Bass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees.
[1][2] Serre's original motivation was to understand the structure of certain algebraic groups whose Bruhat–Tits buildings are trees.
Subsequent work of Bass[3] contributed substantially to the formalization and development of basic tools of the theory and currently the term "Bass–Serre theory" is widely used to describe the subject.
Mathematically, Bass–Serre theory builds on exploiting and generalizing the properties of two older group-theoretic constructions: free product with amalgamation and HNN extension.
Graphs of groups, which are the basic objects of Bass–Serre theory, can be viewed as one-dimensional versions of orbifolds.
Apart from Serre's book,[2] the basic treatment of Bass–Serre theory is available in the article of Bass,[3] the article of G. Peter Scott and C. T. C. Wall[4] and the books of Allen Hatcher,[5] Gilbert Baumslag,[6] Warren Dicks and Martin Dunwoody[7] and Daniel E.
The algebraic definition is easier to state: First, choose a spanning tree T in A.
The fundamental group of A with respect to T, denoted π1(A, T), is defined as the quotient of the free product where F(E) is a free group with free basis E, subject to the following relations: There is also a notion of the fundamental group of A with respect to a base-vertex v in V, denoted π1(A, v), which is defined using the formalism of groupoids.
It turns out that for every choice of a base-vertex v and every spanning tree T in A the groups π1(A, T) and π1(A, v) are naturally isomorphic.
The group G = π1(A, T) defined above admits an algebraic description in terms of iterated amalgamated free products and HNN extensions.
First, form a group B as a quotient of the free product subject to the relations This presentation can be rewritten as which shows that B is an iterated amalgamated free product of the vertex groups Av.
Then the group G = π1(A, T) has the presentation which shows that G = π1(A, T) is a multiple HNN extension of B with stable letters
for i = 0, ..., n. Suppose that g = 1 in G. Then The normal forms theorem immediately implies that the canonical homomorphisms Av → π1(A, T) are injective, so that we can think of the vertex groups Av as subgroups of G. Higgins has given a nice version of the normal form using the fundamental groupoid of a graph of groups.
[9] This avoids choosing a base point or tree, and has been exploited by Moore.
[10] To every graph of groups A, with a specified choice of a base-vertex, one can associate a Bass–Serre covering tree
The vertex set of X is the set of cosets Two vertices gK and fH are adjacent in X whenever there exists k ∈ K such that fH = gkH (or, equivalently, whenever there is h ∈ H such that gK = fhK).
For an edge [gH, ghK] of X its G-stabilizer is equal to ghα(C)h−1g−1.
Similarly, every vertex of type gK has degree [K:ω(C)] in X.
Then T = v is a spanning tree in A and the fundamental group π1(A, T) is isomorphic to the HNN extension with the base group B, stable letter e and the associated subgroups H = α(C), K = ω(C) in B.
is an isomorphism and the above HNN-extension presentation of G can be rewritten as In this case the Bass–Serre tree
Then π1(A,v) is equal to the fundamental group π1(A,v) of the underlying graph A in the standard sense of algebraic topology and the Bass–Serre covering tree
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 Av = π1(A, A).
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.
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.
[11] 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.
The length-function ℓX : G → Z is said to be abelian if it is a group homomorphism from G to Z and non-abelian otherwise.
In general, an action of G on a tree X without edge-inversions is said to be minimal if there are no proper G-invariant subtrees in X.
An important fact in the theory says that minimal non-abelian tree actions are uniquely determined by their hyperbolic length functions:[13] Let G be a group with two nonabelian minimal actions without edge-inversions on trees X and Y.
Suppose that the hyperbolic length functions ℓX and ℓY on G are equal, that is ℓX(g) = ℓY(g) for every g ∈ G. Then the actions of G on X and Y are equal in the sense that there exists a graph isomorphism f : X → Y which is G-equivariant, that is f(gx) = g f(x) for every g ∈ G and every x ∈ VX.