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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 seminal 1977 monograph (developed in collaboration with Hyman Bass) on the subject.〔J.-P. Serre. ''Arbres, amalgames, SL2''. Rédigé avec la collaboration de Hyman Bass. Astérisque, No. 46. Société Mathématique de France, Paris, 1977〕〔J.-P. Serre, (''Trees''. ) (Translated from the French by John Stillwell). Springer-Verlag, 1980. ISBN 3-540-10103-9〕 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 Hyman Bass〔H. Bass, (''Covering theory for graphs of groups.'' ) Journal of Pure and Applied Algebra, vol. 89 (1993), no. 1–2, pp. 3–47〕 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 Scott and Wall〔Peter Scott and Terry Wall. ''Topological methods in group theory.'' in: "Homological group theory (Proc. Sympos., Durham, 1977)", pp. 137–203, London Mathematical Society Lecture Notes Series, vol. 36, Cambridge University Press, Cambridge-New York, 1979; ISBN 0-521-22729-1〕 and the books of Hatcher,〔A. Hatcher. (''Algebraic topology.'' ) Cambridge University Press, Cambridge, 2002. ISBN 0-521-79160-X; 0-521-79540-0〕 Baumslag,〔G. Baumslag. (''Topics in combinatorial group theory.'' ) Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1993. ISBN 3-7643-2921-1〕 Dicks and Dunwoody〔W. Dicks, and M. J. Dunwoody. (''Groups acting on graphs.'' ) Cambridge Studies in Advanced Mathematics, 17. Cambridge University Press, Cambridge, 1989. ISBN 0-521-23033-0〕 and Cohen.〔Daniel E. Cohen. ''Combinatorial group theory: a topological approach.'' London Mathematical Society Student Texts, 14. Cambridge University Press, Cambridge, 1989. ISBN 0-521-34133-7〕
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