Examples of residually-finite groups

Here's a few examples in line with classical combinatorial group theory.

1. Though small cancellation groups as a whole have already been mentioned, one important subclass of these are the one-relator groups with torsion $langle A mid w^n = 1 rangle$, which were shown to be linear rather recently. Baumslag conjectured in the 60s that they would be residually finite, and Allenby-Tang resolved a good number of special cases of this quite some time before the recent work of Agol-Wise-Haglund settled it completely.
2. Polycyclic groups (due to Hirsch) and f.g. abelian-by-nilpotent groups (due to P. Hall).
3. Not all one-relator groups are residually finite, as the example non-Hopfian Baumslag-Solitar group $langle a, b mid b^{-1} a^2 b = a^3 rangle$ shows. However, an important class is the class of cyclically pinched one-relator groups, i.e. those which admit a presentation of the form $langle A cup B mid U = Vrangle$, where $U$ is a word over the generators $A$ and their inverses, and $V$ is one over $B$ and their inverses (and $A cap B = varnothing$). Baumslag, again, showed that these are residually finite in 1969. Some familiar examples, which have already appeared in other answers, are the fundamental groups of compact surfaces.
4. Baumslag showed in 1963, in an exceptionally short and beautiful paper (the proof is one paragraph long, and the remainder of the paper is a couple of paragraphs of applications) that the automorphism group of a finitely generated residually finite group is again residually finite. In particular, the automorphism group of the automorphism group of a free group is residually finite -- I do not know whether there is any other proof of this fact available using only the known presentations for these groups.
5. Finally, a non-example, but perhaps of some relevance/interest: it was once conjectured (by G. Lallement in 1974, in a paper on semigroup theory!) that positive one-relator groups are residually finite. Here a positive one-relator group is one $langle A mid w = 1 rangle$ where $w in A^ast$ is a word not including any inverse symbols. However, as noted by Perrin-Schupp, the non-Hopfian Baumslag-Solitar group $BS(2, -3)$ admits the positive presentation $langle a, b mid (ab)^2(ba)^3 = 1 rangle$.

I just wanted to mention another class of residually finite groups that have lots of strange properties, even though probably Yiftach knows about them.

This is the class of Generalised Golod-Shafarevich (GGS) groups. These are groups with "sparse enough relations", so that the group can be at the same time be small and have lots of quotients. They were used by Misha Ershov and Andrei Jaikin to solve many open questions on abstract and pro-p groups.

A few properties, a f.g. GGS group: has exponential word growth, has fast subgroup growth, has a GGS quotient with property (T), has a GGS torsion quotient, has a GGS hereditarily just infinite torsion quotient.

About finite presentation: these groups can be finitely presented, but it is conjectured that a GGS f.p. abstract group contains a free subgroup.

As a consequence of Thurston's paper, one proves that knot groups are residually finite, see the second reference. This was conjectured in Neuwirth's book.

References:

• Hempel, John. Residual finiteness for $3$-manifolds. Combinatorial group theory and topology (Alta, Utah, 1984), 379--396, Ann. of Math. Stud., 111, Princeton Univ. Press, Princeton, NJ, 1987. MR0895623
• Neuwirth, L. P. Knot groups. Annals of Mathematics Studies, No. 56 Princeton University Press, Princeton, N.J. 1965 vi+113 pp. MR0176462
• Thurston, William P. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357--381. MR0648524

It's perhaps worth mentioning a powerful construction of finitely presented residually finite groups due to Bridson--Grunewald (though the residual finiteness comes from work of Wise).

Wise produced a residually finite version of the Rips construction, which has the following statement.

For every finitely presented group $Q$ there is a residually finite hyperbolic group $Gamma$ and a short exact sequence of groups

$1to KtoGammato Qto 1$

so that $K$ is finitely generated.

The usefulness of this statement lies in the fact that pathologies of $Q$ translate into pathologies of $K$. For instance, if $Q$ has unsolvable word problem then $K$ has unsolvable membership problem.

Unfortunately, the subgroup $K$ is almost never finitely presented, but one can partially rectify this using the "1--2--3 theorem" of Baumslag--Bridson--Miller--Short.

If

$1to KtoGammastackrel{f}{to} Qto 1$

is a short exact sequence of groups with $K$ finitely generated and $Q$ of type $F_3$ (this is a higher-dimensional analogue of finite presentability, meaning that $Q$ has an Eilenberg--Mac Lane space with finite 3-skeleton) then the fibre product

$P={(gamma_1,gamma_2)inGammatimesGammamid f(gamma_1)=f(gamma_2)}$

is finitely presented.

Putting these together, we obtain a machine for translating pathologies of an arbitrary $F_3$ group $Q$ into pathologies of a finitely presented subgroup $P$ of the residually finite group $GammatimesGamma$. In fact, nowadays, one can do even better, since Haglund--Wise proved a linear version of the Rips construction. Here are some sample applications.

• Bridson--Grunewald found a proper finitely presented subgroup $Q$ of a finitely presented residually finite group $GammatimesGamma$ so that the inclusion map induces an isomorphism on the profinite completion.

• A related argument of Bridson--Miller, combined with the Haglund--Wise version of the Rips construction, shows that the isomorphism problem is unsolvable for finitely presented linear groups.

• Martino--Minasyan used this construction to give examples of conjugacy separable finitely presented groups with non-conjugacy separable subgroups of finite index.

• Bridson and I showed that the isomorphism problem is undecidable for profinite completions of finitely presented, residually finite groups.

Mapping class group of surfaces are examples of residually finite groups (Theorem 6.11 of "A Primer on Mapping class groups"). Although except in some small genus cases, it is not known whether they are linear or not.

Here are some examples of finitely presented residually finite groups. All these can be easily found in arXiv.

1. Small cancellation groups (Agol and Wise) although these turned out to be linear.

2. Ascending HNN extensions of free groups (Borisov - Sapir). Some of them are not linear.

3. Some complicated solvable residually finite groups (Kharlampovich-Myasnikov-Sapir). These are not linear. At the beginning of that paper, there is a survey of various constructions of finitely presented residually finite (and non-residually finite) groups.

Talking about finitely generated, residually finite groups the first result that comes to my mind (apart the well-known fact that every free-group is residually finite) is the following powerful result:

Theorem (Malcev, 1940) Every finitely-generated linear group is residually finite.

See

A. I. Malcev: On the faithful representation of infinite groups by matrices, Mat. Sb. 8 (50) (1940), 405-422; English transl., Amer. Math. Soc. Transl. (2) 45 (1965), 1-18. MR 2, 216.

Other interesting examples are of finitely generated, residually finite groups are:

• the one-relator group $$langle a, , t; | ; a^{t^2}=a^2 rangle,$$ which is residually finite but not linear (Drutu-Sapir);
• fundamental groups of type $pi_1(S)$, where $S$ is a compact surface;
• Baumlag-Solitar groups of type $B(m, , m)$, $B(n, , 1)$ and $B(1, , m)$, where $$B(m, , n) = langle a, , b ; | ; a^{-1}b^ma=b^nrangle.$$

See this blog post on the work on Bausmlag, that also contain many other interesting results, mostly about the relationships between residually finite and Hopfian groups.