Volume-preserving actions of simple algebraic -groups on low-dimensional manifolds
We prove that has no nontrivial, , volume-preserving action on any compact manifold of dimension strictly less than . More generally, suppose is a connected, isotropic, almost-simple algebraic group over , such that the simple factors of every localization of have rank . If there does not exist a nontrivial homomorphism from to , then every , volume-preserving action of on any compact -dimensional manifold must factor through a finite group.
July 16, 2019
Keywords: group action; algebraic group; volume-preserving; manifold.
AMS Subject Classification: 37C85; 20G30, 22F99, 57S99.
The second author has conjectured that if is a simple algebraic -group, and , then every , volume-preserving action of the arithmetic group on an compact manifold of small dimension must be finite. (This means that the action factors through the action of a finite group. See [?] for a precise statement of the conjecture and a survey of progress on this problem.) In this paper, we show that known results imply the analogue of the conjecture with in the place of . For example, we establish:
has no nontrivial, , volume-preserving action on any compact manifold of dimension strictly less than .
contains large finite subgroups whenever is large (such as an elementary abelian group of order ). Therefore, topological arguments imply that if is any compact manifold, then there is some , such that has no nontrivial, action on (see [?, Thm. 2.5]). However, unlike in section 1, the value of depends on details of the topology of , not just its dimension, because every finite group acts freely on some compact, connected, 2-dimensional manifold [?, Thm. 7.12].
The nontrivial part of section 1 (namely, when ) is a special case of the following much more general result:
is an isotropic, almost-simple, linear algebraic group over , such that, for every place of , the -rank of every simple factor of is at least two,
, such that there are no nontrivial, continuous homomorphisms from to , and
is a subgroup of finite index in .
Then every , volume-preserving action of on any -dimensional compact manifold is finite.
Remark 1.4 (anisotropic groups).
Assume, for simplicity, that is connected. Then the assumption that is isotropic can be eliminated if we add two hypotheses on the universal cover :
is projectively simple, and
sufficiently large -arithmetic subgroups of have the Congruence Subgroup Property.
Both of these hypotheses are known to be true unless is anisotropic of type , , or . See creftype 2.8 for more details.
To satisfy the requirement that the -rank of every simple factor of is at least two, it suffices to let be an absolutely almost-simple algebraic group over , such that . In particular, we can take with . This yields section 1.
The assumption that the subgroup has finite index can be replaced with the weaker assumption that it contains the commutator subgroup .
Our bound on the dimension of is probably not sharp. In particular, we conjecture that has no volume-preserving action on any compact manifold of dimension strictly less than . In the general case, it should suffice to assume that has no simple factor of dimension .
2 Proof of section 1
Assume the situation of section 1. By passing to a subgroup of finite index, we assume that is connected.
is the universal cover of . (We may realize as a Zariski-closed subgroup of , for some [?, Thm. 8.6, p. 63], so is defined for any integral domain of characteristic zero.)
is the natural homomorphism.
is the kernel of (so is a finite, central -subgroup of ).
If is any finite set of prime numbers:
is the ring of -integers. That is, , where .
, so is an -arithmetic subgroup of .
is the profinite completion of .
is the ring of -adic integers, for any prime .
We begin by recalling a few well-known facts about :
Every proper, normal subgroup of is contained in the center of , and is therefore finite.
is an abelian group whose exponent divides . (In particular, is a normal subgroup of .)
(2) See [?, Thm. 8.1]. This relies on our assumption that is isotropic.
(2) We have the following long exact sequence of Galois cohomology groups [?, (1.11), p. 22]:
In other words,
Since is central in , it is easy to see that the connecting map is a group homomorphism. Therefore, the desired conclusion follows from the observation that multiplication by annihilates the abelian group . ∎
Since , and acts on , section 2(2) provides an action of on (for any ). The following theorem about this action requires our assumption that there are no nontrivial, continuous homomorphisms from to . It also uses our assumption that simple factors of have rank at least two. (This implies that has Kazhdan’s property .)
Theorem 2.3 ([?, Cor. 1.3]).
If is any finite set of prime numbers, then there exist
a continuous action of a compact group on a compact metric space , and
a homomorphism ,
such that the resulting action of on is measurably isomorphic (a.e.) to the action of on .
We may assume that is dense in . This implies:
Lemma 2.4 (cf. [?, Cor. 1.5]).
It is an easy consequence of the Peter-Weyl Theorem that every compact group is a projective limit of compact Lie groups [?, Cor. 2.43, p. 51]. However, since has no compact factors, the Margulis Superrigidity Theorem [?, Thm. B(iii), pp. 258–259] tells us that any homomorphism from into a compact Lie group must have finite image. Since is dense in , this implies that is a projective limit of finite groups, as desired. ∎
Therefore, we may assume is the profinite completion of . We have the following well-known description of (because is isotropic).
Theorem 2.5 (Congruence Subgroup Property [?,?,?]).
If is nonempty, then the natural inclusion extends to an isomorphism .
Fix a prime number . The inclusion provides us with an action of on , but this must be isomorphic to the action of on (since both are isomorphic to the action on ). Therefore, the action of on must factor through (a.e.). Furthermore, if we use creftype 2.5 to identify with , then it is obvious that is in the kernel of the homomorphism . Therefore, acts trivially on (a.e.).
Since the subgroups generate a dense subgroup of , we conclude that acts trivially (a.e.). Therefore, acts trivially on (not just a.e., because acts continuously on ), so the action of has an infinite kernel. Hence, section 2(2) implies that the kernel is all of . This means that acts trivially on .
Theorem 2.6 ([?, Thm. 2.5]).
If is any abelian group of prime exponent, then every action of on any compact manifold is finite.
If is any abelian group of finite exponent, then every action of on any compact manifold is finite.
We can assume the exponent of is a power of a prime (because is the direct product of its finitely many Sylow subgroups). We can also assume that the action of is faithful, so the theorem tells us that has only finitely many elements of order . This means the kernel of the homomorphism is finite, so it is easy to prove by induction that has only finitely many elements of any order . Since has finite exponent, this implies that is finite. ∎
The projective simplicity is known to be true unless is anisotropic of type , , or [?, pp. 513–515]. (Projective simplicity obviously fails if there is a nonarchimedean place , such that has a compact factor [?, pp. 510–511]. However, compact nonarchimedean factors cannot arise unless is of type [?, Thm. 6.5, p. 285]. In any case, we have ruled out compact factors by requiring the simple factors of to have rank at least .) When there are no compact factors, projective simplicity is also known to be true for inner forms of type [?, p. 180].
For the Congruence Subgroup Property, it suffices to assume that every prime number is contained in a finite set of prime numbers, such that the congruence kernel is central. (This condition is known to be true unless is anisotropic of type , , or [?, Thms. 9.23 and 9.24, pp. 568–569]. In fact, for our purposes, it would suffice to know that is abelian.) To see that this assumption suffices, note that, for any finite set of prime numbers, Strong Approximation [?, Thm. 7.12, p. 427] tells us . In particular, , so, for each prime , we may let be the inverse image of in . The homomorphism must map into for all . If is abelian, this implies that the image of the commutator subgroup is trivial. Since this is true for all , we conclude that acts trivially on . This is sufficient to show that acts trivially.
D. W. M. would like to thank the mathematics departments of Indiana University and the University of Chicago for their excellent hospitality while this paper was being written, and would especially like to thank David Fisher for very helpful conversations about this material.
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