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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 $$\mathcal{A}$$.

## By Ashani Dasgupta

Pursuing Ph.D. in Geometric Group Theory at University of Wisconsin, Milwaukee