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?

MathOverflow Asked by Yiftach Barnea on November 26, 2021

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

- 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.
- Polycyclic groups (due to Hirsch) and f.g. abelian-by-nilpotent groups (due to P. Hall).
- 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. - 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.
- 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 $langle a, b mid (ab)^2(ba)^3 = 1 rangle$.

Answered by Carl-Fredrik Nyberg Brodda on November 26, 2021

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.

Answered by Matteo Vannacci on November 26, 2021

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

Answered by Leandro Vendramin on November 26, 2021

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.

Answered by HJRW on November 26, 2021

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.

Answered by Cusp on November 26, 2021

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

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

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

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.

Answered by user6976 on November 26, 2021

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.

Answered by Francesco Polizzi on November 26, 2021

1 Asked on January 19, 2021 by adittya-chaudhuri

cech cohomology ct category theory grothendieck topology nonabelian cohomology

0 Asked on January 17, 2021 by harry-gindi

0 Asked on January 17, 2021 by manfred-weis

1 Asked on January 16, 2021 by learning-math

geometric probability pr probability probability distributions st statistics

18 Asked on January 16, 2021

big list mathematics education open problems soft question thesis

3 Asked on January 16, 2021 by jochen-glueck

fa functional analysis linear algebra matrix analysis oa operator algebras operator theory

1 Asked on January 15, 2021

1 Asked on January 15, 2021 by no_idea

10 Asked on January 14, 2021 by edmund-harriss

big list ho history overview mathematical philosophy sociology of math

1 Asked on January 13, 2021 by david-jones

computational number theory conjectures nt number theory prime numbers

0 Asked on January 13, 2021 by jianrong-li

1 Asked on January 12, 2021

0 Asked on January 12, 2021 by francesco-polizzi

combinatorial group theory finite groups gap geometric group theory gr group theory

2 Asked on January 11, 2021 by user839372

ac commutative algebra coalgebras hopf algebras reference request

0 Asked on January 10, 2021 by goulifet

duality fa functional analysis integration schwartz distributions

0 Asked on January 10, 2021 by stabilo

cones derived categories modules quasi isomorphism triangles

0 Asked on January 9, 2021 by longyearbyen

0 Asked on January 9, 2021 by user164237

1 Asked on January 9, 2021 by bill-bradley

ca classical analysis and odes convex analysis convex optimization matrices permanent

1 Asked on January 9, 2021 by lrnv

Get help from others!

Recent Questions

- MouseLook Script “Pops” back to the last value when the script is enabled after being disabled or destroyed
- Unity app crashes when using unmodified custom Android manifest (didn’t find class “UnityPlayerActivity”)
- How do i draw a ray in unity
- How to test consistency of responses?
- How can I understand these variograms?

Recent Answers

- DMGregory on MouseLook Script “Pops” back to the last value when the script is enabled after being disabled or destroyed
- eric_kernfeld on How to test consistency of responses?
- Philipp on How do i draw a ray in unity
- kjetil b halvorsen on How to test consistency of responses?
- Justin Markwell on Unity app crashes when using unmodified custom Android manifest (didn’t find class “UnityPlayerActivity”)

© 2022 AnswerBun.com. All rights reserved.