# Examples of residually-finite groups

One of the main reasons I only supervised one PhD student is that I find it hard to find an appropriate topic for a PhD project. A good approach, in my view, is to have on the one hand a list of interesting questions and on the other hand rich enough family of examples so that the student would be able to find answers to some of these questions for some of the examples. I have many questions that I would like to know the answer for interesting examples of residually-finite groups, in particular, for finitely presented ones. Unfortunately, I am very far from being an expert on this topic.

Is there anywhere I can find a list of examples of families of residually-finite groups, preferably, including some of their interesting properties?

• A group is residually finite if and only if it can be embedded inside the direct product of a family of finite groups (from wikipedia).
– Name
Commented Jan 11, 2018 at 10:09
• Automata groups are residually finite and fairly concrete. Commented Jan 11, 2018 at 10:38
• Thanks @BenjaminSteinberg. Can you suggest a good survey? Commented Jan 11, 2018 at 11:03
• There is a nice survey by Grigorchuk, Nekrashevych and Suschsnskii in the proceedings of the Steklov Institute. I think Grigorchuk has it in his homepage. Grigorchuk and Sunic have a survey called self-similar groups and there is Nekrashevch's book on self similar groups for more advanced learners. Bartholdi and Silva also have a survey. Commented Jan 11, 2018 at 13:06
• @BenjaminSteinberg you might like to put it all as an answer, just for completness. Commented Jan 11, 2018 at 13:16

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.

• (1) is Wise + Haglund-Wise + Agol, understanding it as referring to $C'(1/6)$ small cancelation groups. All three contributions are substantial (quickly summarized in the next comment). And of course one could also include contributions of the previous century, including: Dehn's pioneering work in the 1910s, introduction of $C'(1/6)$ small cancelation groups by Greendlinger in 1960, introduction by Gromov of Gromov-hyperbolic groups in 1987 and his result that these include $C'(1/6)$-small cancelation groups. (...)
– YCor
Commented May 1, 2020 at 9:41
• Wise (GAFA 2004) proved that these are cubulable (act properly cocompactly on a CAT(0) cube complex). Haglund-Wise (GAFA 2008) invented the notion of special-cubulable groups (= those with a special proper cocompact action on a CAT(0) cube complex — in their words the fundamental groups of finite special locally CAT(0) cube complexes), and proved that these groups can be embedded into RAAGs and in particular are linear over $\mathbf{Z}$. Agol (Doc Math 2013) proved that Gromov-hyperbolic groups are special-cubulable, and the result follows.
– YCor
Commented May 1, 2020 at 9:41

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

$$1\to K\to\Gamma\to Q\to 1$$

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

$$1\to K\to\Gamma\stackrel{f}{\to} Q\to 1$$

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

$$P=\{(\gamma_1,\gamma_2)\in\Gamma\times\Gamma\mid f(\gamma_1)=f(\gamma_2)\}$$

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 $$\Gamma\times\Gamma$$. 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 $$\Gamma\times\Gamma$$ 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.

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 $$B(m, \, n) = \langle a, \, b \; | \; a^{-1}b^ma=b^n\rangle.$$

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.

• Of course not, we cannot even classify finite groups. I am just looking for a list of interesting families for which it would be interesting to examine questions. Finitely generated linear groups are residually finite, but this is probably too general to any specifc question I have in mind. Commented Jan 11, 2018 at 9:48
• @Yiftach, it's worse than that. Finite groups are at least recursively enumerable. The class of fg residually finite groups is uncountable, and probably the class of fp ones is not recursively enumerable. (Certainly its complement isn't.)
– HJRW
Commented Jan 11, 2018 at 12:56
• I have edited the post, adding more examples and further references. Commented Jan 11, 2018 at 14:46

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.

• While (outer) automorphism groups of free groups (again, except in small cases) are residually finite (for the same reason as mapping class groups) and are known not to be linear (by a result of Formanek--Procesi).
– HJRW
Commented Jan 11, 2018 at 16:14

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
• In fact, Thurston and Hempel proved residual finiteness for the fundamental group of any Haken 3-manifold, and that can now be improved to any compact 3-manifold, following Perelman's proof of the geometrization conjecture.
– HJRW
Commented Jan 11, 2018 at 16:11

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.

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 $$\langle A \cup B \mid U = V\rangle$$, 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 $$\langle a, b \mid (ab)^2(ba)^3 = 1 \rangle$$.
• Hall proved that f.g. abelian-by-nilpotent groups are residually finite. But it's false for f.g. nilpotent-by-nilpotent groups, and Hall himself constructed examples.
– YCor
Commented Jun 19, 2020 at 13:22
• @YCor Thank you. The claim that f.g. metanilpotent groups are RF is stated exactly as such in the Baumslag article from 1963 on automorphism groups, so this seems an oversight from him. I'll edit to reflect this. Commented Jun 19, 2020 at 13:29
• Actually Hall's result was extended by Roseblade to f.g. abelian-by-polycyclic, which was apparently very hard.
– YCor
Commented Jun 19, 2020 at 13:30