478 E. Groups and Graphsconclude that
kl!_.1! 111~1 L A(s)llJIB(£2(rp,q)) = 1.
sErpk(S)
That is, the Baumslag-Solitar group r p,q is not "uniformly nonamenable."Groups acting on trees.
Example E.12. Let r = I'1 A I'2 be the amalgamated free product of I'1
and r 2 with a common subgroup A. Then the associated Bass-Serre tree is
the graph T whose vertex set is r /r1 LJ r /r2 and whose edge set is r I A,
where sA connects sr 1 to sr 2. It is instructive to check that T is indeed
a tree. (Check that it is connected and has no nontrivial closed paths.)
The group r acts on T by left multiplication. The vertex stabilizers are
isomorphic to either r1 or r2, and the edge stabilizers are isomorphic to A.
Example E.13. Let r = (r, z I z-^1 az = O(a) for a E A) be the HNN-
extension associated with (r, A, {)). Then the associated Bass-Serre tree is
the graph T whose vertex set is I' /r and whose edge set is I'/ A, where sA
connects sr to szr. It is again instructive to check that T is a tree, that I'*
acts on T by left multiplication and the vertex stabilizers are isomorphic to
r, while the edge stabilizers are isomorphic to A.
These are the two archetypes of actions on trees. Indeed, Bass-Serre
theory provides a recipe for reconstructing a group acting on a tree from
its vertex and edge stabilizers by successive amalgamated free products and
HNN-extensions.
References. For more on amalgamated free products and Bass-Serre the-
ory see [13, 121, 172]. Expanders receive more attention in [122].