MathOverflow Asked by Yiftach Barnea on November 26, 2021
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?
Here's a few examples in line with classical combinatorial group theory.
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:
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:
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
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