2 open problems about natural groups

The first problem is whether every additive group of every field is natural. Lets call this a field group. All field groups of cardinality not exceeding the continuum are natural. As typical in the naturality habitat, one can see this in various way. Start for example with R, which is a connected Lie group. It is in general true that all connected Lie groups are natural. But for R one can note that every isometry must be either a translation or reflection. That dihedral group however has only one connected subgroup and these are the translations. Unlike in the case Z, where a translation invariant metric also admits the dihedral group which can be implemented on Z, this is not possible on R. Then one can go to the rational numbers. The are dense in R with respect to the natural metric $(\mathbb{R},d)$ on $\mathbb{R}$ . Restricting the metric to $\mathbb{Q}$ now produces a natural metric on $\mathbb{Q}$ . Any other group action which is translation invariant would produce a different group action also on R. We can not reverse this argument as the natural Q is dense in the dyadic group of intgers $\mathbb{Z}_2$ which is not natural. On the other hand, the completion of $\mathbb{Q}$ with respect to any non-archimedean norm is a field $\mathbb{Q}_p$ that is natural, simply because the additive group $(\mathbb{R},+)$ and $(\mathbb{Q}_p,+)$ are isomorphic as abstract groups. They are both vector spaces over $\mathbb{Q}$ with the same cardinality. Affine rigidity and Mazur-Ulam can then show that for any vector field V over a field of characteristic 0, the additive group $(V,+)$ is natural. Taking quotients then leads to the theorem that all abelian Banach Lie groups are natural independent of the cardinality. When moving to characteristic 2, one an also go beyond the cardinality of the continuum using a neat Cayley graph trick. But already for characteristic 3, this trick does not work any more.

The second open problem is also quite attractive. It is whether every group generated by reflections is natural. Also here the result can be done with Cayley graphs if the group does not have cardinality larger than the continuum. The result that groups generated by reflections are natural is probably the nicest. It immediately implies quite powerful statements like that O(n) is natural (by Cartan-Dieudonne) or that $Diff_0(M)$ is natural for any manifold $M$ . First of all, this group is not larger than the continuum, if the manifold is second countable. Second, it is generated by involutions (which was shown by Rybicki in 2002) which builds on the fact that diffeomorphisms can be broken down into local components, as well as a theorem of Thurston telling that the group $Diff_0(M)$ is simple. It is quite fun here to follow a bit the story of mapping class groups and go through lists of manifolds where one can lift the naturality from $Diff_0(M)$ to $Diff(M)$ , starting with 2-manifolds, tori and spheres, where the case $S^4$ appears to be a tough obstacle. Fortunately, there are other paths to naturality and one has not to ponder difficult mapping class problems to answer the question for $Diff(M)$ . What helps is a strong structure theorem telling that an non-abelian group of cardinality not exceeding the continuum is natural if and only if it is not generalized dicyclic. So, if a group satisfies this cardinality condition and is not metabilian (metabelian is a the notion that the second commutator G”=1, it is weaker than abelian, which is the notion that G’=1) , then it can not be generalized dicyclic and so must be natural. The fact that $G=Diff(M)$ is not metabelian follows from the fact that its commutator is the identity component which is a perfect group so that $G''=G'$ prevents G to be metabelian and so can not be generalized dicyclic and so is natural. I love it if things can be resolved using simple arguments. The structure theorem relies on a deeper theorem of Leemann and de la Salle. They prove more general theory (but the part which is needed for the structure theorem can be condensed down to two pages). I should have said in the video that of course, Leemann and de la Salle do not talk about naturality. The paper I use was written in 2021 but I only learned in 2026 about it.

I hope soon have a short report about this follow-up adventure on natural groups. In retrospect, I was spot on with the choice of the topic of natural groups. There are many different angles to look at it. The problem can be tackled with many different tools. It reaches into topics related to graphical regular representations, combinatorial group theory, topology, Le group theory, descriptive set theory in the case of large cardinals. It is nice also for an older mathematician like me to go back to review some mathematics learned as a student. We had excellent courses on all these topics. As a student at ETH, I felt like in a candy shop and it was difficult to chose later which field to go into. I preferred then more logic and group theory or number theory but the analysis folks and physics were the most exciting. it is also fun to look at old group friends again like the Rubik cube (which are all quickly seen to be natural from the structure theorem) which mattered to me in high school in particular or finitely presented groups, to look through Fuchsian groups (and groups that appeared in a course given by Baumslab) , sweep through other classes like wallpaper groups (which are very accessible and relevant in art). I had mentioned original motivations for studying the topic of natural groups already early on, like to study how “time (group structure) emerges from geometry (metric space)”. There is therefore also a bit of philosophy involved. But I have personally a bit of a problem with philosophy. We had at ETH to take some additional “GenEd course”, courses which had no exams. I went to every class even so I found all the ramblings about Habermass or Adorno rather strange (to say it politely). Today I would probably call it bullshit (if that were not politically incorrect). But Philosophy has always been a good driving force to study things more rationally like with mathematics (where one can call out BS with logic). The problem with many topics in phillosophy is that definitions are often not precise enough. It is typical that critical or post marxists like Adorno or Habermass deliberately change language and meaning of things.

Unambiguous language is important. I learned that in my “naturlity project” again the hard way. Originally, in my 2022 paper, I had been under the spell that adding “left and right translation invariance” is not a big deal. It is more natural to assume “left and right invariance” rather than “right invariance”. But the consequences are dramatic. It is actually quite an interesting question whether natural groups necessarily lead to topological groups. It is not obvious as the following two statements show (I will give the full argument elsewhere): 1) Every natural group (G,*) admits a natural metric (G,d) that is discrete! This in particular implies that we have also left translation continuous (it is not an isometry in general of course) but it is continuous as any permutation of a discrete topological space is continuous! The second statement is much harder and it appears to be tough to come up with counter examples. 2) There are natural groups (G,*) which admit a natural metric (G,d) for which (G,*,d) is not a topological group. I still battle with that but it is much, much harder as most examples which have come up so far actually lead to natural metric spaces (G,d) that induce the discrete topology. Or then, in the wake of Mazur-Ulam, that the natural metric is related to the usual metric considered, like a symmetry broken distance metric on $R^n$ . The $l^4$ metric does the job but on $\mathbb{R}^n$ r compact Lie groups we always have a biinvariant Riemannian metric.

After mentioning that groups and metric spaces are champions among algebraic (single operation structures) and topological structures, and the topic of “naturality” makes them “friends”, we can think about them also while listening to Queen and modify it to “We are the champions, and friends!”. I mentioned on this blog quite a few of my former teachers (mostly mathematicians) but also in philosophy like Markus Werner (high school). I should mention here Werner Geissberger, with whom we was our music teacher during our high school time. I also had participated in the “Kanti Chor”, we even performed an opera once, Purcel’s Dido and Aeneas which we performed at various places, once in Herisau (in Appenzell), in Lugano (in the Tessin) and then of course in the Stadttheater at Schaffhausen. In his music classes, Geissberger would not only cover classical music, we also learned about more modern composers like Stockhausen (a particularly interesting composer as he wrote pieces with the 5th root of 5 geometric series not the 12th root of 2, like our 12 tone music. And we also learned about Pop music like Queen, I think we analyzed the Bohemian rhapsody. Anyway, this also illustrates that we had excellent high school teachers. I remembered that piece during the pandemic and used it for cheering up a bit my 1A class . it was a depressive time, especially to see how reasonable folks could lose common sense. Anyway, the video is here (Nothing really matters to me! ).

[“Nothing really matters to me!” was a good moto during an insane time of 2020/2021. Only a few weeks later after singing that song for the 1A class in 2021, I would almost die with a PE, as my blood had developed temporarily clots. The good doctors at the Mt Auburn hospital were able to clear the stuck piece in my lung with some serious blood thinners and I’m fine since (I ran a 10 K already the day after released from the hospital and a half marathon a week later already) and blood values have gone completely back to normal in the spring of 2022 already . Today, in 2026, it is undisputed that PE spiked significantly. The official reason given is still to blame it entirely on the virus (a simple google search confirms this). There are other reasons which could matter of course, one of them being having a disfunctional health care system (in 2020) the the other with revolutionary experimental new treatments based on new mRNA technology (2021). Many (including me) do not trust science any more as pre-2020. Many now well documented lies were told then under the slogan “trust the science”. At the moment, viral infections are linked to clots formation via Virchow’s triad. I myself consider this single official explanation a bit of a stretch because these type of corona viruses have not gone away, and we do not see these levels of PE before 2020 or after 2023, 2020 was a time where the health care systems collapsed having many die “with Covid” not “from Covid”. 2021 was a time where new medications appeared. I’m also aware that we still live in a time, where litigations could bring down the entire economy, so that it is only reasonable to keep the narrative, as it is. This is simply economics and common sense. On the other hand, one has to see that lying for the “common good” has long term consequences. It might in the short term benefit, but be ultra contra productive in the long term. Any action has to balance risks. In 2020 it appeared (and I myself agreed with that at that time as I must shamefully admit and I changed my mind close to death) that forcing everybody to take a medication that is pretty much untested is a lesser evil than having the economy completely deteriorate. Everybody was in panic mode then, very few were able to keep thinking straight. The truth is probably more complicated. Several factors matter. The problem is that today, we just don’t trust scientists any more with data, especially if they depend on funds from governments and so have to fall in line with any official narratives if they want to keep receiving funds, independent which political party is in charge. The 2020/2021 case was even worse, because dissent or refusal to participate in the global trial would terminate your job, and many did, especially in the health care industry. The time was a disaster for the reputation of science and classical legacy media and technology firms which filtered dissent. It will probably take decades, until trust is regained. ].

In any case, I still graded the second midterm for my Summer 2021 class on my tablet in the ET (It was also quite fun for example to watch the folks at Mt Auburn to scan my heart with some cool tomography, I could see my heart move live on the screen thanks to the wonders of tomography which involves some neat mathematics! (*)) and as I started to contemplate about the possibility of death, which prompted me, still then, on the hospital bed, to pick up again my old project on “natural groups” (I took my trusty ipad with me when the ambulance picked me up at home when I had the attack) and then continue to work on it for a few months (my wife Ruth brought me my laptop), to send it off to the Arxiv in 2022. Section 20.8 in that working document tells that story a bit (I say “minor health scare” there, but that is a bit of an euphemism; it is scary if you can suddenly almost not breathe any more!). In any case, this project is very close to my heart. [(*) I just remember the great movie “the inlaws”, with Michael Douglas, where the good foot doctor (played by Albert Brooks) meets the mafia boss “Cherkasov” (2003 was a time, when Holywood still was a lot of fun and could pull off to used names like “Cherkasov” and still be have a movie that can also be seen by children because they do not get it), and they start to chat: Cherkasov: “isn’t it great to hold a human’s heart in your hand” (meaning to kill to person) where the foot doctor (who has an other kind of situation in mind, where this matters, like surgery) agrees with “Yes, a great feeling”. ]

Author: oliverknill