Bass–Serre theory is a part of the
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
subject of
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
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 (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
In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topo ...
, particularly the study of
3-manifold
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent ...
s. 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
orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space that is locally a finite group quotient of a Euclidean space.
D ...
s.
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
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
. 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''(''e'') is called the ''origin'' or the ''initial vertex'' of ''e'' and the vertex ''o''() 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'', belongs to ''E''
+ and the other belongs to ''E''
−.
Graphs of groups
A ''
graph of groups'' A consists of the following data:
* A connected graph ''A'';
* An assignment of a ''vertex group'' ''A''
''v'' to every vertex ''v'' of ''A''.
* An assignment of an ''edge group'' ''A''
''e'' to every edge ''e'' of ''A'' so that we have
for every ''e'' ∈ ''E''.
* ''Boundary monomorphisms''
for all edges ''e'' of ''A'', so that each
is an
injective
In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
group homomorphism
In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
whe ...
.
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
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
(as a certain iterated application of
amalgamated free products and
HNN extensions), and the second using the language of
groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
* '' Group'' with a partial fu ...
s.
The algebraic definition is easier to state:
First, choose a
spanning tree
In the mathematical field of graph theory, a spanning tree ''T'' of an undirected graph ''G'' is a subgraph that is a tree which includes all of the vertices of ''G''. In general, a graph may have several spanning trees, but a graph that is no ...
''T'' in ''A''. The fundamental group of ''A'' with respect to ''T'', denoted π
1(A, ''T''), is defined as the quotient of the
free product
In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, an ...
:
where ''F''(''E'') is a
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''− ...
with free basis ''E'', subject to the following relations:
*
for every ''e'' in ''E'' and every
. (The so-called ''Bass–Serre relation''.)
*''e'' = 1 for every ''e'' in ''E''.
*''e'' = 1 for every edge ''e'' of the spanning tree ''T''.
There is also a notion of the fundamental group of ''A'' with respect to a base-vertex ''v'' in ''V'', denoted π
1(A, ''v''), which is defined using the formalism of
groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
* '' Group'' with a partial fu ...
s. It turns out that for every choice of a base-vertex ''v'' and every spanning tree ''T'' in ''A'' the groups π
1(A, ''T'') and π
1(A, ''v'') are naturally
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
.
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'' = π
1(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
*''e''
−1α
''e''(''g'')''e'' = ω
''e''(''g'') for every ''e'' in ''E
+T'' and every
.
*''e'' = 1 for every ''e'' in ''E''
+''T''.
This presentation can be rewritten as
:
which shows that ''B'' is an iterated
amalgamated free product of the vertex groups ''A
v''.
Then the group ''G'' = π
1(A, ''T'') has the presentation
:
which shows that ''G'' = π
1(A, ''T'') 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 (e.g. finite, cyclic, abelian, etc.), 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''(''X'') is a
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''− ...
with free basis ''X'' = ''E''
+(''A''−''T'') consisting of all positively oriented edges (with respect to some orientation on ''A'') in the complement of some spanning tree ''T'' of ''A''.
The normal forms theorem
Let ''g'' be an element of ''G'' = π
1(A, ''T'') represented as a product of the form
:
where ''e''
1, ..., ''e
n'' is a closed edge-path in ''A'' with the vertex sequence ''v''
0, ''v''
1, ..., ''v
n'' = ''v''
0 (that is ''v''
0=''o''(''e''
1), ''v
n'' = ''t''(''e
n'') and ''v
i'' = ''t''(''e
i'') = ''o''(''e''
''i''+1) for 0 < ''i'' < ''n'') and where
for ''i'' = 0, ..., ''n''.
Suppose that ''g'' = 1 in ''G''. Then
*either ''n'' = 0 and ''a''
0 = 1 in
,
*or ''n'' > 0 and there is some 0 < ''i'' < ''n'' such that ''e''
''i''+1 = and
.
The normal forms theorem immediately implies that the canonical homomorphisms ''A
v'' → π
1(A, ''T'') are injective, so that we can think of the vertex groups ''A''
''v'' as subgroups of ''G''.
Higgins has given a nice version of the normal form using the fundamental
groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
* '' Group'' with a partial fu ...
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
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S.
Many sets of transformations form a group under ...
of the fundamental group π
1(A, ''v'') 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 (that is, so that for every edge ''e'' of ''X'' and every ''g'' in ''G'' we have ''ge'' ≠ ), 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'' = π
1(A, ''v'') 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 π
1(A, ''v'') and there is an equivariant isomorphism between the tree ''X'' and the Bass–Serre covering tree
. More precisely, there is a
group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a bijection between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the ...
σ: ''G'' → π
1(A, ''v'') 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''(''gx'') = ''g'' ''j''(''x'') and ''j''(''ge'') = ''g'' ''j''(''e'').
This result is also known as the ''structure theorem''.
One of the immediate consequences is the classic
Kurosh subgroup theorem describing the algebraic structure of subgroups of
free product
In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, an ...
s.
Examples
Amalgamated free product
Consider a graph of groups A consisting of a single non-loop edge ''e'' (together with its formal inverse ) with two distinct end-vertices ''u'' = ''o''(''e'') and ''v'' = ''t''(''e''), vertex groups ''H'' = ''A
u'', ''K'' = ''A
v'', an edge group ''C'' = ''A
e'' and the boundary monomorphisms
. Then ''T'' = ''A'' is a spanning tree in ''A'' and the fundamental group π
1(A, ''T'') 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
coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s
:
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'').
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
'gH'', ''ghK''of ''X'' its ''G''-stabilizer is equal to ''gh''α(''C'')''h''
−1''g''
−1.
For every ''c'' ∈ ''C'' and ''h'' ∈ 'k'' ∈ ''K' the edges
'gH'', ''ghK''and
'gH, gh''α(''c'')''K''are equal and the degree of the vertex ''gH'' in ''X'' is equal to the
index
Index (: indexes or indices) may refer to:
Arts, entertainment, and media Fictional entities
* Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index''
* The Index, an item on the Halo Array in the ...
'H'':α(''C'') Similarly, every vertex of type ''gK'' has degree
'K'':ω(''C'')in ''X''.
HNN extension
Let A be a graph of groups consisting of a single loop-edge ''e'' (together with its formal inverse ), a single vertex ''v'' = ''o''(''e'') = ''t''(''e''), a vertex group ''B'' = ''A
v'', an edge group ''C'' = ''A
e'' and the boundary monomorphisms
. Then ''T'' = ''v'' is a spanning tree in ''A'' and the fundamental group π
1(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''. 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
coset
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
s ''VX'' = .
Two vertices ''gB'' and ''fB'' are adjacent in ''X'' whenever there exists ''b'' in ''B'' such that either ''fB'' = ''gbeB'' or ''fB'' = ''gbe''
−1''B''. 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
'B'' : ''H''nbsp;+
'B'' : ''K''
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 ''π''
1(A,''v'') is equal to the
fundamental group ''π''
1(''A'',''v'') of the underlying graph ''A'' in the standard sense of
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
and the Bass–Serre covering tree
is equal to the standard
universal covering space of ''A''. Moreover, the action of ''π''
1(A,''v'') on
is exactly the standard action of ''π''
1(''A'',''v'') on
by
deck transformations.
Basic facts and properties
*If A is a graph of groups with a spanning tree ''T'' and if ''G'' = π
1(A, ''T''), then for every vertex ''v'' of ''A'' the canonical homomorphism from ''A
v'' to ''G'' is injective.
*If ''g'' ∈ ''G'' is an element of finite order then ''g'' is conjugate in ''G'' to an element of finite order in some vertex group ''A
v''.
*If ''F'' ≤ ''G'' is a finite subgroup then ''F'' is conjugate in ''G'' to a subgroup of some vertex group ''A
v''.
*If the graph ''A'' is finite and all vertex groups ''A
v'' are finite then the group ''G'' is ''virtually free'', that is, ''G'' contains a free subgroup of finite index.
*If ''A'' is finite and all the vertex groups ''A
v'' are
finitely generated then ''G'' is finitely generated.
*If ''A'' is finite and all the vertex groups ''A
v'' are
finitely presented and all the edge groups ''A
e'' are finitely generated then ''G'' is finitely presented.
Trivial and nontrivial actions
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 ''A
v'' = π
1(A, ''A''). This is equivalent to the condition that ''A'' is a tree and that for every edge ''e'' =
'u'', ''z''of ''A'' (with ''o''(''e'') = ''u'', ''t''(''e'') = ''z'') such that ''u'' is closer to ''v'' than ''z'' we have
z'' : ω''e''(''Ae'')">'Az'' : ω''e''(''Ae'')nbsp;= 1, that is ''A
z'' = ω
''e''(''A
e'').
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 (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. The theorem states that a
finitely generated group
In algebra, a finitely generated group is a group ''G'' that has some finite generating set ''S'' so that every element of ''G'' can be written as the combination (under the group operation) of finitely many elements of ''S'' and of inverses o ...
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 (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.
Hyperbolic length functions
Let ''G'' be a group acting on a tree ''X'' without edge-inversions.
For every ''g''∈''G'' put
:
Then ''ℓ
X''(''g'') 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
*For ''g'' ∈ ''G'' exactly one of the following holds:
:(a) ''ℓ
X''(''g'') = 0 and ''g'' fixes a vertex of ''G''. In this case ''g'' is called an ''elliptic'' element of ''G''.
:(b) ''ℓ
X''(''g'') > 0 and there is a unique bi-infinite embedded line in ''X'', called the ''axis'' of ''g'' and denoted ''L
g'' which is ''g''-invariant. In this case ''g'' acts on ''L
g'' by translation of magnitude ''ℓ
X''(''g'') and the element ''g'' ∈ ''G'' is called ''hyperbolic''.
*If ''ℓ
X''(''G'') ≠ 0 then there exists a unique minimal ''G''-invariant subtree ''X
G'' of ''X''. Moreover, ''X
G'' is equal to the union of axes of hyperbolic elements of ''G''.
The length-function ''ℓ
X'' : ''G'' → Z is said to be ''abelian'' if it is a
group homomorphism
In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
whe ...
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 ''ℓ''
''X'' and ''ℓ''
''Y'' on ''G'' are equal, that is ''ℓ''
''X''(''g'') = ''ℓ''
''Y''(''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
In graph theory, an isomorphism of graphs ''G'' and ''H'' is a bijection between the vertex sets of ''G'' and ''H''
: f \colon V(G) \to V(H)
such that any two vertices ''u'' and ''v'' of ''G'' are adjacent in ''G'' if and only if f(u) and f(v) a ...
''f'' : ''X'' → ''Y'' which is ''G''-equivariant, that is ''f''(''gx'') = ''g'' ''f''(''x'') 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:
*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 group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
s (that is discrete subgroups of
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
s 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 foldings and Nielsen methods for approximating group actions on trees and analyzing their subgroup structure.
[R. Weidmann. ''The Nielsen method for groups acting on trees.'' Proceedings of the London Mathematical Society (3), vol. 85 (2002), no. 1, pp. 93–118]
*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.
Generalizations
There have been several generalizations of Bass–Serre theory:
*The theory of
complexes of groups (see Haefliger, Corson Bridson-Haefliger) provides a higher-dimensional generalization of Bass–Serre theory. The notion of a
graph of groups is replaced by that of a
complex of groups, where groups are assigned to each cell in a
simplicial complex
In mathematics, a simplicial complex is a structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...
, together with monomorphisms between these groups corresponding to face inclusions (these monomorphisms are required to satisfy certain compatibility conditions). One can then define an analog of the fundamental group of a graph of groups for a complex of groups. However, in order for this notion to have good algebraic properties (such as embeddability of the vertex groups in it) and in order for a good analog for the notion of the Bass–Serre covering tree to exist in this context, one needs to require some sort of "non-positive curvature" condition for the complex of groups in question (see, for example ).
*The theory of isometric group actions on
real trees (or R-trees) which are
metric space
In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
s generalizing the graph-theoretic notion of a
tree (graph theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by path, or equi ...
. The theory was developed largely in the 1990s, where the
Rips machine of
Eliyahu Rips on the structure theory of ''stable'' group actions on R-trees played a key role (see Bestvina-Feighn). This structure theory assigns to a stable isometric action of a finitely generated group ''G'' a certain "normal form" approximation of that action by a stable action of ''G'' on a simplicial tree and hence a splitting of ''G'' in the sense of Bass–Serre theory. Group actions on
real trees arise naturally in several contexts in
geometric topology
In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topo ...
: for example as boundary points of the
Teichmüller space (every point in the Thurston boundary of the Teichmüller space is represented by a measured geodesic lamination on the surface; this lamination lifts to the universal cover of the surface and a naturally dual object to that lift is an R-tree endowed with an isometric action of the fundamental group of the surface), as
Gromov-Hausdorff limits of, appropriately rescaled,
Kleinian group
In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...
actions,
and so on. The use of R-trees machinery provides substantial shortcuts in modern proofs of
Thurston's Hyperbolization Theorem for
Haken 3-manifolds.
Similarly, R-trees play a key role in the study of
Culler-
Vogtmann's Outer space as well as in other areas of
geometric group theory; for example,
asymptotic cones of groups often have a tree-like structure and give rise to group actions on
real trees. The use of R-trees, together with Bass–Serre theory, is a key tool in the work of Sela on solving the isomorphism problem for (torsion-free)
word-hyperbolic groups, Sela's version of the JSJ-decomposition theory and the work of Sela on the Tarski Conjecture for free groups and the theory of
limit groups.
*The theory of group actions on ''Λ-trees'', where ''Λ'' is an ordered
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
(such as R or Z) provides a further generalization of both the Bass–Serre theory and the theory of group actions on R-trees (see Morgan, Alperin-Bass,
[R. Alperin and H. Bass. ''Length functions of group actions on Λ-trees.'' in: Combinatorial group theory and topology (Alta, Utah, 1984), pp. 265–378, Annals of Mathematical Studies, 111, ]Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large.
The press was founded by Whitney Darrow, with the financial ...
, Princeton, NJ, 1987; Chiswell
).
See also
*
Geometric group theory
References
{{DEFAULTSORT:Bass-Serre Theory
Group theory
Geometric group theory