Bass–Serre Theory - Important Developments in Bass–Serre Theory

Important Developments in Bass–Serre Theory

Important developments in Bass–Serre theory in the last 30 years include:

• Various accessibility results for finitely presented groups that bound the complexity (that is, the number of edges) in a graph of groups decomposition of a finitely presented group, where some algebraic or geometric restrictions on the types of groups considered are imposed. These results include:
  • Dunwoody's theorem about accessibility of finitely presented groups stating that for any finitely presented group G there exists a bound on the complexity of splittings of G over finite subgroups (the splittings are required to satisfy a technical assumption of being "reduced");
  • Bestvina–Feighn generalized accessibility theorem stating that for any finitely presented group G there is a bound on the complexity of reduced splittings of G over small subgroups (the class of small groups includes, in particular, all groups that do not contain non-abelian free subgroups);
  • Acylindrical accessibility results for finitely presented (Sela, Delzant) and finitely generated (Weidmann) groups which bound the complexity of the so-called acylindrical splittings, that is splittings where for their Bass–Serre covering trees the diameters of fixed subsets of nontrivial elements of G are uniformly bounded.

• The theory of JSJ-decompositions for finitely presented groups. This theory was motivated by the classic notion of JSJ decomposition in 3-manifold topology and was initiated, in the context of word-hyperbolic groups, by the work of Sela. JSJ decompositions are splittings of finitely presented groups over some classes of small subgroups (cyclic, abelian, noetherian, etc., depending on the version of the theory) that provide a canonical descriptions, in terms of some standard moves, of all splittings of the group over subgroups of the class. There are a number of versions of JSJ-decomposition theories:
  • The initial version of Sela for cyclic splittings of torsion-free word-hyperbolic groups.
  • Bowditch's version of JSJ theory for word-hyperbolic groups (with possible torsion) encoding their splittings over virtually cyclic subgroups.
  • The version of Rips and Sela of JSJ decompositions of torsion-free finitely presented groups encoding their splittings over free abelian subgroups.
  • The version of Dunwoody and Sageev of JSJ decompositions of finitely presented groups over noetherian subgroups.
  • The version of Fujiwara and Papasoglu, also of JSJ decompositions of finitely presented groups over noetherian subgroups.
  • A version of JSJ decomposition theory for finitely presented groups developed by Scott and Swarup.

• The theory of lattices in automorphism groups of trees. The theory of tree lattices was developed by Bass, Kulkarni and Lubotzky by analogy with the theory of lattices in Lie groups (that is discrete subgroups of Lie groups of finite co-volume). For a discrete subgroup G of the automorphism group of a locally finite tree X one can define a natural notion of volume for the quotient graph of groups A as

The group G is called an X-lattice if vol(A)< ∞. The theory of tree lattices turns out to be useful in the study of discrete subgroups of algebraic groups over non-archimedean local fields and in the study of Kac–Moody groups.

• Development of foldinds and Nielsen methods for approximating group actions on trees and analyzing their subgroup structure.
• The theory of ends and relative ends of groups, particularly various generalizations of Stallings theorem about groups with more than one end.
• Quasi-isometric rigidity results for groups acting on trees.