C is called regular if to any pair of vertices v~,v~ E C' with p(vD = p(v~) there is one (and only one) covering transformation g with g( vD = v~. The general example of a regular covering is that described prior to the definition. It follows easily from the definition that a covering transformation fixing any vertex, edge or face is the identity.
Lifting back to X, we may regard Y as a subgraph of X. Let Gv = Stabc(v), Gw = Stabc(w) and Ga = Gv n G w = Stabc(a). Then the claim is that G = G v *c" Gw . The fact that X is a tree gives the required normal form for elements of G. 20. Let G = 7fl (9, Y) and let H be a subgroup of G. 22. Since H is a subgroup of G, it also acts on the tree X. 21, H = 7fl (Ti, Z) for some graph (Ti, Z) of groups. A vertex group Hz of (Ti, Z) is of the form StabH(z) where z is a vertex of X lying over z. Now z = (gGv,v) for some 9 E G and some v E V(X) and so Hz = StabH(z) = H n Stabc(z) = H n gG vg- 1 .