph of spaces and since is simply connected.

Applying to everything, we have a graph of groups . Its underlying graph is . Its vertex groups are the vertex stabilizers of , its edge groups are the edge stabilizers, and the edge maps are the inclusions. Also, .

Question for next time: Does every graph of groups arise in this way?

## 3 comments

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14 February 2010 at 5.41 pm

AdeelThe definition of graph of groups is not the same as it is given in Hatcher. it seems that by subdivision, the definition above would reduce to Hatcher’s version. However, what not clear is how the edge homomorphism will be modified. Moreover, Hatcher used mapping cylinder to construct graph of spaces.

14 February 2010 at 7.12 pm

Henry WiltonAdeel,

The standard definition of a graph of groups is given in Serre’s book

Arbres. The above definition is modelled on the one given by Scott and Wall in their article ‘Topological methods in group theory’ (Homological group theory (Proc. Sympos., Durham, 1977), pp. 137–203, London Math. Soc. Lecture Note Ser., 36, Cambridge Univ. Press, Cambridge-New York, 1979), though it may differ in some details. In practice, it coincides both with Serre’s and theirs.I haven’t looked at Hatcher’s definition recently, but I have no doubt that with some thought you will see that it is essentially the same.

Note that the definition of graph of spaces above is a generalisation of a mapping cylinder! Just take to be the connected graph with one edge and two vertices; label one vertex by the domain of your favourite map and the other by the range; label the edge by the domain; take one edge map to be the identity, and the other to be your favourite map.

16 February 2010 at 10.37 pm

AdeelThat is what I thought, but need some confirmation. I was bit uncomfortable choosing one map to be identity. Thanks!

\Adeel