12 1. Introduction

One can also define algebraic groups such as GLn over the p-adic ratio-

nals Qp; thus for instance GLn(Qp) is the group of invertible n × n matrices

with entries in the p-adics. This is still a locally compact group, and is

certainly not Lie.

Exercise 1.1.3 (Solenoid). Let p be a prime. Let G be the solenoid group

G

:= (Zp×R)/ZΔ,

where

ZΔ :=

{(n, n) : n ∈ Z} is the diagonally embedded

copy of the integers in Zp × R. (Topologically, G can be viewed as the set

Zp ×[0, 1] after identifying (x+1, 1) with (x, 0) for all x ∈ Zp.) Show that G

is a compact additive group that is connected but not locally connected (and

thus not a Lie group). Thus one cannot eliminate p-adic type behaviour from

locally compact groups simply by restricting attention to the connected case

(although we will see later that one can do so by restricting to the locally

connected case).

We have now seen several examples of locally compact groups that are

not Lie groups. However, all of these examples are “almost” Lie groups in

that they can be turned into Lie groups by quotienting out a small compact

normal subgroup. (It is easy to see that the quotient of a locally compact

group by a compact normal subgroup is again a locally compact group.) For

instance, a group with the trivial topology becomes Lie after quotienting out

the entire group (which is “small” in the sense that it is contained in every

open neighbourhood of the origin). The infinite-dimensional torus

(R/Z)N

can be quotiented into a finite-dimensional torus

(R/Z)d

(which is of course

a Lie group) by quotienting out the compact subgroup

{0}d ×(R/Z)N;

note

from the definition of the product topology that these compact subgroups

shrink to zero in the sense that every neighbourhood of the group identity

contains at least one (and in fact all but finitely many) of these subgroups.

Similarly, with the p-adic group Zp, one can quotient out by the compact

(and open) subgroups

pjZp

(which also shrink to zero, as discussed above) to

obtain the cyclic groups

Z/pjZ,

which are discrete and thus Lie. Quotienting

out Qp by the same compact open subgroups

pjZp

also leads to discrete

(hence Lie) quotients; similarly for algebraic groups defined over Qp, such

as GLn(Qp). Finally, with the solenoid group G := (Zp ×

R)/ZΔ,

one can

quotient out the copy of

pjZp

× {0} in G for j = 0, 1, 2,... (which are

another sequence of compact subgroups shrinking to zero) to obtain the

quotient group

(Z/pjZ

×

R)/ZΔ,

which is isomorphic to a (highly twisted)

circle R/Z and is thus Lie.

Inspired by these examples, we might be led to the following conjecture:

if G is a locally compact group, and U is a neighbourhood of the identity,

then there exists a compact normal subgroup K of G contained in U such

that G/K is a Lie group. In the event that G is Hausdorff, this is equivalent

to asserting that G is the projective limit (or inverse limit) of Lie groups.