Bass–Serre theory


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. The theory relates group actions on trees with decomposing groups as iterated applications of the operations of free product with amalgamation and HNN extension, via the notion of the fundamental group of a graph of groups. Bass–Serre theory can be regarded as one-dimensional version of the orbifold theory.

History

Bass–Serre theory was developed by Jean-Pierre Serre in the 1970s and formalized in Trees, Serre's 1977 monograph on the subject. Serre's original motivation was to understand the structure of certain algebraic groups whose Bruhat–Tits buildings are trees. However, the theory quickly became a standard tool of geometric group theory and geometric topology, particularly the study of 3-manifolds. Subsequent work of Bass 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. However, unlike the traditional algebraic study of these two constructions, Bass–Serre theory uses the geometric language of covering theory and fundamental groups. 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, the basic treatment of Bass–Serre theory is available in the article of Bass, the article of G. Peter Scott and C. T. C. Wall and the books of Allen Hatcher, Gilbert Baumslag, Warren Dicks and Martin Dunwoody and Daniel E. Cohen.

Basic set-up

Graphs in the sense of Serre

Serre's formalism of graphs is slightly different from the standard formalism from graph theory. Here a graph A consists of a vertex set V, an edge set E, an edge reversal map such that ≠ e and for every e in E, and an initial vertex map. Thus in A every edge e comes equipped with its formal inverse. The vertex o is called the origin or the initial vertex of e and the vertex o is called the terminus of e and is denoted t. Both loop-edges = t) and multiple edges are allowed. An orientation on A is a partition of E into the union of two disjoint subsets E+ and E− so that for every edge e exactly one of the edges from the pair e, belongs to E+ and the other belongs to E−.

Graphs of groups

A graph of groups A consists of the following data:
For every the map is also denoted by.

Fundamental group of a graph of groups

There are two equivalent definitions of the notion of the fundamental group of a graph of groups: the first is a direct algebraic definition via an explicit group presentation, and the second using the language of groupoids.
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, is defined as the quotient of the free product
where F 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, 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 and π1 are naturally isomorphic.
The fundamental group of a graph of groups has a natural topological interpretation as well: it is the fundamental group of a graph of spaces whose vertex spaces and edge spaces have the fundamental groups of the vertex groups and edge groups, respectively, and whose gluing maps induce the homomorphisms of the edge groups into the vertex groups. One can therefore take this as a third definition of the fundamental group of a graph of groups.

Fundamental groups of graphs of groups as iterations of amalgamated products and HNN-extensions

The group G = π1 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 has the presentation
which shows that G = π1 is a multiple HNN extension of B with stable letters.

Splittings

An isomorphism between a group G and the fundamental group of a graph of groups is called a splitting of G. If the edge groups in the splitting come from a particular class of groups, the splitting is said to be a splitting over that class. Thus a splitting where all edge groups are finite is called a splitting over finite groups.
Algebraically, a splitting of G with trivial edge groups corresponds to a free product decomposition
where F is a free group with free basis X = E+ consisting of all positively oriented edges in the complement of some spanning tree T of A.

The normal forms theorem

Let g be an element of G = π1 represented as a product of the form
where e1,..., en is a closed edge-path in A with the vertex sequence v0, v1,..., vn = v0, vn = t and vi = t = o and where for i = 0,..., n.
Suppose that g = 1 in G. Then
The normal forms theorem immediately implies that the canonical homomorphisms Av → π1 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. This avoids choosing a base point or tree, and has been exploited by Moore.

Bass–Serre covering trees

To every graph of groups A, with a specified choice of a base-vertex, one can associate a Bass–Serre covering tree, which is a tree that comes equipped with a natural group action of the fundamental group π1 without edge-inversions.
Moreover, the quotient graph is isomorphic to A.
Similarly, if G is a group acting on a tree X without edge-inversions, one can define the natural notion of a quotient graph of groups A. The underlying graph A of A is the quotient graph X/G. The vertex groups of A are isomorphic to vertex stabilizers in G of vertices of X and the edge groups of A are isomorphic to edge stabilizers in G of edges of X.
Moreover, if X was the Bass–Serre covering tree of a graph of groups A and if G = π1 then the quotient graph of groups for the action of G on X can be chosen to be naturally isomorphic to A.

Fundamental theorem of Bass–Serre theory

Let G be a group acting on a tree X without inversions. Let A be the quotient graph of groups and let v be a base-vertex in A. Then G is isomorphic to the group π1 and there is an equivariant isomorphism between the tree X and the Bass–Serre covering tree. More precisely, there is a group isomorphism σ: G → π1 and a graph isomorphism such that for every g in G, for every vertex x of X and for every edge e of X we have j = g j and j = g j.
One of the immediate consequences of the above result is the classic Kurosh subgroup theorem describing the algebraic structure of subgroups of free products.

Examples

Amalgamated free product

Consider a graph of groups A consisting of a single non-loop edge e with two distinct end-vertices u = o and v = t, vertex groups H = Au, K = Av, an edge group C = Ae and the boundary monomorphisms. Then T = A is a spanning tree in A and the fundamental group π1 is isomorphic to the amalgamated free product
In this case the Bass–Serre tree can be described as follows. 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.
The G-stabilizer of every vertex of X of type gK is equal to gKg−1 and the G-stabilizer of every vertex of X of type gH is equal to gHg−1. For an edge of X its G-stabilizer is equal to ghαh−1g−1.
For every c ∈ C and h ∈ 'k ∈ K' the edges and are equal and the degree of the vertex gH in X is equal to the index . Similarly, every vertex of type gK has degree in X.

HNN extension

Let A be a graph of groups consisting of a single loop-edge e, a single vertex v = o = t, a vertex group B = Av, an edge group C = Ae and the boundary monomorphisms. Then T = v is a spanning tree in A and the fundamental group π1 is isomorphic to the HNN extension
with the base group B, stable letter e and the associated subgroups H = α, K = ω in B. The composition is an isomorphism and the above HNN-extension presentation of G can be rewritten as
In this case the Bass–Serre tree can be described as follows. The vertex set of X is the set of cosets VX =.
Two vertices gB and fB are adjacent in X whenever there exists b in B such that either fB = gbeB or fB = gbe−1B. The G-stabilizer of every vertex of X is conjugate to B in G and the stabilizer of every edge of X is conjugate to H in G. Every vertex of X has degree equal to + .

A graph with the trivial graph of groups structure

Let A be a graph of groups with underlying graph A such that all the vertex and edge groups in A are trivial. Let v be a base-vertex in A. Then π1 is equal to the fundamental group π1 of the underlying graph A in the standard sense of algebraic topology and the Bass–Serre covering tree is equal to the standard universal covering space of A. Moreover, the action of π1 on is exactly the standard action of π1 on by deck transformations.

Basic facts and properties

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. This is equivalent to the condition that A is a tree and that for every edge e = of A = u, t such that u is closer to v than z we have = 1, that is Az = ωe.
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 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 subgroups 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 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.

Hyperbolic length functions

Let G be a group acting on a tree X without edge-inversions.
For every g∈G put
Then ℓX is called the translation length of g on X.
The function
is called the hyperbolic length function or the translation length function for the action of G on X.

Basic facts regarding hyperbolic length functions

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. Similarly, the action of G on X is said to be abelian if the associated hyperbolic length function is abelian and is said to be 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:

Uniqueness theorem

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 = ℓY 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 = g f for every g ∈ G and every x ∈ VX.

Important developments in Bass–Serre theory

Important developments in Bass–Serre theory in the last 30 years include:
There have been several generalizations of Bass–Serre theory: