Bass–Serre Theory - Trivial and Nontrivial Actions

Trivial and Nontrivial Actions

A graph of groups A is called trivial if A = T is already a tree and there is some vertex v of A such that A_v = π₁(A, A). This is equivalent to the condition that A is a tree and that for every edge e = of A (with o(e) = u, t(e) = z) such that u is closer to v than z we have = 1, that is A_z = ω_e(A_e).

An action of a group G on a tree X without edge-inversions is called trivial if there exists a vertex x of X that is fixed by G, that is such that Gx = x. It is known that an action of G on X is trivial if and only if the quotient graph of groups for that action is trivial.

Typically, only nontrivial actions on trees are studied in Bass–Serre theory since trivial graphs of groups do not carry any interesting algebraic information, although trivial actions in the above sense (e. g. actions of groups by automorphisms on rooted trees) may also be interesting for other mathematical reasons.

One of the classic and still important results of the theory is a theorem of Stallings about ends of groups. The theorem states that a finitely generated group has more than one end if and only if this group admits a nontrivial splitting over finite subroups that is, if and only if the group admits a nontrivial action without inversions on a tree with finite edge stabilizers.

An important general result of the theory states that if G is a group with Kazhdan's property (T) then G does not admit any nontrivial splitting, that is, that any action of G on a tree X without edge-inversions has a global fixed vertex.