# Properties of linear groups with restricted unipotent elements

###### Abstract

We consider linear groups which do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic, and show that this class of groups has good properties which resemble those held by groups of non positive curvature and which do not hold for arbitrary characteristic zero linear groups. In particular if such a linear group is finitely generated then centralisers virtually split and all finitely generated abelian subgroups are undistorted. If further the group is virtually torsion free (which always holds in characteristic zero) then we have a strong property on small subgroups: any subgroup either contains a non abelian free group or is finitely generated and virtually abelian, hence also undistorted. We present applications, including that the mapping class group of a surface having genus at least 3 has no faithful linear representation which is complex unitary or over any field of positive characteristic.

## 1 Introduction

Knowing that an abstract group is actually linear (here meaning that embeds in the general linear group for some and field of arbitrary characteristic) is, or at least should be, an indication that is a well behaved group. For instance if is also finitely generated then it is residually finite (but this is not true of linear groups in general), whereas the Tits alternative states that if has characteristic zero, or if again is finitely generated (but not necessarily for arbitrary linear groups in positive characteristic) then either contains a non abelian free subgroup or is a virtually solvable group. In fact if is finitely generated and linear in any characteristic then all subgroups of , whether finitely generated or not, also satisfy this alternative in that either contains the non abelian free group or is virtually solvable (this is outlined in Section 4).

Now on sticking to the case where is both linear and finitely generated, we can try and go further with those subgroups of that do not contain by asking whether has to be finitely generated and also whether is always virtually polycyclic, virtually nilpotent or virtually abelian. Moreover we can think about the geometry of the embedding in the cases where does turn out to be finitely generated and ask if is undistorted in . For positive results in a different context, suppose we take any word hyperbolic group . In this case will be finitely generated, indeed finitely presented, but it need not be linear over any field. However we have a very strong property held by any “small” subgroup of , namely that if is a subgroup not containing then is virtually cyclic and is undistorted in .

In this paper we will
certainly not want to eliminate groups containing , so the correct
analogy here
for the “strong behaviour of small subgroups” seen in the
word hyperbolic case is that
every subgroup of either contains a non abelian free subgroup or
is virtually abelian, finitely generated and undistorted in .
However let us consider the following
examples, all of which are finitely generated linear groups
or subgroups thereof.

Example 1.1: The wreath product , which can be
thought of as a semidirect product
, is
finitely generated
and a subgroup of via the matrices
is transcendental, so that this group is actually
linear in two dimensions over the field for an arbitrary element
which is transcendental over . Clearly this group is solvable, even
metabelian, but not virtually abelian and it contains the subgroup
which is abelian but not finitely
generated.

Example 1.2: We can do the same in any positive characteristic
to obtain the “-lamplighter group” , using the
matrices . Again this is a semidirect product
and has the
property that it is finitely generated and metabelian, but it
contains an abelian group which is torsion and not finitely
generated. Thus fails to be virtually torsion free.
over the field and
which is credited to Philip Hall. In fact here we only use the
fact that and

We now consider distortion of cyclic and abelian subgroups. The next two
cases are by far the most commonly given examples of distorted infinite
cyclic subgroups.

Example 1.3: Consider the finitely presented Baumslag-Solitar
group

. This is also solvable
and metabelian and is even linear over using the matrices
for the infinite order element ,
the subgroup is distorted in . In fact the
behaviour is even worse because is an example of a group
which is not balanced: namely there exists an element of
infinite order
such that is conjugate to but (whereupon
is distorted in if is finitely generated).
Such groups do not have good properties in general, for instance they
cannot be linear over .

Example 1.4: Consider the Heisenberg group
(where ) which is finitely
presented and nilpotent of class 2 (so every subgroup is also
finitely presented) but which is not virtually abelian. It is
linear over using the matrices .
Again it has an abelian subgroup which is not finitely generated. Moreover
as we clearly have and

but as , we have that
is distorted in . (Here is said to
have polynomial distortion, whereas in the previous
example has exponential distortion but in this paper we will not be concerned
with the type of distortion, rather just whether the subgroup is distorted
or not.)
Note also that if is any finite index subgroup of and we have
with then is a commutator in , so
no power of has infinite order in the abelianisation of .

Example 1.5: Consider the semidirect product
given by the automorphism of . This is polycyclic,
so again every subgroup is finitely presented, but
not virtually nilpotent (and thus not virtually abelian). Moreover
is linear over , where is the golden ratio,
via the matrices

Now the abelian subgroup is distorted in because for the th Fibonacci number. However every cyclic subgroup of is undistorted (for instance by seeing as the fundamental group of a closed Sol 3-manifold).

These examples, which were all in very small dimension and used basic fields, certainly demonstrate that small subgroups of finitely generated linear groups need not have the strong behaviour that we see elsewhere. However surely a striking feature of all of these examples is that the “trouble” comes from non identity matrices of the form

which happen to be unipotent.

In this paper we will present two classes of linear groups, defined by restricting the unipotent matrices which are allowed to occur. One of our classes contains the other but both have considerable range in all dimensions and in any characteristic. Indeed the more general class includes all linear groups in positive characteristic and in zero characteristic it includes all real orthogonal and complex unitary linear groups. The stricter class is actually the same as the more general class in characteristic zero whereas it certainly contains all linear groups in positive characteristic which are virtually torsion free. We will show in Theorem 5.2 that if is a finitely generated group in the more general class then every finitely generated virtually abelian subgroup is undistorted in , and in Corollary 4.5 that a finitely generated group in the stricter class has the “small subgroups” property that every subgroup either contains or is both finitely generated and virtually abelian (thus also undistorted).

In fact the motivation for these results came not quite from hyperbolic groups, but more from non positive curvature. Our stricter class of linear groups, or at least the finitely generated groups in this class, corresponds reasonably well to the class of CAT(0) groups (those groups which act geometrically, namely properly and cocompactly by isometries, on a CAT(0) space). Although neither class in this correspondence contains the other, we will see that they share some group theoretic properties. As for our wider class of linear groups, in this paper they will be compared to groups having an isometric action on a CAT(0) metric space where the action is both proper and, as a weakening of being cocompact, is semisimple (meaning that for any the displacement function attains its infimum over ). Again the two classes being compared are not equal, for instance the Burger-Mozes groups are CAT(0) and even have a geometric action on a 2 dimensional CAT(0) cube complex, but are infinite simple finitely presented groups, so are as far from being linear as can be imagined. However this time we will show that the correspondence between the group theoretic properties of these two classes is extremely close indeed. We now give more details of our results.

First for CAT(0) groups, we have the following theorem in [9] Part III Chapter Section 1:

###### Theorem 1.1

([9] Theorem 1.1 Part 1)
A CAT(0) group has the following properties:

(1) is finitely presented.

(2) has only finitely many conjugacy classes of finite
subgroups.

(3) Every solvable subgroup of is virtually abelian.

(4) Every abelian subgroup of is finitely generated.

(5) If is torsion-free, then it is the fundamental group of
a compact cell complex whose universal cover is contractible.

Now none of these five points hold in general on replacing CAT(0) group by finitely generated linear group, regardless of the characteristic. For instance is linear in any characteristic but it has finitely generated subgroups which are not finitely presented, thus (1) and (5) fail here on being applied to one of these subgroups. Next Proposition 4.2 of [15] displays a finitely generated subgroup of (also linear in any characteristic) with infinitely many conjugacy classes of order 4 (extended in [7] to produce finitely presented subgroups of with the same property). Moreover we have already seen the failure of (3) and (4) for the wreath products in Examples 1.1 and 1.2. Indeed the counterexamples being given here for Properties (1), (2) and (5) all belong in our stricter class of linear groups. Nevertheless a consequence of Theorem 4.4 in our paper is that (3) and (4) do actually hold for all finitely generated linear groups in our stricter class, so some of this correspondence does go through.

However there is a Part 2 of the theorem in [9] as follows:

###### Theorem 1.2

([9] Theorem 1.1 Part 2)
If is a finitely generated group that acts properly (but not necessarily
cocompactly) by semisimple isometries on the CAT(0) space , then:

(i) Every polycyclic subgroup of is virtually abelian.

(ii) All finitely generated abelian subgroups of are
undistorted in .

(iii) does not contain subgroups of the form
for non zero with .

(iv) If is central in then there exists a subgroup
of finite index in that contains as a direct factor.

Again all four of these points fail for finitely generated linear groups overall, as already seen in the examples above. But in this paper we will show that these four results hold verbatim for any finitely generated group in our more general class of linear groups. Consequently they all hold for any finitely generated group which is linear in positive characteristic. In fact we have already observed that Property (iii) is a consequence of Property (ii). The rest of the paper is arranged as follows: in the next section we introduce our two classes, where the more general class will be known as NIU-linear groups (for no infinite order unipotents) and the stricter class as VUF-linear groups (virtually unipotent free). We show in this section that both classes of groups have good closure properties, indeed the same as for linear groups in general. Section 3 is about centralisers where we establish Property (iv) for finitely generated NIU-linear groups in Corollary 3.2.

Section 4 is about small subgroups. Property (i) is readily established for NIU-linear groups in Proposition 4.3 but we get much stronger behaviour for VUF-linear groups, as Corollary 4.5 states that a finitely generated VUF-linear group (in fact here finite generation is required) has the property that every subgroup of either contains or is virtually abelian and finitely generated. Then in Section 5 we show that for any finitely generated NIU-linear group, all finitely generated abelian subgroups of are undistorted in . This extends a result in [19] which established this for cyclic subgroups.

Of course both and the subgroup must be finitely generated for this concept to be defined, but combining this with Section 4 tells us that if is any finitely generated VUF-linear group then any subgroup of is virtually abelian, finitely generated and undistorted in . Consequently we end up showing that finitely generated VUF-linear groups are extremely well behaved: not only do they have the very strong properties on small subgroups enjoyed by word hyperbolic groups, but they also have much better closure properties than word hyperbolic groups.

We provide applications of each of these results in Section 6. In particular, by using the ideas in [8] on centralisers of Dehn twists in mapping class groups, we show in Corollary 6.3 that for genus at least 3 the mapping class group cannot be linear over any field of positive characteristic, nor embed in the complex unitary group of any finite dimension (indeed any such representation in either case sends all Dehn twists to elements of finite order). We then consider how our classes of linear groups correspond to the fundamental groups of 3-manifolds of non positive curvature. Though we only consider closed 3-manifolds with a geometric structure, we can show quickly in Theorem 6.4 that for such an with not virtually cyclic, the fundamental group belongs in either of our two classes of linear groups if and only if admits a Riemannian metric of non positive curvature.

We then consider Euclidean groups, meaning arbitrary abstract subgroups of the Euclidean isometry group in a particular dimension, without any condition on their geometric properties. These need not be in either of our two classes of linear groups, although by using our earlier results we can reduce this to dealing with their translation subgroup. We establish Property (iv) for any finitely generated Euclidean group in Corollary 6.5 and a similar result on small subgroups in Corollary 6.6 (though even in dimension two we can have finitely generated Euclidean groups which are solvable but not virtually abelian). We also prove in Corollary 6.7 that all infinite cyclic subgroups of finitely generated Euclidean groups are undistorted, in contrast to [11] which shows that finitely generated abelian subgroups are often distorted.

We finish by looking at the group which is not linear (at least for ). The paper [1] shows that all infinite cyclic subgroups of are undistorted and this was recently extended to abelian groups in [25], though in Theorem 6.8 we show that this also follows by combining the original result in [1] and a quick argument using translation lengths. We also note that the property of undistorted abelian subgroups then extends immediately to and free by cyclic groups .

## 2 Properties of linear groups without infinite order unipotents

If is any field and any positive integer then we say an element of the general linear group is unipotent if all its eigenvalues (considered over the algebraic closure of ) are equal to 1, or equivalently some positive power of is the zero matrix.

###### Proposition 2.1

(i) If is a field of characteristic and is unipotent then has finite order equal to some power of .

Conversely if is any element of with order
which is a multiple of then is a
non identity unipotent element.

(ii) If has zero characteristic then the only unipotent element
having finite order in is .

Proof. For (i) there is with for , where . If we take to be any power of which is at least then

But because and modulo for as is a power of , thus has order dividing .

We now assume for the rest of the proof that is algebraically closed. As we know that has order exactly and hence has minimum polynomial dividing , but any eigenvalue of a matrix must be a root of its minimum polynomial so is unipotent.

For (ii), if has finite order then the minimum polynomial
of is for some and in characteristic zero this has
no repeated roots, so is diagonalisable over but has all eigenvalues
equal to 1, so is the identity.

We now come to the two key definitions of the paper.

###### Definition 2.2

If is any field and any dimension then we say that a subgroup of is NIU-linear (standing for linear with No Infinite order Unipotents) if every unipotent element of has finite order.

Note: by Proposition 2.1, if has positive characteristic then
is automatically NIU-linear. If has characteristic zero then the
definition says that the only unipotent element of is the identity.

Example: If is any subgroup of the real orthogonal group or
of the complex unitary group in any dimension then is
NIU-linear because all orthogonal or unitary matrices are diagonalisable
over .

###### Definition 2.3

If is any field and any dimension then we say that a subgroup of is VUF-linear (standing for linear and Virtually Unipotent Free) if has a finite index subgroup where the only unipotent element of is the identity.

Note: clearly VUF-linear implies NIU-linear and they are the same in
characteristic zero. As for the case when is linear in positive
characteristic, clearly if is also virtually torsion free then
it is VUF-linear. Although this need not true the other
way round, it does hold if is finitely generated, say by [24]
Corollary 4.8. This states that any
finitely generated linear group
has a finite index subgroup whose elements of finite order are
all unipotent (which might be thought of as “Selberg’s theorem in
arbitrary characteristic”).

When we have a group which is only given in abstract form then to
say is NIU-linear or VUF-linear will mean that there exists
some field and dimension
such that has a faithful representation in and the
image of this representation has the respective property.
We now begin by examining the closure properties of these two classes of
linear groups, which turn out to be the same as for
arbitrary linear groups. More precisely the property
of being linear in a given
characteristic is known to be preserved under subgroups, commensurability
classes, direct and free products, so we now show that the same is true
for either of these two classes when restricted to a particular characteristic.

###### Proposition 2.4

If and are groups which are both NIU-linear over fields
having the same characteristic then:

(i) Any subgroup of (or ) is NIU-linear.

(ii) Any group commensurable with (or ) is NIU-linear.

(iii) The direct product is NIU-linear.

(iv) The free product is NIU-linear.

The same holds with NIU-linear replaced throughout by VUF-linear.

Proof. We will proceed in the following order. First, NIU-linear groups in a given positive characteristic just mean arbitrary linear groups in this characteristic, in which case the closure properties are already known. We then argue for NIU-linear groups in characteristic zero, which here are the same as VUF-linear groups. We then make the necessary adjustments in our proof for VUF-linear groups in positive characteristic.

Part (i) is immediate for NIU-linear groups and follows straight away for VUF-linear groups because if is a subgroup of and is the given finite index subgroup which is unipotent free then has finite index in . This now reduces (ii) to saying that if is NIU-linear and is also a finite index subgroup of the group then is NIU-linear too (and the same for VUF-linearity but this is immediate). We certainly know that is linear over the same field as by induced representations (though probably of bigger dimension). But if is a unipotent element of infinite order then so are all its positive powers and some of these will lie in .

For (iii) and (iv), we first observe that there is a field containing both and . (As they have the same prime subfield , we can adjoin enough transcendental elements to which are all algebraically independent, resulting in a field where all elements of and are algebraic over , so that and both embed in the algebraic closure of .) We then see that the direct product of is linear over in the usual way by combining the two blocks representing and . Then we note that the eigenvalues of an element are just the union of the eigenvalues in each of the two blocks. Thus a unipotent element of is unipotent in both the and the blocks, hence in characteristic zero it is the identity in if and are both NIU-linear. If instead they are VUF-linear with finite index subgroups respectively that are unipotent free then so is , which has finite index in .

Free products of linear groups over the same characteristic were shown to be linear in [20], which of course implies NIU-linearity in positive characteristic. For the remaining cases we can assume that we are in an algebraically closed field which has infinite transcendence degree over its prime subfield. This is the setting for the proofs in [21] which rediscovered the result on free products. (Actually that paper works in but the proofs go through in as well.)

We first assume, by increasing the size of the matrices and adding 1s on the diagonal if needed, that the NIU-linear groups and both embed in with neither subgroup containing any scalar matrices apart from the identity. Moreover these embeddings will still be NIU-linear. Then Lemma 2.2 of the above paper says there is a conjugate of (which will henceforth be called ) in such that no non identity element of this conjugate has zero in its top right hand entry. Similarly by taking a conjugate of we can assume that no non identity element of has zero in its bottom left hand entry. Next [21] Proposition 1.3 shows that embeds in for any element which is transcendental over . This is achieved by conjugating by a diagonal matrix made up of powers of and then showing that in the resulting linear representation of , any element which is not conjugate into or will have a trace which is a Laurent polynomial in with at least two non trivial terms and so is not the identity. But this also means that the trace of is transcendental over , thus cannot equal and so is not unipotent. Now being unipotent is a conjugacy invariant and as there are no unipotents in or either, the resulting faithful linear representation of is NIU-linear.

Thus we are done in characteristic zero for both our classes of linear groups. As for preservation of VUF-linearity under free products in characteristic , we can use a trick: by dropping down further if necessary we can assume that both the unipotent free finite index subgroups of and of are normal. This then gives us two homomorphisms (for ) onto finite groups and these can be both be extended from to with kernels which we will call and . Now note that is also normal and has finite index in , and that both maps from to are retractions so we have .

So suppose that there is a non identity unipotent element .
By Proposition 2.1 (i) we can assume that has order . Thus
in the free product we must have that is conjugate into
or . This conjugate also lies in and is
unipotent, so if it is
in then it is also in and the same for and . But
both and are unipotent free, so either way we are done.

Note: To see that we cannot mix and match different characteristics
in (iii) and (iv), even for finitely generated groups, the lamplighter
group is linear in characteristic 2,
but only in this characteristic, whereas the “trilamplighter” group
is linear only in characteristic 3. Thus any group
containing them both (such as their direct or free product) is not
NIU-linear, or even linear over any field.

We finish this section with a few words on how these closure properties work for the three geometric classes of groups that were mentioned in the introduction. First, being word hyperbolic is well known to be preserved under commensurability classes and free products, but hardly ever under direct products. Moreover it is certainly not preserved under passage to subgroups in general, but even if we only pass to finitely presented subgroups then there are examples of Noel Brady which are not word hyperbolic.

Next CAT(0) groups are preserved under both free and direct products, but again we can have finitely presented subgroups of CAT(0) groups which are not themselves CAT(0), for instance by using finiteness properties of direct products of free groups and their subgroups (see [9] Chapter III. Section 5). Also commensurability is unclear because of the problem of lifting the action of a finite index subgroup to the whole group in a way that preserves cocompactness.

However we do obtain all four closure properties for the class of groups acting properly and semisimply on a CAT(0) space, provided we impose the mild extra condition that the CAT(0) space is complete (indeed some authors refer to this as a Hadamard space). This geometric property is preserved by free and direct products, and clearly also by passage to arbitrary subgroups. But less obviously, if a group has finite index in and has a proper action on the CAT(0) space then we can use this to induce an action of on the direct product of copies of (also a CAT(0) space) and this action will be proper. Moreover if is complete then this induced action of will also be semisimple, by [9] Chapter II Proposition 6.7 and Part (2) of Theorem 6.8 (we thank Martin Bridson for helpful correspondence on this point).

Consequently, as for the analogies we have mentioned between the various classes of linear and geometric groups, the strongest one by far is between our NIU-linear groups and groups acting properly and semisimply on a complete CAT(0) space. Having seen that the closure properties above all hold in both of these cases, the next sections are about other group theoretic properties of NIU-linear and VUF-linear groups.

We finish this section by remarking on two papers containing related results: first [17] considers basically the same class of groups (which they call Hadamard groups) as those acting properly and semisimply on complete CAT(0) spaces (their definition of a proper action, which they call discrete, is that of acting metrically properly, though if the space is itself a proper metric space then the two definitions are equivalent). Also [2] shows that in characteristic zero a finitely generated linear group has finite virtual cohomological dimension if and only if there is a finite upper bound on the Hirsch ranks of its finitely generated unipotent subgroups. Thus our finitely generated NIU-linear groups in characteristic zero always have finite virtual cohomological dimension, though as this also holds for the last three examples in Section 1, we see that this property on its own is not enough to rule out bad behaviour.

## 3 Abelianisation of centralisers in linear groups with restricted unipotent elements

We first show the following result.

###### Theorem 3.1

Suppose that is a linear group over some field
of arbitrary
characteristic and is a abelian subgroup which is
central in . Let be the homomorphism from to
its abelianisation . We have:

(i) If is NIU-linear then is a torsion group.

(ii) If is VUF-linear then is a finite group.

Proof. We first replace our field by its algebraic closure, which we will also call . Then it is true that any abelian subgroup of is conjugate in to an upper triangular subgroup of , for instance by induction on the dimension and Schur’s Lemma.

For any and we have . This means that must map not just each eigenspace of to itself, but each generalised eigenspace

and together these span, so that if has distinct eigenvalues then is a -invariant direct sum of .

We now take a particular (but arbitrary) non identity element of and restrict to the first of these generalised eigenspaces , so that here only has the one eigenvalue . If this property also holds on for every other then we proceed to , and so on. Otherwise there is another such that we can split further into pieces where has only one eigenvalue on each piece. Moreover this decomposition is also -invariant because it can be thought of as the direct sum of the generalised eigenspaces of when is restricted to .

We then continue this process on all of the pieces and over all elements of until it terminates (essentially we can view it as building a rooted tree where every vertex has valency at most and of finite diameter). We will now find that we have split into a -invariant sum of blocks, where any element of has a single eigenvalue when restricted to any one of these blocks.

Now we conjugate within each of these blocks so that the restriction of to this block is upper triangular, using the comment at the start of this proof. Under this basis so obtained for , we have that any will now be of the form

where each block is an upper triangular matrix with all diagonal entries equal (as these are the eigenvalues of within this block). More generally any will be of the form

for various matrices which are the same size as the respective matrices because we know preserves this decomposition.

Consequently we have available as homomorphisms from to the multiplicative abelian group not just the determinant itself but also the “subdeterminant” functions , where for the function is defined as the determinant of the th block of when expressed with respect to our basis above, and these are indeed homomorphisms as is

As is a homomorphism from to an abelian group, it factors through the homomorphism from to its abelianisation because this is the universal abelian quotient of . This means that is contained in and so we can replace with for the rest of the proof.

Thus suppose that there is some which is in the kernel of . We know that

for upper triangular matrices and as the diagonal entries of are constant, say for , we conclude that where . In other words a power of is unipotent, thus if is NIU-linear then has finite order.

In the case where is VUF-linear with a finite index subgroup
having no non trivial unipotent elements, we have that has
finite index in so we will
take the restriction of to
and show that this has finite kernel.
If we have elements with exactly
the same diagonal entries then must be unipotent and so
. But on considering the diagonal entries
of an upper triangular
element in , we see they are all roots of unity
with bounded exponent
and so there are only finitely many possibilities, thus also only
finitely many possibilities for elements of which are also
in the kernel of .

In particular any infinite order element of also has infinite order in the abelianisation of . We now adapt this to obtain the same conclusion of Theorem 1.2 Part (iv) in the case of NIU-linear groups.

###### Corollary 3.2

If is a finitely generated NIU-linear group and is central in then there exists a subgroup of finite index in that contains as a direct factor.

Proof. Theorem 3.1 gives us a homomorphism from to some abelian group which is injective on . By dropping to the image, we can assume that is onto without loss of generality and so is also finitely generated. By the classification of finitely generated abelian groups, we have that Torsion for and we can compose with a homomorphism from to in which still injects, so will have finite index.

Thus if we set then the pullback
has finite index in . Also and are
normal subgroups of with , giving .

## 4 Small subgroups of NIU-linear and VUF-linear groups

Suppose that is an algebraically closed field of any characteristic, and that is a solvable subgroup of .

A consequence of the Lie - Kolchin Theorem, or alternatively results of Malce’ev, is that there is a finite index subgroup of which is upper triangularisable, namely conjugate in to a subgroup of where every element is upper triangular. Thus on assuming that we have conjugated within so that is in this upper triangular form, we immediately see there is a homomorphism from to the abelian group given by the diagonal elements of an element . As the kernel of consists only of upper unitriangular matrices, meaning that it must be a nilpotent group, we obtain the well known:

###### Proposition 4.1

A solvable linear group over an arbitrary field is virtually (nilpotent by abelian), meaning that possesses a finite index subgroup which has an abelian quotient with nilpotent kernel.

To improve on this result we will first use NIU-linearity and VUF-linearity, then further strengthen it by assuming that , which might not be finitely generated, is in fact a subgroup of a finitely generated linear group.

###### Corollary 4.2

Suppose that is a solvable group and is NIU-linear. Then is virtually (torsion by abelian) and if we are in characteristic zero then is in fact virtually abelian (although need not be finitely generated in any characteristic).

Proof. The kernel of in Proposition 4.1 consists entirely of unipotent elements. But as is NIU-linear, we have by Proposition 2.1 that is a torsion group in positive characteristic and in zero characteristic.

We have already seen that the wreath product is
linear in characteristic and this contains the infinitely
generated abelian group .
As for characteristic zero,
we can take all diagonal matrices over say in any particular
dimension to get an NIU-linear and VUF-linear
group which is countable and abelian but not finitely generated.

Next we show the equivalent result for NIU-linear groups of Theorem 1.2 Part (i).

###### Proposition 4.3

Suppose that is polycyclic and NIU-linear, then is virtually abelian.

Proof.
We are done in characteristic zero by Corollary 4.2 above.
Moreover being polycyclic means that all subgroups of are
finitely generated, in particular which is also
a solvable torsion group and thus is finite.
Hence is finite by abelian as well as finitely generated, so by
standard results it is virtually abelian (for instance: is
certainly residually finite, so we can take finitely many
finite index subgroups of , each missing an element of ,
and their intersection injects under so is abelian).

We now move to considering solvable groups which are VUF-linear. We can obtain strong results if is finitely generated, but in fact it is enough to assume that is merely contained in some linear group which is finitely generated. This is because we can then utilise the fact that can be thought of as a subgroup not only of for a finitely generated field, but (by taking the ring generated by all entries of a generating set for the group which is closed under inverses) also of where is an integral domain which is finitely generated as a subring of . This approach is exploited in [24] Chapter 4 and allows us here to obtain a much better result on small subgroups in line with word hyperbolic or CAT(0) groups.

###### Theorem 4.4

Suppose that the solvable group is VUF-linear. Then is virtually abelian. If further we have that is a subgroup of where is finitely generated then is also finitely generated.

Proof. We first replace with the appropriate finite index subgroup which is unipotent free. We next assume that is algebraically closed and proceed as in Proposition 4.1 and Corollary 4.2 by conjugating in so that has a finite index subgroup which is upper triangular. Now the homomorphism from to has kernel consisting only of unipotent elements but is unipotent free, so is abelian with and being virtually abelian.

Now suppose that is contained in the subgroup of
with finitely generated. We then follow
[24] Lemma 4.10 and see that, as is a subgroup of
for a finitely generated subring of the field
obtained from the entries of a symmetric generating set for ,
our homomorphism actually has image
in for the group of units of , which
happens to be a finitely generated abelian group,
so and thus and are also finitely generated.

We can now make this result on small subgroups of VUF-linear groups definitive by bringing in the Tits alternative.

###### Corollary 4.5

Suppose that is a finitely generated VUF-linear group. Then any subgroup of (whether finitely generated or not) satisfies the following alternative: either contains a non abelian free subgroup or is virtually abelian and finitely generated.

Proof. The Tits alternative in characteristic zero tells us that any linear group in this characteristic either contains a non abelian free subgroup or is virtually solvable, thus Theorem 4.4 applies.

This does not quite work in positive characteristic, because although
the Tits alternative still holds as above for linear groups in positive
characteristic which are finitely generated, for an arbitrary
linear group we might only conclude that it is solvable by locally
finite (for instance for and
the algebraic closure of the finite field ). However we
can again evoke finite generation of the ring in this situation
to conclude that if is a subgroup of a finitely generated group
which is linear in positive characteristic then either contains
a non abelian free subgroup or is indeed virtually solvable.
For instance [24] Lemma 10.12 states that if is a
finitely generated integral domain and is a subgroup of
, which is the case for here, such that every 2-generator
subgroup of is virtually solvable, which is also the case for
here if it does not contain a non abelian free subgroup, then
itself is virtually solvable.

We note that in other settings we do not always have a Tits alternative available. For instance it is currently unknown whether every finitely generated subgroup of a CAT(0) group must either contain or be virtually solvable, whereas there are finitely generated groups acting properly and semisimply on CAT(0) spaces which fail this alternative.

## 5 Undistorted abelian subgroups

In this section we prove Theorem 5.2 which states that for a finitely generated NIU-linear group , all finitely generated abelian subgroups are undistorted in . Proposition 2.4 of [19] showed this for cyclic subgroups (in fact their statement concludes that cyclic subgroups do not have exponential distortion but the proof given works for arbitrary distortion). They use a fact from the paper [23] establishing the Tits alternative which is that for any element of infinite multiplicative order in a finitely generated field, we can find an absolute value on the field which is not equal to 1 on this element. We will argue in this way but will need to work with many absolute values simultaneously, as well as needing a separate argument for units in number fields.

Suppose we have a group which is finitely generated but otherwise arbitrary for now, along with a finitely generated abelian subgroup of . We will repeatedly use the fact that showing a finite index subgroup of is undistorted in also establishes this for itself, so without loss of generality we will say that for some .

Suppose further that we have a function with the following two properties:

Lower bound on : There exists a real number such that for | ||||

Then on taking any finite generating set for and denoting for the standard word length of an element with respect to , we have by (i) that on setting . Now and indeed unless , in which case (ii) cannot hold. Thus if is an arbitrary element of then

so is undistorted in .

For linear groups, at least over , we have a natural candidate for this function . Indeed if we now suppose that is a subgroup of then we can put the max norm on and the corresponding operator norm

on which satisfies for (or even for two by matrices over ), so the operator norm is a map from to which is submultiplicative. Of course this can be made subadditive by taking logs, but it might return negative values. Therefore we actually define

This will be at least zero because if is an eigenvalue with eigenvector , so that , then , but if then is an eigenvalue of with .

Now subadditivity for is easily checked using the submultiplicity of , so we first review the proof of Proposition 2.4 in [19] for the case when our abelian group is infinite cyclic: if possesses an eigenvalue with then for we get , and if then so that

and we can swap and when , hence for . Thus when we will have and so is undistorted in .

However it is quite possible that all eigenvalues of every element of have modulus 1, for instance if were a group of orthogonal or unitary matrices (which is exactly one of the cases we are considering). We also need a generalisation of this argument to arbitrary fields, not just the characteristic zero case. In order to complete the proof for cyclic subgroups, [19] refers back to the famous proof [23] of the Tits alternative, and specifically Lemma 4.1 of this paper which states that for any finitely generated field and for any element of infinite multiplicative order, there is an absolute value on with . Here if is any field then an absolute value satisfies

Thus to finish the cyclic case, suppose that our group is a subgroup of for an arbitrary field. We first note that we can take to be a finitely generated field without loss of generality because is a finitely generated group. Then for any element we extend to by adjoining the eigenvalues of , so that is still finitely generated. Now if all eigenvalues of are roots of unity then a power of is unipotent, which we are specifically excluding. Thus there must be an eigenvalue of and an absolute value on such that , so we do indeed have that in the above and is undistorted in .

Now we move to the case when is a free abelian subgroup of rank in the
finitely generated group . However, although still following
the same path, the argument requires more work than the cyclic case.
In particular, difficulties are caused by the fact that
we can have in general as the next example shows:

Example: Let and consider . Then on trying the obvious absolute value
for and setting

Property (ii) above would require such that . But so this is impossible because is dense in . However we also have -adic evaluations on , in particular the 2-adic evaluation and the 3-adic evaluation . Let us now define as above but using either of these two evaluations in place of the usual Euclidean absolute value. Then is still subadditive but on setting , we have

However in the first case we have and in the second, so Property (ii) fails badly here. Moreover by Ostrowski’s Theorem, every absolute value on is equivalent either to the modulus or to a -adic evaluation for some prime , all of which give rise to our function being subadditive but all fail Property (ii). The key now is to see that we can combine different functions obtained from separate evaluations, because the sum of two subadditive functions is also subadditive.

In particular, here we can set to be the function

which (as ) is at least , thus providing the required lower bound on . Of course as we have merely shown that is undistorted in itself, but we have used Properties (i) and (ii) to establish this.

We can now proceed with the general argument. We first introduce the functions that we will be using here.

###### Proposition 5.1

Let any field, with any finite list of absolute values on .

Then for any dimension there exists a function which is subadditive and with the following property: if has an eigenvalue then for all we get .

Proof. We first let be the max norm on the vector space with respect to , so that given we have

We then let be the corresponding operator norm on , namely

and define like before as

for . Then each is subadditive, thus so is . Now if is an eigenvalue for then for each we certainly have

thus .

We now come to the proof of the equivalent of Theorem 1.2 Part (ii).

###### Theorem 5.2

If is any finitely generated group which is NIU-linear and is any finitely generated abelian subgroup of then is undistorted in .

Proof. We take to be a subgroup of , where is some field which we initially assume to be algebraically closed. By dropping to a finite index subgroup of if required, we can assume that is free abelian of rank . Moreover we can conjugate within so that every element of is upper triangular. Having done this, we henceforth assume that is the finitely generated field over the relevant prime subfield which is generated by the entries of .

By taking logs, any absolute value on can also be thought of as a group homomorphism from to . Moreover from a single absolute value , we obtain functions from to where by setting for the th diagonal entry of the element . As any is an upper triangular matrix, we see that is actually a homomorphism.

###### Lemma 5.3

There exist absolute values on such that if we let be the homomorphism obtained from the coordinate functions then is injective.

Proof. We proceed by induction on . We first write for and we can assume that there exist absolute values such that the resulting homomorphism , obtained by combining the relevant coordinate functions , is injective on . If it is injective on too then we are happy and can just add any to the list of absolute values. Otherwise vanishes on some element of which must be of the form for and . We now pretend that actually was the finite index subgroup which is also equal to (with the subgroup unchanged).

Next we use Tit’s Lemma 4.1 in [23]: as is assumed finitely generated, we have that for any of infinite order, there exists an absolute value on a locally compact field containing , and thus on itself by restriction, such that . We now apply this to each diagonal element of : if every one has finite order then the eigenvalues of are all roots of unity and so some power of is unipotent, which is excluded by the NIU-linear hypothesis unless has finite order but is free abelian of rank .

Consequently there is some where
the th diagonal element of has infinite
order, thus there is also an absolute value
with
. We set so that
. Hence on letting
be the extension of from to obtained
by including as extra coordinate
functions, we have that is injective on : suppose
vanishes on an element of , which can also be written as for
some , so that
. Then
looking at the first coordinate functions tells
us that is the identity (because we suppose that here vanishes
under but that injects), whereas
implies .

An absolute value on a field is called discrete if the image of is a discrete set, which is equivalent to saying that there exists such that the image of the function in is . We now proceed on the assumption that all absolute values obtained in Lemma 5.3 are discrete (thus in this would mean we have taken the -adic evaluations but not the modulus) and we will remove this assumption at the end of the proof.

We have the coordinate functions on and we can now assume that there exists real numbers with . Thus by Lemma 5.3 embeds via in the subgroup

which is clearly a lattice in , namely a discrete subgroup of which spans. Thus on letting be the vector subspace of spanned by , which is itself a lattice in , we see that for and for any norm on , we have such that

Thus if we put the norm on induced from the max norm on , we obtain

But for the th diagonal entry of . As is upper triangular, this is an eigenvalue of . Hence by applying Proposition 5.1 with , we obtain our subadditive function where for any and , we have so that

Thus satisfies both Property (i) and Property (ii) and hence is undistorted in .

We must now consider when we are able to use absolute values which are discrete, thus we turn again to [23] Lemma 4.1 but this time we examine the proof rather than just the statement. Our field is finitely generated over its prime subfield (namely in characteristic zero and in characteristic ) and we suppose we are given a non zero element which is not a root of unity (so has infinite order in ). Let us first assume that is transcendental over , in which case this proof proceeds as follows. We set to be the subfield of consisting of those elements which are algebraic over , with being finitely generated over