View source for Baumslag-Solitar group

''Solitar–Baumslag group''

The Baumslag–Solitar groups are a particular class of two-generator one-relator groups which have played a surprisingly useful role in combinatorial and, more recently (the 1990s), geometric group theory. In a number of situations they have provided examples which mark boundaries between different classes of groups and they often provide a testbed for theories and techniques.

These groups have a deceptively simple definition. For each pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b1300701.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b1300702.png" /> of non-zero integers, there is a corresponding group defined by the presentation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b1300703.png" /></td> </tr></table>

where, as usual, this notation means that the group is the quotient of the [[Free group|free group]] on the two generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b1300704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b1300705.png" /> by the normal closure (cf. also [[Normal subgroup|Normal subgroup]]) of the single element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b1300706.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b1300707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b1300708.png" /> is the fundamental group of the [[Torus|torus]] or Klein bottle (cf. also [[Klein surface|Klein surface]]), both of which have been long familiar and well understood — and can therefore be regarded as standing somewhat apart from the remaining groups.

The Baumslag–Solitar groups were introduced in [[#References|[a1]]] to provide some simple examples of so-called non-Hopfian groups. A group is called Hopfian (or nowadays Hopf) if every [[Epimorphism|epimorphism]] from the group to itself is an [[Isomorphism|isomorphism]]. (The name is derived from the topologist H. Hopf and is thought to reflect the fact that whether fundamental groups of manifolds are  "Hopfian"  is of interest; cf. also [[Hopf group|Hopf group]]; [[Non-Hopf group|Non-Hopf group]].) The condition itself is an instance of what are often referred to as finiteness conditions, i.e. conditions satisfied by finite groups which may or may not hold for infinite groups (cf. also [[Group with a finiteness condition|Group with a finiteness condition]]). It is of interest to see what implications there are among different finiteness conditions within classes of infinite groups.

The main result concerning finiteness conditions and the Baumslag–Solitar groups is as follows:

a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b1300709.png" /> is residually finite (i.e. the intersection of all its subgroups of finite index is trivial) if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007010.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007011.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007012.png" />. See also [[Residually-finite group|Residually-finite group]].

b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007013.png" /> is Hopfian if and only if it is residually finite or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007015.png" /> denotes the set of prime divisors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007016.png" />. Note that a simple well-known argument shows that a finitely-generated residually-finite group is Hopfian.

Most famously, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007017.png" /> is non-Hopfian; the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007018.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007020.png" />, is an epimorphism that is not an isomorphism. It is elementary to show the first (and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007021.png" /> is well-defined) and standard techniques provide an easy verification that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007022.png" /> is a non-trivial element of the kernel. An illustration of the weakness of the Hopfian property as a finiteness condition is provided by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007023.png" />, which is Hopfian but has a (normal) subgroup of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007024.png" />, the normal closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007025.png" />, which is non-Hopfian.

(Note that there is a minor error in the actual statement in the original [[#References|[a1]]] (which is basically an announcement of results), since it asserts that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007026.png" /> is residually finite if one of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007028.png" /> divides the other. The observation of this error and its correction are due to S. Meskin [[#References|[a16]]].)

The above result has been extended and generalized in various ways, but it appears to be difficult to go beyond results that are somewhat particular and limited in scope.

Although the above theorem divides the Baumslag–Solitar groups into three classes: those that are residually finite, those that are Hopfian but not residually finite and those that are non-Hopfian, the most marked contrast is between those Baumslag–Solitar groups where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007029.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007030.png" /> and those for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007031.png" />.

For a group

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007032.png" /></td> </tr></table>

there is an obvious [[Homomorphism|homomorphism]] onto the infinite [[Cyclic group|cyclic group]], obtained by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007033.png" />, and standard techniques show that the kernel is isomorphic to the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007034.png" />-adic rational numbers. Thus, such groups are meta-Abelian (cf. [[Meta-Abelian group|Meta-Abelian group]]) and have strong structural properties; in particular, they do not contain a free subgroup of rank two. Moreover, these groups have a particularly simple normal form for their elements: Each element is uniquely represented by a word of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007035.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007036.png" /> and, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007038.png" /> is not divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007039.png" />.

When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007040.png" />, there is still a strong normal form result since all Baumslag–Solitar groups are examples, indeed are the simplest possible examples, of an [[HNN-extension|HNN-extension]] (see also [[#References|[a3]]] or [[#References|[a15]]] for a definition). Hence the following result holds for Baumslag–Solitar groups: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007041.png" /> be a freely reduced word of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007042.png" /> which represents the identity element. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007043.png" /> has a subword of the form either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007044.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007045.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007046.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007047.png" />.

This result shows, for example, that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007048.png" /> the word <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007049.png" /> does not represent the identity, and can also be used to show that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007050.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007051.png" /> contains a free subgroup of rank two.

==Baumslag–Solitar groups as examples and counterexamples.==
Below, a number of results concerning Baumslag–Solitar groups will illustrate their role as testbeds in combinatorial and geometric group theory.

===Automorphisms.===
The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007052.png" />, and more generally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007053.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007054.png" /> and one of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007056.png" /> divides the other, has an infinitely-generated automorphism group [[#References|[a4]]]. A generating set, no finite part of which will generate the automorphism group, consists of the automorphisms

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007057.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007058.png" /></td> </tr></table>

plus the inner automorphisms associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007060.png" />.

A different kind of failure of finite generation is illustrated by the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007061.png" />, where the automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007062.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007064.png" /> has fixed subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007065.png" /> that is not finitely generated. This is easily established using the normal form specified above.

===Subgroups.===
Groups defined by generators and relations arise from topological and geometric contexts. It is perhaps therefore not altogether surprising that the Baumslag–Solitar groups play a role in questions concerning groups defined by particular topological or geometric conditions. A reasonably classical illustration of this is the fact that only the residually finite Baumslag–Solitar groups can be realized as fundamental groups of  "nice"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007067.png" />-manifolds (cf. also [[Three-dimensional manifold|Three-dimensional manifold]]; [[Fundamental group|Fundamental group]]). An exceptionally strong result [[#References|[a13]]] shows that those Baumslag–Solitar groups that are not residually finite cannot even occur as subgroups of nice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007068.png" />-manifolds, thereby providing a  "negativity test"  for deciding when a group is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007070.png" />-manifold group.

A contrasting result found in [[#References|[a14]]] is that the meta-Abelian Baumslag–Solitar groups are the only finitely-generated (non-cyclic) solvable subgroups of one-relator groups (cf. also [[Solvable group|Solvable group]]). In the torsion-free case (when the single relator is not a proper power), this result is a corollary of the theorem in [[#References|[a11]]] that the meta-Abelian Baumslag–Solitar groups are the only finitely-generated solvable groups of cohomological dimension two. A further result [[#References|[a9]]], more akin to the exclusion result for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007071.png" />-manifold groups, is that, for a torsion-free one-relator group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007072.png" />, all maximal Abelian subgroups are malnormal if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007073.png" /> excludes as subgroups both the direct product of a free group of rank two with the infinite cyclic group, and all the meta-Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007075.png" />.

===Geometric group theory.===
Observations with a flavour similar to that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007076.png" />-manifold groups hold. For example, no Baumslag–Solitar group can be a subgroup of a word hyperbolic group (cf. also [[Hyperbolic group|Hyperbolic group]]). For automatic groups, the position is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007077.png" /> is automatic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007078.png" /> but is otherwise not automatic. It is unknown whether  "subgroup exclusion"  occurs whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007079.png" />. A variant condition, weaker than straight automaticity and known as  "asynchronous automaticity"  is, however, satisfied by all Baumslag–Solitar groups.

An alternative view of this area is provided by the concept of isoperimetric inequality. This concerns, for a given group presentation, the relationship between the length of an otherwise arbitrary word which represents the identity element and the number of relators required to express this fact. The relationship is linear if and only if the group is word hyperbolic, and for automatic groups it is quadratic in the sense that the number of relators needed is no more than a quadratic function of the length of the word. It is not known (as of 2000) whether the converse to this last statement is true or false, the Baumslag–Solitar groups having, on this occasion, failed to play their traditional role of providing discriminating examples by requiring a number of relators that can always be exponential in the length of the word. (See [[#References|[a10]]].)

===Normal forms.===
The study of normal forms for elements of a group goes back to the roots of combinatorial group theory and Dehn's introduction of the word problem (cf. also [[Identity problem|Identity problem]]). Two ideas relating to normal forms are of interest here. One is that of describing, in terms of the concept of a regular language (cf. also [[Formal languages and automata|Formal languages and automata]]; [[Grammar, regular|Grammar, regular]]) properties of particular sets of normal forms and how effectively they can be computed; it was the essential impetus for the study of automatic groups. The second is the idea of a growth function for a group, which is the complex power series whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007080.png" />th coefficient is the number of elements of (minimal) length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007081.png" /> relative to some fixed generating set (cf. also [[Polynomial and exponential growth in groups and algebras|Polynomial and exponential growth in groups and algebras]]). All automatic groups have a growth function which is rational, see [[#References|[a5]]], while the residually-finite Baumslag–Solitar groups have rational growth [[#References|[a2]]], [[#References|[a6]]] but, as noted earlier, are not in general automatic. Interestingly, the argument in [[#References|[a2]]] establishing this fact uses finite-state automata, although it is actually the case [[#References|[a12]]] that there is no regular set of length-minimal normal forms for the meta-Abelian Baumslag–Solitar groups.

===Rigidity and convexity.===
Baumslag–Solitar groups continue to be used as test beds for theories and techniques, in particular those derived from [[Metric space|metric space]] concepts applied to the [[Word metric|word metric]] for a group.

Two illustrations are as follows:

1) An important classification tool for metric spaces is the concept of quasi-isometry: Two metric spaces are quasi-isometric if there are bijective mappings between them in which distortion of distances is uniformly linearly bounded (cf. also [[Quasi-isometric spaces|Quasi-isometric spaces]]). A considerable amount of successful classification has been done on groups which arise in specific geometric and topological settings and also on groups which have a nilpotent subgroup of finite index (cf. also [[Nilpotent group|Nilpotent group]]). It has been shown [[#References|[a7]]], [[#References|[a8]]] that for the meta-Abelian Baumslag–Solitar groups, the position is essentially rigid; namely that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007083.png" /> are quasi-isometric if and only if they have isomorphic subgroups of finite index. Furthermore, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007084.png" /> is quasi-isometric to some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007085.png" />, then, ignoring a finite normal subgroup, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007086.png" /> has an index-one or -two subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007087.png" /> quasi-isometric to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007088.png" />. This work is of particular interest since the meta-Abelian Baumslag–Solitar groups do not have a natural geometric setting nor do they satisfy any kind of nilpotency condition.

2) Another metric space concept that has applications to groups is that of  "almost convexity" : A presentation of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007089.png" /> is almost convex if, whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007091.png" /> lie within a specific distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007092.png" /> of the origin and are at most a given distance apart, then there is a path of uniformly bounded length which strays no further than distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007093.png" /> from the origin. The motivation is that almost convexity implies efficient computation of the [[Cayley graph|Cayley graph]] and this has been studied with successful but not always positive outcomes for the fundamental groups of closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007094.png" />-manifold groups with uniform geometries. In its standard presentation,the meta-Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007096.png" />, is not almost convex [[#References|[a18]]].

==Further reading.==
General reference texts in combinatorial and geometric group theory, such as [[#References|[a3]]], [[#References|[a5]]], [[#References|[a15]]], [[#References|[a17]]], provide background reading for non-specialists.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Baumslag,   D. Solitar,   "Some two generator one-relator non-Hopfian groups"  ''Bull. Amer. Math. Soc.'' , '''689'''  (1962)  pp. 199–201</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Brazil,   "Growth functions for some nonautomatic Baumslag–Solitar groups"  ''Trans. Amer. Math. Soc.'' , '''342'''  (1994)  pp. 137–154</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D.J. Collins,   R.I. Grigorchuk,   P.F. Kurchanov,   H. Zieschang,   "Combinatorial group theory and applications to geometry" , ''Encyclopaedia Math. Sci.'' , '''58''' , Springer  (1993)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  D.J. Collins,   F. Levin,   "Automorphisms and Hopficity of certain Baumslag–Solitar groups"  ''Archiv Math.'' , '''40'''  (1983)  pp. 385–400</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D.B.A. Epstein,   J.W. Cannon,   D.F. Holt,   S.V.F. Levy,   M.S. Paterson,   W.P. Thurston,   "Word processing in groups" , Jones &amp; Bartlett  (1992)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M. Edjvet,   D.L. Johnson,   "The growth of certain amalgamated free products and HNN-extensions"  ''J. Austral. Math. Soc.'' , '''52'''  (1992)  pp. 285–298</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  B. Farb,   L. Mosher,   "A rigidity theorem for the solvable Baumslag–Solitar groups (With an appendix by Daryl Cooper)"  ''Invent. Math.'' , '''131'''  (1998)  pp. 419–451</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  B. Farb,   L. Mosher,   "Quasi-isometric rigidity for the solvable Baumslag–Solitar groups II"  ''Invent. Math.'' , '''137'''  (1999)  pp. 613–649</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  D. Gildenhuys,   O. Kharlampovich,   A. Myasnikov,   "CSA-groups and separated free constructions"  ''Bull. Austral. Math. Soc.'' , '''52'''  (1995)  pp. 63–84</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  S.M. Gersten,   "Dehn functions and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007097.png" />-norms of finite presentations"  G. Baumslag (ed.)  C.F. Miller, III (ed.) , ''Algorithms and Classification in Combinatorial Group Theory'' , Springer  (1992)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  D. Gildenhuys,   "Classification of soluble groups of cohomological dimension two"  ''Math. Z.'' , '''166'''  (1979)  pp. 21–25</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  J.M.J. Groves,   "Minimal length normal forms for some soluble groups"  ''J. Pure Appl. Algebra'' , '''114'''  (1996)  pp. 51–58</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  W.H. Jaco,   P.B. Shalen,   "Seifert fibered spaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130070/b13007098.png" />-manifolds" , ''Memoirs'' , '''192''' , Amer. Math. Soc.  (1979)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  A. Karrass,   D. Solitar,   "Subgroups of HNN groups and one-relator groups"  ''Canad. Math. J.'' , '''23'''  (1971)  pp. 627–643</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  R.C. Lyndon,   P.E. Schupp,   "Combinatorial group theory" , ''Ergebn. Math. Grenzgeb.'' , Springer  (1977)</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  S. Meskin,   "Non-residually finite one-relator groups"  ''Trans. Amer. Math. Soc.'' , '''64'''  (1972)  pp. 105–114</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  W. Magnus,   A. Karrass,   D. Solitar,   "Combinatorial group theory" , Wiley  (1966)</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  C.F. Miller III,   M. Shapiro,   "Solvable Baumslag–Solitar groups are not almost convex"  ''Geom. Dedicata'' , '''72''' :  2  (1998)  pp. 123–127</TD></TR></table>