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Abstract

We study totally disconnected locally compact second countable (t.d.l.c.s.c.) groups that contain a compact open subgroup with finite rank. We show such groups that additionally admit a pro- compact open subgroup for some finite set of primes are virtually an extension of a finite direct product of topologically simple groups by an elementary group. This result, in particular, applies to l.c.s.c. -adic Lie groups. We go on to obtain a decomposition result for all t.d.l.c.s.c. groups containing a compact open subgroup with finite rank. In the course of proving these theorems, we demonstrate independently interesting structure results for t.d.l.c.s.c. groups with a compact open pro-nilpotent subgroup and for topologically simple l.c.s.c. -adic Lie groups.

T.d.l.c. groups locally of finite rank]Totally disconnected locally compact groups locally of finite rank Phillip Wesolek]PHILLIP WESOLEK
Université catholique de Louvain, Louvain-la-Neuve, Belgium e-mail: phillip.wesolek@uclouvain.be

1 Introduction

There are a number of theorems that point to a deep relationship between the structure of the compact open subgroups and the global structure of a totally disconnected locally compact (t.d.l.c.) group; cf. [1], [5], [8], [9], [23]. These global structural consequences of local properties are, following M. Burger and S. Mozes [5], often called local-to-global structure theorems. In the work at hand, we contribute to the body of local-to-global structure theorems by proving results for t.d.l.c. groups that have a compact open subgroup of finite rank.

  • A profinite group has rank if every closed subgroup contains a dense -generated subgroup. When a profinite group has rank , we say it has finite rank.

These t.d.l.c. groups are of wide interest as locally compact -adic Lie groups have a compact open subgroup with finite rank.

  • We study t.d.l.c. groups that are also second countable (s.c.). The second countability assumption is natural and mild. T.d.l.c.s.c. groups belong to the robust and natural class of Polish groups studied in descriptive set theory. T.d.l.c.s.c. groups are also the correct generalization of countable discrete groups studied by geometric group theorists. More pragmatically, most natural examples of t.d.l.c. groups are second countable. As for the mildness of our assumption, t.d.l.c. groups may always be written as a directed union of compactly generated open subgroups, and these subgroups are second countable modulo a compact normal subgroup [12, (8.7)]. The study of t.d.l.c. groups therefore reduces to the study of second countable groups and profinite groups, so little generality is lost.

11 Statement of results

  • A t.d.l.c. group is said to be locally for a property of profinite groups, if contains a compact open subgroup with property . In the case has a compact open subgroup with finite rank, we say is locally of finite rank.

We consider t.d.l.c.s.c. groups that are locally of finite rank. Via a deep result of M. Lazard [13] and later work of A. Lubotzky and A. Mann [14], l.c.s.c. -adic Lie groups, e.g. , are examples. Our first theorem is a structure result for locally of finite rank groups that are also locally pro- for some finite set of primes . The statement requires the notion of an elementary group: The collection of elementary groups is the smallest class of t.d.l.c.s.c. groups that contains the second countable profinite groups and the countable discrete groups, is closed under group extension, and is closed under countable increasing union. This class is identified and investigated in [22].

Theorem 14

Suppose is a t.d.l.c.s.c. group that is locally of finite rank and locally pro- for some finite set of primes . Then either is elementary or there is a series of closed characteristic subgroups

such that

  1. is elementary and is finite; and

  2. for where each is a non-elementary compactly generated topologically simple -adic Lie group of adjoint simple type for some .

A -adic Lie group is of adjoint simple type if it is isomorphic to for some adjoint -simple isotropic -algebraic group .

As a corollary, we obtain a decomposition result for l.c.s.c. -adic Lie groups. We call attention to the similarity with the solvable-by-semisimple decomposition for connected Lie groups.

Corollary 15

Suppose is l.c.s.c. -adic Lie group. Then either is elementary or there is a series of closed characteristic subgroups

such that

  1. is elementary and is finite; and

  2. for where each is a non-elementary compactly generated topologically simple -adic Lie group of adjoint simple type.

For a non-elementary l.c.s.c. -adic Lie group , it follows has a semisimple Lie algebra. The aforementioned similarity with the connected setting is thus quite deep.

We then relax the locally pro- assumption. We first note an immediate corollary of our results for locally pro- groups.

Corollary 16

Suppose is a t.d.l.c.s.c. group that is locally of finite rank. Then either is elementary or there is an -increasing exhaustion of by compactly generated open subgroups such that for each there is a series of closed characteristic subgroups

such that

  1. is elementary and is finite; and

  2. for where each is a non-elementary compactly generated topologically simple -adic Lie group of adjoint simple type for some prime .

We go on to obtain a more detailed structure result.

Theorem 17

Suppose is a t.d.l.c.s.c. group that is locally of finite rank. Then either is elementary or there is a series of closed characteristic subgroups

such that

  1. and are elementary; and

  2. there is a possibly infinite set of primes so that is a quasi local direct product of where is a finite direct product of non-elementary compactly generated topologically simple -adic Lie groups of adjoint simple type and is a compact open subgroup of .

A group is a quasi local direct product of if each is a closed normal subgroup of and the multiplication map is a well-defined injective homomorphism with dense image. The group is the local direct product of over .

The proofs of the above theorems require two independently interesting lines of inquiry. We first study locally pro-nilpotent groups. Here we obtain a compelling decomposition result.

Theorem 18

Suppose is a t.d.l.c.s.c. group that is locally pro-nilpotent. Then either is elementary or there is a series of closed characteristic subgroups

so that

  1. and are elementary; and

  2. there is a possibly infinite set of primes so that where is a locally pro- t.d.l.c.s.c. group and is a compact open subgroup of .

Our principal local-to-global results additionally require a study of topologically simple l.c.s.c. -adic Lie groups. We prove three results for such groups.

Proposition 19

If is a non-elementary topologically simple l.c.s.c. -adic Lie group, then is isomorphic to a closed subgroup of for some and , the Lie algebra of , is simple.

Via the proposition, a close relationship with -algebraic groups emerges.

Theorem 110

Suppose is a non-elementary topologically simple l.c.s.c. -adic Lie group. Then with an adjoint -simple isotropic -algebraic group.

This relationship gives strong restrictions on the automorphism group.

Theorem 111

Suppose is a non-elementary topologically simple l.c.s.c. p-adic Lie group, let be the group of topological group automorphisms of , and let be the collection of inner automorphisms. Then is finite.

   Acknowledgements.

Many of the results herein form part of author’s thesis work at the University of Illinois at Chicago. The author thanks his thesis adviser Christian Rosendal and the University of Illinois at Chicago. The author also thanks Pierre-Emmanuel Caprace and Ramin Takloo-Bighash for their many thoughtful comments and helpful suggestions. The author finally thanks the anonymous referee for suggesting a generalization that gave Proposition 31 and for pointing out results in the literature that streamlined many of the proofs.

2 Generalities on t.d.l.c. groups

We begin with a brief overview of necessary background. The notations, definitions, and facts discussed here are used frequently and, typically, without reference.

21 Notations

All groups are taken to be Hausdorff topological groups and are written multiplicatively. Topological group isomorphism is denoted . We use “t.d.”, “l.c.”, and “s.c.” for “totally disconnected”, “locally compact”, and “second countable”, respectively.

For a topological group , and denote the collection of closed subgroups of and the collection of compact open subgroups of , respectively. All subgroups are taken to be closed unless otherwise stated. We write and to indicate is an open subgroup of and is a cocompact subgroup of , respectively. Recall is cocompact if the quotient space is compact in the quotient topology.

For any subset , is the collection of elements of that centralize every element of . We denote the collection of elements of that normalize by . The topological closure of in is denoted by . For , we put

For , denotes the -th Cartesian power. For , .

Ordinal numbers are used in this work. The first countable transfinite ordinal is denoted by ; as we assume the natural numbers contain zero, as linear orders. We use and interchangeably.

We use to denote the prime numbers. For , we put . When , the set is written .

22 Basic theory

The foundation of the theory of t.d.l.c. groups is an old result of D. van Dantzig:

Theorem 21 (van Dantzig [12, (7.7)])

A t.d.l.c. group admits a basis at of compact open subgroups.

It follows the compact open subgroups in a t.d.l.c. groups given by van Dantzig’s theorem are profinite, i.e. inverse limits of finite groups. Profinite groups will be discussed at length later. For the moment, we remark that compact groups and profinite groups are one and the same in the category of t.d.l.c groups. We thus use “compact group” and “profinite group” interchangeably. We remark further that a profinite group is pro- for some set of primes if every finite continuous quotient is a -group - that is the order is divisible by only primes in .

Pro- groups with a finite set of primes play an important role in the study of compactly generated groups.

Theorem 22 (Caprace, see [9, Proposition 4.9])

If is a compactly generated t.d.l.c. group, then for every compact open subgroup there is a compact normal so that and that is locally pro- for some finite set of primes .

Topological analogues of the familiar isomorphism theorems hold for t.d.l.c.s.c. groups. The first isomorphism theorem requires non-trivial modification, hence we recall its statement.

Theorem 23 ([12, (5.33)])

Let be a t.d.l.c.s.c. group, be a closed subgroup, and be a closed normal subgroup. If is closed, then as topological groups.

In the category of t.d.l.c.s.c. groups, care must be taken with infinite unions. Suppose is an increasing sequence of t.d.l.c.s.c. groups such that for each . The group is then a t.d.l.c.s.c. group under the inductive limit topology: is defined to be open if and only if is open in for each .

Using our notion of an infinite union, we may define an infinite direct product that stays in the category of t.d.l.c.s.c. groups. This definition goes back to J. Braconnier.

  • Suppose is a countable set, is a sequence of t.d.l.c.s.c. groups, and for each there is a distinguished . Letting enumerate , put

    1. and give the product topology.

    2. and give the product topology.

    The local direct product of over is defined to be

    with the inductive limit topology.

Since for each , the group is a t.d.l.c.s.c. group with as a compact open subgroup. The isomorphism type of a local direct product is also independent of the enumeration of used in the definition.

There is a weakening of the notion of a direct product: A t.d.l.c.s.c. group is a quasi-product with quasi-factors if each is a closed normal subgroup of and the multiplication map is injective with dense image. This notion naturally generalizes to local direct products: A group is a quasi local direct product of if is a closed normal subgroup of , , and the multiplication map is a well-defined injective homomorphism with dense image.

There are two important concepts concerning subgroups of a t.d.l.c.s.c. group . First, following P-E. Caprace, C. Reid, and G. Willis [8], a subgroup is called locally normal if is compact and is open. Second, two subgroups and are commensurate, denoted , if and are finite.

We require a few additional facts around commensurated subgroups. It is easy to check is an equivalence relation on and is preserved under the action by conjugation of on . The commensurability relation gives rise to an additional subgroup: For , the commensurator subgroup of in is

If , we say is commensurated.

We shall make frequent use of two canonical normal subgroups of a t.d.l.c.s.c. group . Generalizing the notion of the centre of a group, the quasi-centre, defined in [5], of is

The group is a characteristic but not necessarily closed subgroup. For the second canonical normal subgroup, a closed subgroup of is locally elliptic if every finite subset generates a relatively compact subgroup. V.P. Platonov [17] shows there is a unique maximal closed normal subgroup of that is locally elliptic; this subgroup is called the the locally elliptic radical and is denoted by . The same work demonstrates that a t.d.l.c.s.c. group is locally elliptic if and only if it is a countable increasing union of compact open subgroups.

The locally elliptic radical along with Theorem 22 give a somewhat canonical decomposition for non-compactly generated groups.

Corollary 25

Suppose is a t.d.l.c.s.c. group. Then there is an increasing exhaustion of by compactly generated open subgroups so that for each , is locally pro- for some finite set of primes .

We stress that in the above corollary depends on and, in general, grows as increases.

We conclude by recalling a general technique for producing normal but not necessarily closed subgroups of a t.d.l.c.s.c. group . A subset is conjugation invariant if is fixed setwise under the action by conjugation of on . If is fixed setwise by all topological group automorphisms of , we say is invariant. We say is hereditary if for all , . The -core, denoted , is the collection of such that for all , .

By results of [22] or as an easy verification, if is (invariant) conjugation invariant and hereditary, then is a (characteristic) normal subgroup of . A subgroup of of the form for some conjugation invariant and hereditary is called a synthetic subgroup of . It is easy to see all closed normal subgroups are synthetic subgroups. However, the collection of synthetic subgroups of often strictly contains the collection of closed normal subgroups of . For example, is a synthetic subgroup and is rarely closed.

23 Profinite groups

We now recall a few facts and definitions from the theory of profinite groups; our discussion falls well short of comprehensive. We direct the interested reader to the excellent texts [19] and [24].

Profinite groups admit a basis at of open normal subgroups. For a profinite group , we say is a normal basis at for , if , is -decreasing with trivial intersection, and for each , .

The group is said to be pro- for some property of finite groups if is an inverse limit of finite groups with property . For example, may be pro- for some prime or pro-nilpotent.

We say is topologically (finitely generated) -generated if admits a dense (finitely generated) -generated subgroup. Central to this work,

  • A profinite group has rank if for every closed , is topologically -generated. If a profinite group has rank for some , we say has finite rank.

Profinite groups with finite rank have a well understood structure.

Theorem 27 ([24, Theorem 8.4.1])

If is a profinite group with finite rank, then has a series of normal subgroups such that is pro-nilpotent, is solvable, and is finite.

Profinite groups have a Sylow theory arising from the inverse limit construction. For a prime , a -Sylow subgroup of a profinite group , denoted , is a maximal pro- subgroup of . Analogous to the finite setting,

Proposition 28 ([24, 2.2.2])

Let be a profinite group and a prime. Then

  1. has -Sylow subgroups.

  2. All -Sylow subgroups are conjugate.

  3. Every pro- subgroup is contained in a -Sylow subgroup.

Proposition 29 ([19, Proposition 2.3.8])

Suppose is a profinite group that is pro-nilpotent. Then . In particular,

  1. For each prime , has a unique normal -Sylow subgroup .

  2. For primes , .

The Frattini subgroup of , denoted , is the intersection of all maximal proper open subgroups of .

Proposition 210 ([24, Proposition 2.5.1])

Let be a profinite group. If and , then .

Proposition 211 ([19, Proposition 2.8.11])

If is pro-supersolvable and pro- for some finite set of primes , then is topologically finitely generated if and only if is open.

The -core of a profinite group , , is the closed subgroup generated by all subnormal pro- subgroups of where is a possibly infinite set of primes. In [18, Lemma 2.4], is shown to be pro- and normal. Under certain assumptions on , the -core behaves nicely where denotes the collection of primes different from .

Theorem 212 (Reid [18, Corollary 5.11])

If is a profinite group such that has a topologically finitely generated -Sylow subgroup, then is virtually pro-.

Finitely generated -Sylow subgroups have strong structural consequences by work of O.V. Melnikov; we include a proof via Theorem 212 for completeness.

Theorem 213 (Melnikov, [16])

If is a profinite group that is pro- for some finite set of primes and for which every -Sylow subgroup is topologically finitely generated, then is virtually pro-nilpotent.

  • Let list and form . Since is pro-,

    hence the diagonal map is injective. In view of Theorem 212, there is so that is pro-. The group

    is thus an open subgroup of , and . Since is pro-nilpotent, we conclude is also pro-nilpotent verifying the theorem.    

Melnikov’s theorem implies a useful corollary.

Corollary 214

If is a finite rank profinite group that is pro- for some finite set of primes , then is virtually pro-nilpotent.

Associated to a profinite group is the group of continuous automorphisms of , denoted . There is a natural topology on : For , put

By declaring the sets as varies over open normal subgroups of to be a basis at , becomes a topological group. Following L. Ribes and P. Zalesskii [19], we call this topology the congruence subgroup topology of . In the case is topologically finitely generated, is a profinite group under the congruence subgroup topology [19, Corollary 4.4.4].

24 Elementary groups

Elementary groups play a central role in this work. The class of elementary groups is, intuitively, the class of all t.d.l.c.s.c. groups that can reasonably be built by hand from second countable profinite groups and countable discrete groups. Formally,

  • The class of elementary groups is the smallest class of t.d.l.c.s.c. groups such that

    1. contains all second countable profinite groups and countable discrete groups.

    2. is closed under taking group extensions of second countable profinite or countable discrete groups. I.e. if is a t.d.l.c.s.c. group and is a closed normal subgroup with and profinite or discrete, then .

    3. If is a t.d.l.c.s.c. group and where is an -increasing sequence of open subgroups of with for each , then . We say is closed under countable increasing unions.

The class of elementary groups is surprisingly robust supporting our intuition that is the class of groups “built by hand”.

Theorem 216 ([22, Theorem 3.18])

enjoys the following permanence properties:

  1. is closed under group extension.

  2. If , is a t.d.l.c.s.c. group, and is a continuous, injective homomorphism, then . In particular, is closed under taking closed subgroups.

  3. is closed under taking quotients by closed normal subgroups.

  4. If is a residually elementary t.d.l.c.s.c. group, then . In particular, is closed under inverse limits.

  5. is closed under quasi-products.

  6. is closed under local direct products.

  7. If is a t.d.l.c.s.c. group and there is an -increasing sequence of elementary subgroups of such that is open for each and , then .

We make use of a strong sufficient condition to be elementary.

Theorem 217 ([22, Theorem 8.1])

If is a t.d.l.c.s.c. group and has an open solvable subgroup, then .

The permanence properties of the class of elementary groups give rise to two canonical normal subgroups in an arbitrary t.d.l.c.s.c. group.

Theorem 218 ([22, Theorem 7.9])

Let be a t.d.l.c.s.c. group. Then

  1. There is a unique maximal closed normal subgroup such that is elementary.

  2. There is a unique minimal closed normal subgroup such that is elementary.

We call and the elementary radical and elementary residual, respectively. It is easy to verify has trivial quasi-centre and has trivial locally elliptic radical.

It can be the case and . We give a name to such groups: A t.d.l.c.s.c. group is elementary-free if it has no non-trivial elementary normal subgroups and no non-trivial elementary quotients. Elementary-free groups have a nice property.

Theorem 219 ([22, Corollary 9.12])

For a t.d.l.c.s.c. group , has no non-trivial locally normal abelian subgroups. In particular, elementary-free t.d.l.c.s.c. groups have no non-trivial locally normal abelian subgroups.

The elementary radical and residual may be used to produce a characteristic series. For a t.d.l.c.s.c. group , the ascending elementary series is defined by , , where is the usual projection, and . The ascending elementary series gives a method of reducing to elementary-free groups.

Theorem 220 ([22, Theorem 7.17])

Let be a t.d.l.c.s.c. group. Then the ascending elementary series

is a series of characteristic subgroups with elementary, elementary-free, and elementary.

3 Locally pro-nilpotent t.d.l.c.s.c. groups

Our investigations begin with a general discussion of locally pro-nilpotent t.d.l.c.s.c. groups.

31 Structure theorems

We take as a convention that discrete groups are locally pro- for any finite set of primes .

Proposition 31

Suppose is a t.d.l.c.s.c. group that is locally pro-nilpotent. For each prime , there is a closed characteristic subgroup so that

  1. is locally pro-.

  2. There is a countable increasing exhaustion of by compactly generated open subgroups so that is locally pro-.

  • If is already locally pro-, then satisfies the theorem. Suppose is not locally pro- and consider

    where denotes the set of primes. It is easy to check is invariant and hereditary. We may thus form , the -core. Set and note is a closed characteristic subgroup.

    Fix pro-nilpotent. Letting be a countable dense subset of , form the subgroups

    Certainly, is an -increasing sequence of compactly generated open subgroups of that exhausts . By construction of , we infer that , so there is a pro-nilpotent such that . Therefore, for each . Furthermore, by the uniqueness of in , we see

    and conclude is locally pro-. We have thus verified .

    For , fixing , , and , there is such that . Since , the element centralizes via Proposition 29. Hence,

    It follows , and we conclude . The group is therefore locally pro-.    

Proposition 31 is the essential tool for proving a surprising decomposition result.

Theorem 32

Suppose is a t.d.l.c.s.c. group that is locally pro-nilpotent t.d.l.c.s.c. group. Then either is elementary or the ascending elementary series

is so that

  1. and are elementary; and

  2. there is a possibly infinite set of primes so that where is a locally pro- t.d.l.c.s.c. group and is pro-.

  • Suppose is non-elementary. By passing to , we may assume is elementary free. Fix pro-nilpotent, let list the primes so that has a non-trivial -Sylow subgroup, and form the closed characteristic subgroups as given by Proposition 31 for each .

    For primes from , the group is a closed normal subgroup of . Taking a compactly generated subgroup, Proposition 31 implies is both locally pro- and locally pro-. The group is therefore discrete, and since is elementary, the group is elementary. We conclude is a countable increasing union of elementary subgroups and therefore, is elementary. Since is elementary-free, it must be the case that .

    In view of the proof of Proposition 31, the unique -Sylow subgroup of is a subgroup of . Hence, . On the other hand, if has a non-trivial -Sylow subgroup for , then the uniqueness of implies is non-trivial contradicting that is trivial. Therefore, , and is a compact open subgroup of that is pro-. In particular, is a locally pro- subgroup of .

    We now form the t.d.l.c.s.c. group . In view of Proposition 29, we may identify with , and this allows us to define by

    Since centralizes for and , the map is indeed a continuous homomorphism with an open image. The image of equals , so the image is also normal. Since is elementary-free, we conclude is surjective.

    We now argue is injective. Suppose for contradiction is non-trivial. It follows is non-trivial. The group is contained in , hence there is some prime with so that the -Sylow subgroup of is non-trivial. Since the -Sylow subgroup, , of is unique, it follows contains the -Sylow subgroup of and, in particular, is non-trivial.

    On the other hand, commutes with , so is a central subgroup of . As contains the centre of , we conclude is non-trivial. The subgroup , however, is a characteristic elementary subgroup of , so is a non-trivial elementary normal subgroup of . This contradicts that is elementary-free. Thus, is injective.

    We now conclude verifying the theorem.    

We remark that when is finite in Theorem 32, the local direct product is a direct product.

Corollary 33

A topologically simple locally pro-nilpotent t.d.l.c.s.c. group is either elementary or locally pro- for some prime .

The previous corollary implies an interesting theorem of Y. Barnea, M. Ershov, and T. Weigel.

Corollary 34 (Barnea, Ershov, Weigel [1, Corollary 4.10])

Suppose is a t.d.l.c. group that is non-discrete, compactly generated, and topologically simple. If is locally pro-nilpotent, then is locally pro- for some prime .

  • By [12, (8.7)], is second countable, and by [22, Proposition 6.3], is non-elementary. In view of Corollary 33, we conclude is locally pro- for some prime .    

32 An example

Fixing primes and letting be the -adic integers, we form

and give the product topology. The group is a locally pro-nilpotent and locally pro- t.d.l.c.s.c. group.

We now compute as given by Proposition 31. It is easy to verify . On the other hand,

Letting be the usual projection, is then a closed normal subgroup of . The group is topologically simple, so is either trivial or the entire group. Suppose for contraction . Thus, is a dense subgroup of , and moreover, each normalizes a compact open subgroup of . Results of Willis and H. Glöckner [11, Theorem 5.2] now imply is elementary contradicting [22, Proposition 6.3]. We conclude . This example shows we may not assume is locally pro- if is not elementary free.

It is now easy to verify the ascending elementary series for is so that and . Hence, , which is a direct product of a locally pro- group with a locally pro- group. We have thus computed the decomposition given by Theorem 32.

4 Lie groups over the -adics

We now consider l.c.s.c. -adic Lie groups. The primary references for this section are [3] for Lie group theory and [15] for algebraic group theory.

41 Preliminaries

  • A Lie group is a topological group with a -manifold structure such that the group operations are analytic where is either , , or some non-discrete complete ultrametric field.

A -adic Lie group is a Lie group over , the -adic numbers; note that l.c.s.c. -adic Lie groups are t.d.l.c.s.c. groups. For our purposes, the following characterization of l.c. -adic Lie groups is much more useful:

Theorem 42 (Lazard [13], Lubotzky, Mann [14])

Suppose is a t.d.l.c. group. Then is a -adic Lie group if and only if has a compact open subgroup that is pro- and has finite rank.

A l.c.s.c. -adic Lie group comes with a Lie algebra, denoted . There is a canonical representation with a closed subgroup of for some , [3, III.3.11 Corollary 5].

We also make use of algebraic group theory; in the following denotes a field.

  • An algebraic group is an algebraic variety with group operations given by and such that and are morphisms of algebraic varieties.

An algebraic group is connected if it is connected in the Zariski topology. A subgroup is algebraic if is an algebraic subvariety of . When an algebraic group is a variety defined over a field , we say is a -algebraic group. In general, the prefix “-” is applied to indicate an algebraic object is defined over the field .

There are a number of notions of simplicity in the category of algebraic groups. Since confusing these definitions is easy and dangerous, we enumerate these: An algebraic group is called absolutely simple if is the only proper algebraic normal subgroup of . If all proper algebraic normal subgroups are finite, is called almost absolutely simple. If is the only proper -algebraic normal subgroup, is called -simple. If all proper -algebraic normal subgroups are finite, is said to be almost -simple.

For a -algebraic group, the set of -rational points of is denoted ; the -rational points are the solutions in to the polynomials defining . The set becomes a group under the group operations of . There is a canonical normal subgroup of , denoted . Under the assumption has characteristic zero, is the subgroup generated by all unipotent elements of . (The subgroup exists in the prime characteristic case; the definition, however, is a bit more complicated.) When is a local field, i.e. a non-discrete locally compact topological field, the set of -rational points inherits a topology from the local field . This topology makes a locally compact, -compact, and metrizable topological group. Whenever we say “closed in ”, we mean closed with respect to this topology.

We are now prepared to give a central definition:

  • A -adic Lie group is said to be of simple type if where is an almost -simple isotropic -algebraic group. We say is of adjoint simple type if is also adjoint.

Note adjoint almost -simple -algebraic groups are -simple.

42 Topologically simple l.c.s.c. -adic Lie groups

We now show non-elementary topologically simple l.c.s.c. -adic Lie groups are isomorphic to closed subgroups of and have simple Lie algebras. This is not immediate in the -adic setting because is not necessarily a closed map and there is no longer a bijective correspondence between ideals of the Lie algebra and normal subgroups of the Lie group.

Proposition 45

If is a non-elementary topologically simple l.c.s.c. -adic Lie group, then is isomorphic to a closed subgroup of and is simple.

  • Let be the solvable radical of and suppose for contradiction is non-trivial. Via the Levi-Malcev theorem, see [3, I.6.8 Theorem 5], has a supplement. There is thus and a non-trivial such that ; cf. [3, III.7.1 Proposition 2]. Since is solvable, there is that is solvable. We may find with and . The penultimate term in the derived series of is thus a non-trivial locally normal abelian subgroup of contradicting Theorem 219. We conclude is trivial and, therefore, is semisimple.

    Observe the map is injective. Indeed, else , and applying [11, Proposition 3.1], every element of normalizes a compact open subgroup - such a is called uniscalar. Writing with an -increasing sequence of compactly generated open subgroups of , each is also uniscalar and via [11, Theorem 5.2], elementary. This contradicts that is non-elementary.

    Fix . In view of [3, III.3.8 Proposition 28] and the previous paragraph, as Lie groups, and . Since is semisimple, [20, LA 6.7 Corollary 2] and [3, III.3.11 Corollary 5] imply . Via [3, III.7.1 Theorem 2], is then open in , and it follows and is closed. Since is a closed subgroup of , we conclude is isomorphic to a closed subgroup of .

    To see the simplicity of , say with the simple Lie algebras and let be the collection of such that for each . The subgroup is finite index in , so . Identifying with , we consider . If for each , then is discrete for each which is absurd. Therefore, for some , and since is open in , we infer

    so is simple.    

  • The above argument can easily be adapted to give a proof of a result of Caprace, Reid, and Willis in forthcoming work [7]: An -semisimple l.c.s.c. -adic Lie group has a semisimple Lie algebra.

We now show non-elementary topologically simple l.c.s.c. -adic Lie groups have a close relationship with -algebraic groups. This requires a result from the literature.

Proposition 47 ([10, Proposition 6.5])

Let be a non-compact closed subgroup of such that every proper, closed, characteristic subgroup is compact. Then one of the following cases holds:

  1. The closed subgroup generated by all solvable normal subgroups of is compact open, central in .

  2. is isomorphic to for some .

  3. where is a -adic Lie group of simple type, , and is a central subgroup of that is invariant under a transitive group of permutations of .

With our preliminary work in hand, we now prove the first theorem of this section.

Theorem 48

Suppose is a non-elementary topologically simple l.c.s.c. -adic Lie group. Then with an adjoint -simple isotropic -algebraic group. That is to say, is of adjoint simple type.

  • Via Proposition 45, we may identify with a closed subgroup of and apply Proposition 47. Cases and are impossible since is non-elementary, so for some . Suppose for contradiction and let be the obvious homomorphism. Writing , we see for each , hence for some . We may assume .

    Taking , the group commutes with . It follows commutes with , hence is central in . Since is non-elementary, , and . The group is therefore abelian, and this is absurd as is non-elementary. We conclude where for some almost -simple isotropic -algebraic group.

    Let be the adjoint group of and let be the canonical -isogeny, [15, 1.4.11]. Since is almost -simple and isotropic, is an adjoint -simple isotropic -algebraic group. Further, continuously and surjectively, [15, 2.3.4] and [15, 1.5.5]. By [15, 1.5.6], the group has trivial centre, whereby

    It follows , and we conclude is of adjoint simple type.    

In the next section, -simplicity is not sufficient for our purposes. There is, fortunately, a way to account for this.

Fact 49 ([15, 1.7])

Let be a local field. If is an adjoint -simple isotropic -algebraic group, then there is a finite separable extension of and an adjoint absolutely simple isotropic -algebraic group such that as topological groups.

Corollary 410

If is a non-elementary topologically simple l.c.s.c. -adic Lie group, then as topological groups where is an adjoint absolutely simple isotropic -algebraic group with a finite extension of .

43 Automorphism groups

For a local field and an algebraic group, we consider to be the group of topological group automorphisms of , to be the collection of inner automorphisms, to be the group of algebraic automorphisms of , and to be the group of field automorphisms of . We do not consider the elements of to be continuous.

We now study the automorphism groups of non-elementary topologically simple l.c.s.c. -adic Lie groups. In view of the previous subsection, this is really a question concerning the automorphism group of for a -algebraic group . Conveniently, there is a deep theorem concerning such automorphism groups:

Theorem 411 ([2, Corollaire 8.13], cf. [15, 1.8.2])

Suppose is a finite exten