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. 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. Bass–Serre theory was developed by Jean-Pierre Serre in the 1970s and formalized in Trees, Serre's 1977 monograph (developed in collaboration with Hyman Bass) 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. 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 E → E ,   e ↦ e ¯ {displaystyle E o E, emapsto {overline {e}}} such that e ≠ e and e ¯ ¯ = e {displaystyle {overline {overline {e}}}=e} for every e in E, and an initial vertex map o : E → V. Thus in A every edge e comes equipped with its formal inverse e. The vertex o(e) is called the origin or the initial vertex of e and the vertex o(e) is called the terminus of e and is denoted t(e). Both loop-edges (that is, edges e such that o(e) = t(e)) 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, e belongs to E+ and the other belongs to E−. A graph of groups A consists of the following data: For every e∈E the map α e ¯ : A e → A t ( e ) {displaystyle alpha _{overline {e}}:A_{e} o A_{t(e)}} is also denoted by ωe. 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 (as a certain iterated application of amalgamated free products and HNN extensions), and the second using the language of groupoids.