Chandler and Magnus [CM82, pp97] have observed that the first proof that free groups are residually finite was contained in the solution of the word problem in a free group by Schreier [Schr27, pp169]. This proof involves using a reduced word in the free group, w=s1s2s3...sr (where the si are generators or their inverses) to construct a homomorphism from the free group to Sr+1, the symmetric group on r+1 symbols, such that the image of w is not trivial. Schreier's unintended proof was followed by a paper by Levi [Levi30], which proved that a free group is residually finite. Several proofs that free groups are residually finite may be found in the appendix.
The first treatment of this property as something important in it's own right was by Iwasawa [Iwas43], who proved that free groups are residually finite by proving a similar theorem for finitely generated nilpotent groups, and then combining that result with a result by Magnus [Magn37] that the intersection of the groups of the lower central series of a free group is {1}. The residual finiteness of free groups was again proven independently by Marshall Hall [MHall49], who in fact proved that free groups have the LERF property, a stronger property which will be discussed in section 4 of this paper.
A large set of residually finite groups was revealed by Mal'cev's 1940 result [Mal40] that a finitely generated subgroup of a matrix group over a commutative ring with identity is residually finite. In 1952 K. Hirsch [KHir52] proved that finitely generated polycyclic groups (which he referred to as "S-groups") are residually finite. (This result may now be viewed as a simple consequence of Mal'cev's result [Mal58] that a split extension of a finitely generated residually finite group by a residually finite group is residually finite.) Hirsch also proved [KHir55] that a finitely generated nilpotent group is residually finite.
Groups which are not residually finite were sought once the property was recognized as being of interest. Most of the examples rely on the following result:
Def: A group G is Hopfian if every epimorphism from G to G is an isomorphism.
This concept is connected to residual finiteness by the following result.
Theorem: (Mal'cev [Mal40]) A finitely generated residually finite group is Hopfian.
proof: [LS77] Let G be a finitely generated, residually finite group, and let N be normal in G so that G is isomorphic to G/N. A result of Marshall Hall [MHall50, pp128] is that a finitely generated group has only a finite number of subgroups of a given finite index n. Choose some arbitrary n. If the set of subgroups of index n in G is denoted {Mi}, we can look at their inverse images under the natural projection piN:G->G/N. Call Li=piN-1(Mi). We can see that [G:Mi]=[G:Li]=n. As the sets {Li} and {Mi} have the same cardinality we can see that they coincide, so {Li} is also the set of subgroups of G of index n. But ker(piN) is contained in the intersection of the Li. As our choice of n was arbitrary we see that ker(piN) is contained in the intersection of all finite index subgroups of G. As G is residually finite, this intersection is trivial, so ker(piN)={1}.
Thus a finitely generated non-Hopfian group is not residually finite. (Note, though, that there are examples of finitely generated Hopfian groups which are not residually finite, and do not have solvable word problem [Mill71, pp5-6]). In 1950 B. H. Neumann [BNeum50] produced the following finitely generated, but not finitely presented, non-Hopfian (and hence not residually finite) group:
Example: The group G=<a,b|e2=e3=...=1>, where ei=a-1b-1ab-iab-1a-1bia-1bab-iaba-1bi, is isomorphic to its proper factor group obtained by adding the further relation e1=1.
The first example of a finitely presented non-residually finite group was G. Higman's construction of a 3-generator, 2-relator non- Hopfian group in 1951 [Higm51a].
Example: The group <a,b,c| a-1ca=b-1cb=c2> is isomorphic to the factor group one gets by adding the additional relator aca-1=bcb-1.
G. Baumslag and D. Solitar produced the simplest possible example of a non-residually finite group by demonstrating that the family of 2-generator, 1-relator groups given by G=<a,b|a-1bna=bm> is Hopfian iff m divides n, n divides m, or pi(m)=pi(n), where pi(n) is the set of prime divisors of n. The best known and simplest member of the family is:
Theorem: [BS63] G=<a,b|a-1b2a=b3> is not Hopfian (and hence not residually finite).
proof: Consider the endomorphism h:G->G given by h(a)=a and h(b)=b2. (As h carries the relator to a consequence of the relator, it is a group homomorphism) We now show that h is surjective by noting that a=h(a), and b-1=b2b-3=b2(a-1b2a)-1= h(b)h(a-1ba)-1 are in the image. Lastly, we display a non-trivial element of the kernel by noting that h([a-1ba,b])=[b3,b2]=1, yet [a-1ba,b]=a-1baba-1b-1ab-1, which can be shown to be non-trivial through the use of Britton's Lemma. (G is the HNN extension of <b>, with stable letter a. By [a,b] we denote the commutator, aba-1b-1.)
This class of groups has been extensively studied since then. Collins and Levin [CL83] and Meskin [Mesk72] completely characterized those cases when they are residually finite and Campbell [Camp89] showed that their commutator subgroups are not residually finite when the original groups are non-Hopfian. Stronger finiteness properties have also been proved for many of them. Baumslag later conjectured that any one-relator group with torsion is residually finite [GBaum67b, GBaum73b], which conjecture is still open at the time of this paper.
Some obvious results may be drawn about residually finite groups:
Less trivial results require much more machinery and work, though. The first result of this form was Gruenberg's proof that the free product of residually finite groups is residually finite [Grue57]. In proving this result Gruenberg actually proved it for a whole class of properties he called root properties.
Def: A property P is called a root property if
Examples of root properties are finiteness, solubility, and having order a power of some prime p. Nilpotence, however, fails to satisfy property (iii), and is not a root property. Gruenberg first proved the following theorem as his Theorem 4.1.
Theorem: (4.1) P is a root property. The free product of two residually P groups is residually P iff any free group is residually P.
Gruenberg then goes on to prove a more general result for a regular product [MKS76, pp410-413] of residually P groups. If S is a set of words in a set of variables, he uses the notation V(S,G) to denote the subgroup of G which is generated by all elements representable by replacing the variables by elements of G. This is called the verbal subgroup of the group G generated by the set of words S. (The commutator subgroup of a group G could be viewed as the verbal subgroup generated by the single word xyx-1y-1.)
Theorem: (6.2) P is a root property , S is a given set of words, and V(S,H) is the verbal subgroup of H on the words S. Let G=*lGl be a free product of the residually P groups Gl, and let G* denote the subgroup of G generated by the set of elements of form [x,y], where x and y are in different factors. The regular product G/V(S,G*) is residually P if and only if F/V(S,F) is residually P for any free group F.
The obvious corollary of either of these theorems is that the free product of residually finite groups is residually finite. It is worth our time to pause to note a simple proof of this corollary, written by B. Baumslag and M. Tretkoff [BT78].
Theorem: If A and B are residually finite, then so is A*B.
proof: Assume without loss of generality that A and B are finite. (Say A is residually finite, but infinite. Given an element g=a1b1...anbn in A*B there are representations f1,...,fn such that fi:A->Ai, where Ai is finite and fi(g) does not equal 1. Thus we have f1x...xfn:A->A1x...xAn, finite.) For any integer n we let Sn be the set of all words in A*B of length at most n (for our purposes, if g=a1b1...anbn, where no ai or bi is 1, we define the length of g to be 2n). If s is in Sn and x is in A union B, we define an action of x on Sn by:
s*x=sx if |sx| ≤ n, s otherwise
(This can be seen to be a permutation of Sn by noting that if x is in, say, A, the elements s of Sn for which |sx|>n have the form a1b2...an-1bn, and these are also the elements for which there is no t in Sn such that s=tx.) We may extend this to be an action of A*B on Sn. Now, if we choose an arbitrary element g of A*B of length n, it has the action on the element 1 contained in Sn defined by: 1*g=g. Thus in the homomorphism from A*B into the (finite) permutation group on Sn, g maps to a non-trivial element.
A result of Gilbert Baumslag [GBaum63b] allows one to prove that a number of groups are residually finite.
Theorem: The automorphism group of a finitely generated, residually finite group is residually finite.
proof: Let G be a finitely generated, residually finite group. Consider some non-trivial automorphism h:G->G. There must be some g in G such that g-1h(g) is not equal to 1. And as G is residually finite there is some representation of G onto a finite group, F:G->G, such that F(g-1h(g)) is not equal to 1. If N is the kernel of F it is a normal subgroup of G of some finite index n. We recall Marshall Hall's result [MHall50, pp128] that a finitely generated group has only a finite number of subgroups of any finite index n. Thus the intersection of the subgroups of G of index n is a characteristic subgroup of finite index in G. Call this characteristic subgroup Ñ. Thus we can factor Aut(G)->Aut(G/Ñ) and as G/Ñ is finite, so is Aut(G/Ñ). As g-1h(g) is not in Ñ the image of h under this map is non- trivial and we have our desired finite representation of Aut(G).
In particular, this result can be used to show that the automorphism groups of finitely generated free groups and surface groups are residually finite.
The next constructions worth looking at are those of the HNN extension and the amalgamated free product. While the amalgamated product is an older construction (O. Schreier in 1926 vs. G. Higman, B. H. Neumann and H. Neumann in 1949), the HNN extension is more general. The amalgamated product A*C=DB may be embedded as a subgroup of the HNN extension of A*B given by <A*B,t|t-1Ct=D>. Thus, recalling Gruenberg's result that the free product of residually finite groups is residually finite, and noting that a subgroup of a residually finite group is residually finite, we may draw some of the results on amalgamated products from analogous results for HNN extensions. One may partially reverse this dependence by noting that an HNN extension of a group A may be built as a cyclic extension of an amalgamated product of copies of A. This construction is of limited value to us, though. While cyclic extensions of finitely generated residually finite groups are residually finite, this does not hold if one deletes the requirement that the groups be finitely generated. A more complete description of these constructions may be found in Section 4.2 of the text by Lyndon and Schupp [LS77]. We will cite the results for amalgamated free products before those for HNN extensions, as they were proven first.
Residual finiteness is not preserved under amalgamated products, as was demonstrated by Higman's [Higm51a] example of a free product of two residually finite groups with cyclic amalgamation, which is not residually finite. The earliest positive results were in two papers by Gilbert Baumslag published in 1962. The first paper demonstrated that the fundamental groups of closed compact surfaces are residually finite, producing the following result in the process:
Theorem: [GBaum62] If F is a free group, u in F generates its own centralizer, and f:F->F is an isomorphism, then F*u=f(u)F is residually free. (and hence residually finite, recalling that free groups are residually finite)
The second of Baumslag's two papers used the definition:
Def: A subgroup C<A is closed in A if for all a in A, if, for some n, an is in C, then a is in C.
and produced the result:
Theorem: [GBaum63c] Let A and B be residually finite. Then A*CB is residually finite if any of the following hold:
The same paper also produced a result which showed that conditions (3) and (4) are optimal in some sense.The result showed that for any A and B finitely generated, torsion-free nilpotent there is some amalgamated free product of A and B which is not residually finite. A final set of results from this paper was:
Theorem: [GBaum63c]
The methods used in proving these results was a complex argument based on the existence of filters of finite index subgroups of the factor groups which are compatible with the product.
A simplification of the result that the free product of two residually finite groups with finite amalgamated subgroup is residually finite was accomplished by M. Tretkoff in two papers in 1973 [Tret73a, Tret73b]. The first paper proves this result by explicitly constructing a finite permutation representation for each choice of an element in A*CB. The construction is done by first looking at a set of possible normal forms for the element. The second paper translates the same proof into the language of two-complexes and covering spaces.
A more recent result of this form is the work of Boler and Evans who have shown that :
Theorem: [BE73] The free product of residually finite groups amalgamated along retracts is a residually finite group. (A retract is a subgroup S of G with injection map i and a projection p:G->S, such that pi=idS)
The most recent result on the residual finiteness of amalgamated free products is that of B. A. F. Wehrfritz who was able to get a partial generalization of some of Baumslag's results at the cost of making his conditions more technical:
Theorem: [Wehr81] A and B are residually finite, H<A, K<B, and f:H->K is an isomorphism. Then A*H=f(H)B is residually finite if the following conditions hold:
This result produces as corollaries the results listed above as cases 5, 6, and 7 of Baumslag's theorem on the preceding page. Wehrfritz was also able to draw conclusions in some cases where A and B are subgroups of GL(n,Z).
In 1977 Daniel Cohen [Cohe77], and independently in 1978 B. Baumslag and Marvin Tretkoff [BT78], proved a result for HNN extensions which was analogous to G. Baumslag's earlier result that a free product of residually finite groups, amalgamated over a finite group, is residually finite. This result, in fact, implied the earlier result by using the connection between HNN extensions and amalgamated products mentioned previously.
Theorem: Let G be the HNN extension <A,t|t-1Ct=D>, where A is residually finite and C (and hence D) is finite. Then G is residually finite.
Both proofs use Britton's Lemma to produce a normal form for an arbitrary element of G, and then use this form to construct a homomorphism to a permutation group.
Additionally Baumslag and Tretkoff produced a criterion which, when satisfied, would guarantee that an HNN extension is residually finite. This was again an extension of earlier work by G. Baumslag in the context of amalgamated products. It required that there be a filter of finite index normal subgroups which intersect the associated subgroups in the "same" way.
Theorem: Let A be residually finite with subgroups H and K and an isomorphism f from H to K. Let N be the set {P| P is normal of finite index in A and f(P intersect H)=P intersect K}. Then if:
Then the HNN extension <A,t:t-1Ht=K,f> is residually finite.
This criterion has been reexamined by M. Shirvani [Shir85], who found that the criterion, while sufficient, is not necessary. In particular, only condition (ii) is actually necessary. While a necessary substitute for condition (i) was not produced, some conditions under which condition (i) is necessary were specified.
The most recent work on the residual finiteness of HNN extensions has appeared in a series of articles by S. Andreadakis, E. Raptis, and D. Varsos [ARV86, ARV88, RV87]. Their results are a characterization of when an HNN extension of a finitely generated abelian group is residually finite. They have also produced a simple characterization in particular cases, most notably for an extension of a rank two abelian group with cyclic associated subgroups.
It is reasonable to ask under what conditions an extension of a residually finite group will itself be residually finite. If one thinks in terms of covering spaces it is easy to see that a finite extension of a residually finite group (By this I mean that the quotient group is finite while the normal subgroup is residually finite) is residually finite. If you represent a group G by a 2-complex K whose fundamental group is G, then G being residually finite is equivalent to saying that for any homotopically non-trivial loop on K there is a finite sheeted covering K~ of K, such that the loop does not lift to a closed loop on K~. A finite extension of G may be viewed as the fundamental group of a 2-complex M which has K as a finite sheeted covering. Thus, for any homotopically non-trivial loop in M, either it fails to lift to K or it lifts to K and there is some K~ which is a finite sheeted covering of K, and hence a finite sheeted covering of M, such that the loop does not lift to K~. There are numerous examples of an extension of a residually finite group by a residually finite group which is not residually finite. Some are worth noting:
Theorem: (K. W. Gruenberg [ Grue57, Thm 3.1]) If A and B are residually finite groups then the wreath product of A by B is residually finite iff either A is abelian or B is finite.
This result can, of course, be viewed as being either a source of residually finite or non-residually finite examples, dependent on the taste of the reader.
Theorem: (P. Deligne [ Deli78]) Let Sp(2n,Z)~ be the inverse image of the symplectic group Sp(2n,Z) in the universal covering of Sp(2n,R). It is a central extension of pi1Sp(2n,R)=Z by Sp(2n,Z). If n is greater than or equal to 2, then Sp(2n,Z)~ is not residually finite.
The residual finiteness of Sp(2n,Z) follows from Mal'cev's work, so we have the situation 1->Z->Sp(2n,Z)~->Sp(2n,Z)->1. (Readers of Deligne's paper will note that he phrases Sp(2n,Z)~ as an "extension of Sp(2n,Z) by Z")
A third example of note is an unpublished example by C. Kanta Gupta [Gupt87].
Example: If G=<a,b,t|a2,b2, [a,b]2,[a,b]=tn[a,b]t-n, [a,tnat-n],[b,tnbt-n], [tnat-n,tmbt-m], for all n not equal to m> then we consider the exact sequence 1->K->G->Q->1, where Q is residually finite, and K=<[a,b]>=Z2. While G is a center-by metabelian extension, it is not residually finite.
proof:First we prove that G is not residually finite by exhibiting a subgroup H<=G which is not residually finite and then appealing to the observation that subgroups of residually finite groups are residually finite.
Let H=<tnat-n,tnbt-n, for all n><=G. We note that K is contained in the center of G. Thus any normal subgroup of H of finite index must contain K. Hence H is not residually finite.
We now observe that Q=G/K is the wreath product of (Z2+Z2) by Z. An earlier noted result of Gruenberg [Grue57, Thm 3.2] shows that, as (Z2+Z2) is abelian, the wreath product is residually finite.
The most substantial positive result on extensions was first proved by Mal'cev [Mal58], who showed that a split extension of a finitely generated residually finite group by a residually finite group is itself residually finite. This was extended by Miller [Mill71, Thm III.7] to:
Theorem: Suppose that 1->A->E->B->1 is an exact sequence of groups (ie E is an extension of A by B) where A and B are residually finite and A is finitely generated. Assume that any one of the following conditions holds:
Then E is residually finite.
Beyond this result there are several results on the residual finiteness of specific classes of groups defined as extensions. Gilbert Baumslag showed [GBaum71] that a finitely generated free-by-cyclic group is residually finite (note that the free group itself need not have been finitely generated). Groves showed both an example of a finitely generated center-by-metabelian group which is not residually finite [Grov73] and a proof that any finitely presented center-by-metabelian group is residually finite [Grov78] (In this second paper Groves notes an error in a paper by G. Baumslag [GBaum73a] which attempted to produce an example of a finitely presented center-by-metabelian group which is not residually finite).