Abstract
Ulam asked whether every connected Lie group can be represented on a countable structure. This is known in the linear case. We establish it for the first family of non-linear groups, namely in the nilpotent case. Further context is discussed to illustrate the relevance of nilpotent groups for Ulam’s problem.
1 Introduction
Cayley’s principle states that every group is a permutation group.
In particular, every finite group can be faithfully represented in a symmetric group
For infinite groups, the situation becomes immediately more interesting. We shall investigate it in the light of the following notion.
A group is countably representable if it admits a faithful action on some countable set.
In other words, 𝐺 is countably representable if it can be realised as a subgroup of
Of course, the question of countable representation only arises for groups whose cardinality does not exceed the continuum cardinal
This leaves us with a very rich landscape of groups to investigate because the groups of size 𝔠 include all (non-discrete) Polish groups, thus in particular all (non-discrete) locally compact second countable groups, and still more particularly all (non-trivial) connected Lie groups.
At first sight, the familiar Lie groups have no obvious countable representation; such a representation would be non-continuous, even non-measurable, and indeed, in Solovay’s model [27], they do not have any. But in November 1935, Schreier and Ulam proved the following (see Section 2 for the elementary argument).
Proposition 1 (Schreier–Ulam)
The group 𝐑 is countably representable.
This is Item 95 in the translated Scottish Book [31], although we only found it as a question in the original version [19].
Ulam later asked a question that remains open to this day.
Problem (Ulam)
Is every connected Lie group countably representable?
(See [32, § II.7] and [33, § V.2].)
This question is really the key to the wider world of locally compact groups because the solution to Hilbert’s fifth problem leads to the following fact (see Section 4).
If the answer to Ulam’s problem is affirmative, then every locally compact second countable group is countably representable.
On the ominous side, some of the most familiar Polish groups spectacularly fail to be countably representable.
For instance, it follows from the Rosendal–Solecki automatic continuity theorem [26] that any action of
In the positive direction, Proposition 1 can be generalised to show that, in the abelian case, there is no obstruction at all beyond the obvious size restriction (see Section 4 for two proofs).
Proposition 3 (de Bruijn [6])
Every abelian group is countably representable as long as its cardinality does not exceed 𝔠.
Returning to Lie groups and Ulam’s problem, there is a much stronger result than Proposition 1, namely the following.
This was first established in [29, § 2]; the proof was rediscovered by Kallman [17] and later Ershov–Churkin [8]; see Section 3 for the argument.
Therefore, it remains to settle Ulam’s problem for non-linear Lie groups.
The first example is Birkhoff’s reduced Heisenberg group [1], a central extension of
Every nilpotent connected Lie group is countably representable.
It is fair to ask whether this result just follows formally from the case of abelian groups by taking successive extensions and invoking a general stability property of the class of countably representable groups. It turns out that there is no such general stability: McKenzie [22] has constructed a nilpotent group 𝑀 which is not countably representable, but is a central extension
where 𝐵 is the free
In fact, Churkin conjectured that his group is not a subgroup of any separable (Hausdorff) topological group [4, Remark 3]. We confirm this.
Any homomorphism from McKenzie’s or Churkin’s nilpotent group to any separable Hausdorff topological group is trivial on the centre.
The same statement holds for homomorphisms to Lindelöf Hausdorff topological groups.
This is a strengthening of not being countably representable since
We note that Lindelöf groups include e.g. all compact groups and more generally any countable product of 𝜎-compact groups; see [5, § 8.1 (ii)].
Besides nilpotent groups, perhaps the best-known example of a non-linear Lie group is Cartan’s example [3, § V] of the universal cover of
Let
Recall that
Ulam did not explicitly assume connectedness, but this is only a matter of terminology. Under the convention that Lie groups are second countable, they have at most countably many connected components, and hence we are immediately reduced to the connected case (Lemma 11 below).
If on the other hand no such restriction is made, then we can trivially obtain disconnected Lie groups that are not countably representable by considering a suitably large group as zero-dimensional Lie group.
2 Preliminary observations
We call a subgroup
Then 𝐺 is countably representable if and only if it admits a sequence of cocountable subgroups
We shall often use tacitly the fact that the intersection of a cocountable group with any subgroup remains cocountable in the latter.
Since the (direct) product of any family of symmetric groups acts faithfully on the disjoint union of the corresponding sets, we have the following lemma.
The class of countably representable groups is closed under finite and countable products.∎
The class being closed under passing to subgroups, this further implies the following lemma.
The class of countably representable groups is closed under taking inverse limits of countable inverse systems.∎
The Schreier–Ulam 1935 proof that 𝐑 is countably representable was probably as follows; this argument can also be found in [18, p. 65].
Proof of Proposition 1
By Lemma 8, the countable power
We will establish countable representability for general abelian groups in Section 4 but already record the following.
The circle group
Proof
Viewing 𝐑 again as a 𝐐-vector space, we split off 𝐐 and obtain an isomorphism
Another basic hereditary property relies on the technique of induced actions.
If 𝐺 admits a countably representable cocountable subgroup, then 𝐺 is itself countably representable.
Proof
Suppose that the group 𝐺 admits a cocountable subgroup
3 Proof of Theorem 5
We begin by recalling Thomas’s proof for linear Lie groups.
Let 𝐾 be the field of Puiseux series over
let 𝑉 be its valuation ring.
Then
By the Newton–Puiseux theorem, 𝐾 is algebraically closed.
Since moreover 𝐾 has characteristic zero and cardinality 𝔠, there is a field isomorphism
At this point, it follows already that
Now we give a more explicit expansion of this argument in order to be able to adapt it for some non-linear groups.
Define the decreasing sequence of ideals
The congruence subgroups
Proof
As recalled above,
for some choice of denominator
We will need the following additional observation.
Let
Proof
Viewing everything in 𝐾 rather than in 𝐂, the claim is that 𝑣 is uniformly bounded over
The group 𝐷 is free abelian of finite rank
We turn to the induction step for
Given any element
and consider its residue
(In fact, we will only use the case where 𝐶 is cyclic, but this is still rank
End of proof of Theorem 5
Let 𝐺 be a nilpotent connected Lie group.
We refer to [15, § 11.2] (especially [15, Theorem 11.2.10]) for background on the structure of 𝐺.
We recall in particular the following.
The fundamental group of 𝐺 is a free abelian group of finite rank
Our proof is by induction on 𝑑.
The base case
The case
Since
Let us reformulate for
The lower central series
By the maximality of 𝑖, we can choose 𝑝 such that
Finally, we perform the induction step for
Thus the fundamental group of
Consider the question of whether the double cover
This cannot be proved by using, as above, a cocountable congruence subgroup
The reason is as follows.
The set
4 Auxiliary proofs
The reduction from general locally compact second countable groups to connected Lie groups is standard.
Proof of Proposition 2
Let 𝐺 be a locally compact second countable group.
Denoting the neutral component of 𝐺 by
Since 𝐺 is second countable, we can choose a countable base 𝒰 of neighbourhoods
On the other hand,
Next, we propose two proofs that every abelian group is countably representable, unless its cardinality is too large. Yves Cornulier informed me that the second proof below was already known to de Bruijn; see [6, Theorem 4.3].
Analytic proof of Proposition 3
Let 𝐺 be an abelian group with
Since
To this end, Pontryagin duality ensures that there is a character
The above use of dyadic spaces and of Engelking’s general theorem is convenient but perhaps an overkill.
It has been observed already in the 1940s that continuum powers of separable spaces are separable (the Hewitt–Marczewski–Pondiczery theorem [14, 21, 25]); this applies to
Algebraic proof of Proposition 3
Let 𝐺 be an abelian group with
The 𝑝-torsion part is of the form
5 On McKenzie’s and Churkin’s examples
Churkin [4] defines a group 𝐺 by generators and relations as follows. Let 𝐼 be an index set of cardinality 𝔠. Then
Earlier, McKenzie defined an almost identical group 𝑀 with the same notation as above but imposing in addition that all
Observe that there is an epimorphism from 𝐺 to a free abelian group 𝐴 of continuum rank 𝔠.
Indeed, define 𝐴 to be free on a set
Theorem 15 (McKenzie, Churkin)
Every homomorphism from 𝐺 or from 𝑀 to
Since we found library access to [4] difficult, we reproduce its beautiful argument.
Churkin’s proof of Theorem 15
Let 𝑋 be a countable set with a 𝐺-action.
We need to show that 𝑐 fixes any given
To deduce that 𝐺 is not countably representable, it remains of course to prove that the subgroup generated by 𝑐 in 𝐺 is not trivial. Unfortunately, the argument given in [4], aiming to map 𝐺 onto the integral Heisenberg group, does not work – and indeed Theorem 15 rules out witnessing the non-triviality of 𝑐 in any countable quotient.
There is nonetheless a direct argument. Consider the central extension
given by the following two-cocycle (factor set)
noting that the sum is finite for each
The presentation of 𝐺 implies that there is an epimorphism
This confirms that 𝐺 is not countably representable and further implies that, for all
The above explicit construction can also be used as follows.
The cocycle 𝑓 extends by bilinearity to a 𝐐-valued cocycle on the free 𝐐-module on
containing Churkin’s group.
Defining
There exists a nilpotent group 𝐻 given by a central extension
which is not countably representable.∎
Finally, we present the proof of Theorem 6, establishing that 𝐺 and 𝑀 do not admit any faithful representation to any Hausdorff topological group that is separable or Lindelöf.
This contains Theorem 15 since
Proof of Theorem 6
We keep the notation introduced above for Churkin’s group 𝐺 and for its explicit construction as central extension 𝐸 (the proof for 𝑀 is the same).
We claim that if 𝐺 is covered by countably many left translates of some subset
Indeed, there is an uncountable subset
Consider now a group homomorphism
To that end, choose a symmetric neighbourhood 𝑉 of the identity in 𝐻 such that
In particular,
Thus
We now apply the initial claim to
The reader will have noticed that this proof is inspired by two sources: Churkin’s proof above and the use of the Steinhaus property made in the separable case by Rosendal and Solecki in [26, Proposition 2].
Our proof makes almost no distinction between the separable and the Lindelöf cases, but they are formally independent.
On the one hand, there obviously are non-separable Lindelöf groups since there is no limit to the cardinality of a compact group (while separable regular spaces have cardinality at most
Acknowledgements
Ulam’s problem was first mentioned to me by Pierre de la Harpe. During earlier attempts for higher rank simple groups, I had the benefit and pleasure of conversations with Alex Lubotzky and with Matthew Morrow. I am indebted to Alexander Pinus for sending me the article [4]. I am grateful to Yves Cornulier for his comments on a first version; Yves alerted me to a gap in the earlier proof of Theorem 6 and provided me with additional historical references. The comments of the referee improved the exposition and are gratefully acknowledged.
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Communicated by: Dessislava Kochloukova
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© 2022 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Lower bound on growth of non-elementary subgroups in relatively hyperbolic groups
- Primitivity rank for random elements in free groups
- On the Tits alternative for cyclically presented groups with length-four positive relators
- Lie groups as permutation groups: Ulam’s problem in the nilpotent case
- A pro-2 group with full normal Hausdorff spectra
- On large prime actions on Riemann surfaces
- Powers in wreath products of finite groups
- On powers of conjugacy classes in finite groups
- Conjectural invariance with respect to the fusion system of an almost-source algebra
Artikel in diesem Heft
- Frontmatter
- Lower bound on growth of non-elementary subgroups in relatively hyperbolic groups
- Primitivity rank for random elements in free groups
- On the Tits alternative for cyclically presented groups with length-four positive relators
- Lie groups as permutation groups: Ulam’s problem in the nilpotent case
- A pro-2 group with full normal Hausdorff spectra
- On large prime actions on Riemann surfaces
- Powers in wreath products of finite groups
- On powers of conjugacy classes in finite groups
- Conjectural invariance with respect to the fusion system of an almost-source algebra