G is a finitely presented group.
X is its presentation complex (a simplicial 2-complex). Since G is finitely presented, the number of vertices of X is finite. Suppose \( u_1 , \cdots , u_q \) be the vertices of X.
\( \tilde {X} \) be its universal cover.
Fix lifts of the vertices of X. Suppose they are \( v_1 , \cdots , v_q \).
Choose \( \mathcal {A} \) : a preferred set of subgroups of G, stable under conjugation and taking subgroups. We will be working with trees, whose edge stabilizers come from\( \mathcal {A} \)
Let \( T_1 \leftarrow \cdots \leftarrow T_{k+1} \leftarrow T_{k} \leftarrow \cdots \) be refinements of \( \mathcal {A} \) -trees.
This means there is a collapse map from \( T_{k+1} \to T_k \) for each k.
Collapse Map
A collapse map (is a G-equivariant map) that maps some edges to vertices. On the remaining tree, it is a graph isomorphism.
Pre-image of midpoint of every edge \(e \) of \( T_k \) is the midpoint of an edge e’ in \( T_{k+1} \) that maps isomorphically onto e.
We will construct equivariant maps \( f_k \) from \( \tilde {X} \) to \( T_k \) for all k inductively.
Here is the recipe:
- Map the (favorite) vertices \( v_i \) of \( \tilde {X} \) to vertices of \( T_1 \).
- Extend equivariantly to all vertices of \( \tilde {X} \). The map is now defined on 0- skeleton.
That is if \( v_i \to f_1 (v_i) \) then \( gv_i \to f_1 (gv_i) = g f(v_i) \) - In order to extend the map to 1-skeleton, first note that we know where the endpoints of each edge are going.
- \( \tilde {e} \) be an edge of \( \tilde {X} \). Let v, w be its endpoints. Since \( T_1 \) is a tree, there is an unique edge path connecting \( f_1 (v) \) to \( f_1 (w) \)
- Map \( \tilde {e} \) to that path.
- For \( T_2 \), first send v to any vertex \( t_v \) in the set \( p_1^{-1} ( f_1 (v) ) \). Similarly w is mapped to any vertex \(t_w \) in \( p_1^{-1} (f_1 (w) ) \)
- Restrict \( p_1 \) to the segment \( t_v t_w \). It is a collapse map.
- In particular, the pre-image of the midpoint of an edge \( e_k \) of \( T_k \) is the midpoint of the edge of \(T_{k+1}\) mapping onto \( e_k \).
- Since \( f_1 \) is constant or injective on \( \tilde {e}\) this allows us to define \( f_{2} \) on \( \tilde {e}\) as a map which is either constant or injective.
- Extend \( f_{2} \) equivariantly on the 1-skeleton.
- Finally, we will extend \( f_2 \) to the two simplices. Whenever it is not constant, the image is tripod. Pre-images of midpoints of edges of \(T_2\) are straight arcs joining two sides of the triangle.
Pattern of Dunwoody
The pre-images of midpoints of \( T_k \) is \( \tilde {\tau_k } \subset \tilde {X} \)
Also note that \( \tilde {\tau_k} \subset \tilde {\tau_{k+1}} \)
We will project the Dunwoody pattern to X (the presentation complex we started off with). This is a finite graph.
\( S_k \) be the tree dual to the pattern \( \tilde {\tau_k} \in \tilde {X} \). There is a natural induced map that sends \( S_k \to T_k \), sending edge to edge. Hence edge stabilizers of \( S_k \) are in \( \mathcal{A} \).
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