Fixed points in convex cones
Abstract
We propose a fixed-point property for group actions on cones in topological vector spaces. In the special case of equicontinuous representations, we prove that this property always holds; this statement extends the classical Ryll-Nardzewski theorem for Banach spaces. When restricting to cones that are locally compact in the weak topology, we prove that the property holds for all distal actions, thus extending the general Ryll-Nardzewski theorem for all locally convex spaces.
Returning to arbitrary actions, the proposed fixed-point property becomes a group property, considerably stronger than amenability. Equivalent formulations are established and a number of closure properties are proved for the class of groups with the fixed-point property for cones.
1. Introduction and results
As an art, mathematics has certain fundamental techniques. In particular, the area of fixed-point theory has its specific artistry.
Traiter la nature par le cylindre, la sphère, le cône …
1.A. About fixed points
A large part of fixed-point theory for groups, and perhaps the most widely applied, takes place in the following setting:
A group acts on a compact convex set in a topological vector space .
We will agree that the action on comes from a continuous linear representation on even if we are only given a continuous affine action on ; a suitable , -space can be reconstructed (as the dual of the space of affine continuous functions on ).
The following two themes are of particular interest:
- (I)
General fixed-point theorems without any restriction on the group this requires assumptions on ; , and/or the nature of the action. ,
- (II)
Make no assumptions on or on the action. Then, always having a point becomes a striking property of a group -fixed .
A powerful example of (I) is the original Ryll-Nardzewski theorem Reference RN62. Here the assumptions are that is a Banach space endowed with its weak topology and that the action is (norm-)isometric. It is because compactness is only assumed in the weak topology that this result is so much stronger than earlier theorems of Kakutani Reference Kak38 and Hahn Reference Hah67; Ryll-Nardzewski further generalised his statement to distal actions in general locally convex spaces Reference RN67 (see also Reference GM12 for further extensions).
As for (II), this fixed-point property (considered e.g. by Furstenberg Reference Fur63, Def. 1.4) turns out to be equivalent to amenability Reference Day61, Reference Ric67. Although amenability was originally introduced by von Neumann Reference vN29 in his study of Banach–Tarski paradoxes, it has since become a familiar concept in a number of unrelated areas. The connection to (II) is that finitely additive measures normalised to have total mass one form a compact convex set.
1.B. Compact is too restrictive
The context above is too limited for certain applications. Even in the original case of Banach–Tarski paradoxes, the key question is whether our space carries a finitely additive measure defined on all subsets, invariant under isometries, and normalised for a given subset of interest, e.g. a ball. In particular, such a measure would generally need to be unbounded (and infinite on large subsets). A natural invariant normalisation condition is for all isometries such measures form an invariant convex cone. ;
For the purpose of giving examples of paradoxes, Hausdorff Reference Hau14 and Banach–Tarski Reference BT24 could bypass this issue because they could consider instead subsets of the group of rotations; there, measures can be normalised over the entire group, thus still forming a compact set.
However, the question considered by von Neumann Reference vN29, p. 78 is really whether there exists a finitely additive measure on a set invariant under a group , and normalised for some subset , that need not be indeed be invariant. In fact Tarski proved that this is equivalent to the non-paradoxicality of —or Reference Tar38, end of §3. Groups that always admit such a measure are called supramenable since Rosenblatt Reference Ros72, 4.5.
It is known that supramenability is much stronger than amenability. It fails to hold for metabelian groups such as the Baumslag–Solitar group or the lamplighter group because supramenable groups cannot contain non-abelian free semigroups Reference Ros72, 2.3. In fact, supramenability is equivalent to not containing a Lipschitz-embedded binary tree Reference KMR13, Prop. 3.4.
On the other hand, examples of supramenable groups include all abelian groups, all locally finite groups, and more generally the groups that are subexponential in the sense that for every finite subset of the group, see ;Reference Ros72, 3.5. This argument was in fact already used in 1946 by Sierpiński; see pp. 115–117 in Reference Sie46.
Another situation where compact convex sets are too restrictive is the case of Radon measures on locally compact spaces. Amenability can be characterised in terms of the existence of invariant probability measures on compact topological spaces, but the non-compact case is of course also very interesting (we think e.g. of Haar measures); it has been explored e.g. in Reference KMR13, Reference MR15. The positive Radon measures form, again, a convex cone; it has in general no affine base. A basic example is the affine group of the line, which is amenable but does not fix any non-zero Radon measure on the line.
Many other examples of convex cones without base arise, for instance in non-commutative operator algebras; see e.g. Reference ERS11.
1.C. Convex cones
Our starting point will be the following, to be further refined below (while earlier work is briefly reviewed in Section 10.C):
A group acts on a convex proper cone in a topological vector space The question is whether there exists a non-zero fixed point. .
Recall that proper means or equivalently that the associated pre-order on , is an order. It is understood that a representation of a group on a vector space is an action by linear maps that are assumed continuous (hence weakly continuous) if a topology is given on and order-preserving if a vector space pre-order (equivalently, a convex cone) is given for We then also speak of a representation on the cone, but the ambient linear representation is understood as part of the data. .
The topological condition on shall be that is weakly complete. Choquet has eloquently made the case in his ICM address Reference Cho63 that this condition captures many familiar cones encountered in analysis. Oftentimes, the spaces that we need to consider are weak in the sense that their topology coincides with the associated weak topology; in any case, the weak completeness assumption justifies that we shall only consider locally convex Hausdorff topological vector spaces (as is generally done in the compact convex case too).
For instance, the cone of positive Radon measures on a locally compact space is complete for the vague topology (Proposition 14 in Reference Bou65, III, §1, No. 9), hence weakly complete since this topology is weak.
This setting obviously includes the earlier picture, for if is convex and compact or weakly compact, then the cone
is convex, proper, and weakly complete—indeed even weakly locally compact Reference Bou81, II, §7, No. 3. This special case admits a base, and this base is even , -invariant.
1.D. How groups act on cones
Having given up compactness, we need some bounds on the actions on cones; otherwise, even the group will act without non-zero fixed point on any cone, simply by non-trivial scalar multiplication. We introduce two conditions—which both trivially hold in the above case .
This condition holds in all the examples that we aim at. For instance, if acts on a locally compact space and is the cone of positive Radon measures, then the of any Dirac mass is bounded. Indeed, the closure of the image of -orbit under the canonical map realises the one-point compactification of (Proposition 13 in Reference Bou65, III, §1, No. 9). Notice that typically the orbits admit as an accumulation point.
In this example with the convex subcone of bounded orbits is in fact dense in , Reference Bou81, III, §2, No. 4, a situation that we can often reduce to (Proposition 21 below). However, it is critical to keep in mind that even then, there are usually also unbounded orbits. If for instance we let act by translation on then both the bounded and the unbounded orbits are dense; there are even invariant rays where , acts unboundedly, namely exponentials.
We turn to the second condition, dual to local boundedness. As a motivation, we recall that any infinite group acts on some locally compact space without preserving any non-zero Radon measure Reference MR15, 4.3. By contrast, it was shown in Reference KMR13 that we can characterise supramenability by restricting our attention to cocompact actions on locally compact spaces.
It is easily verified that for this specific example of Radon measures in the vague topology, the following general definition is precisely equivalent to the cocompactness of the on the underlying space -action .
Finally, we can formalise our group-theoretical property:
In light of theme (II) outlined at the beginning of this introduction, this is a strengthening of amenability. As we shall see, all subexponential groups enjoy the fixed-point property for cones (Theorem 83). On the other hand, this property implies supramenability and therefore the metabelian groups indicated above do not have the fixed-point property for cones.
1.E. Groups with the fixed-point property for cones
Just as the fixed-point property defined by Furstenberg turned out to be one of several equivalent incarnations of amenability, we shall establish a few characterisations in the conical setting. These equivalences (Theorem 7) will facilitate our further investigation of the fixed-point property.
Starting from our original motivation, we would like to obtain invariant finitely additive measures normalised for arbitrary subsets of a -space the key case concerns subsets of ; itself. In fact, we can more generally consider invariant “integrals” normalised for functions rather than subsets. To this end, given a bounded function on we denote by , the ordered vector of functions -space by -dominated .
It should not be surprising that such integrals can be obtained as fixed points in cones. The existence of integrals under the stronger assumption that the group is subexponential was established in Reference Jen76, Cor. 3 and Reference Kel14, 2.16.
A simpler task would be to find a functional on the smaller space spanned by the of -orbit However, even in the case of a subset . with it is not clear a priori that such a condition should imply supramenability. This amounts indeed to the question asked in Greenleaf’s classical 1969 monograph ,Reference Gre69, p. 18 (and met with skepticism in Reference Jen80, Remark on p. 369).
We shall give a positive answer to this question in Corollary 20 below. For now, we recall that there is a positive linear form on -invariant normalised for if and only if the following translate property introduced in Reference Ros72 holds for .
This condition is equivalent to the existence of an integral on because it ensures at once that the linear form is both well-defined and positive. It has been known (Reference Gre69, §1, Reference Ros72, §1) that such an integral can be extended to all of provided that is amenable. Therefore, we shall need to show that the translate property implies amenability.
Von Neumann’s investigation of invariant finitely additive measures was not restricted to subsets of acting on itself, but it can readily be reduced to it. Likewise, we shall extend the translate property to abstract ordered vector spaces and consider the abstract translate property (defined in Section 4 below). Finally, in parallel to the finitary characterisations of amenability in terms of Reiter properties, we consider in Section 9 the property corresponding to a function -Reiter on .
We now investigate the class of all groups with the fixed-point property for cones, and its stability under basic group constructions. The case of metabelian groups without this property alerts us to the fact that group extensions will not, in general, preserve the fixed-point property for cones. In fact, we do not even know whether the Cartesian product (of two groups) preserves it. The same question is open for supramenable groups and was first asked in the 1970s (Reference Ros72, p. 49, Reference Ros74, p. 51).
The closure under quotients is obvious, but for the other hereditary properties it will be helpful to be armed with the various characterisations of Theorem 7. We find our proof of point 4 laborious; perhaps the reader will devise a simpler argument.
1.F. Two conical fixed-point theorems
We now return to the first theme (I), namely fixed-point theorems without assumptions on the group Our first fixed-point statement remains in the generality of weakly complete cones . as above but under the assumption that the is equicontinuous on -representation In that case, the local boundedness condition is automatically satisfied (if . ).
In particular, the following contains the Ryll-Nardzewski theorem for isometries of Banach spaces as the special case of a cone .
The reader will have noticed that one inconspicuous assumption on did sneak into Theorem 9, namely that be countable. In the classical compact case, this restriction can immediately be lifted by a compactness argument. For the above statement, however, it is needed even in the most familiar case of Hilbert spaces:
We can nonetheless remove this last hypothesis on if we change the topological assumption on the cone. In fact, the countability of is only used in Theorem 9 to establish that must be weakly locally compact—this is a consequence of equicontinuity which is in strong contrast to the general cones that we had to deal with for the group-theoretical property.
Once we consider cones that are locally compact in the weak topology, we can dispense of any restriction on and deal with the distal case. In particular, the following subsumes the general case of the Ryll-Nardzewski theorem as the particular instance of a cone .
Notice that now the coboundedness assumption has become redundant for this statement, but the local boundedness assumption came back.
For Theorem 9, the coboundedness assumption was needed even in the case of isometric actions on Banach spaces: this is illustrated by the example of given in Remark 3.
2. Proof of the conical fixed-point theorems
We begin with the proof of Theorem 12 and retain its notation. One input for the proof will be established later (Proposition 21) because we shall need it again in a more general setting.
To get started, here is a consequence of the distality of the on -action .
Since is weakly locally compact and proper, it admits a weakly compact base see Exercise 21 in ;Reference Bou81, II, §7 (this result is due to Dieudonné and Klee Reference Kle55, 2.4). We emphasise that this base need not be invariant. This means that we are not in the classical setting, but rather in a “projective” context; the distance between these two settings can be appreciated e.g. in light of the Tychonoff property recalled in Section 10.C. Nevertheless, we will be able to use suitably adapted extensions of some of the known arguments in the proofs of the Ryll-Nardzewski theorem.
Applying a compactness argument to the intersections of with subcones of fixed points in we see that it suffices to find a non-zero fixed point for each countable subgroup of , we can thus assume that ; is countable.
We claim that we can furthermore suppose that is minimal amongst all those invariant closed convex subcones of that contain a non-zero bounded orbit. This is more delicate since the set of bounded orbits is a priori not closed, but we establish this fact in the more general weakly complete setting in Proposition 21 below. We point out that here is trivially of cobounded type as assumed in Proposition 21 because it admits a base. Notice that in particular we have reduced to the case where and hence also , is separable. ,
We now proceed to show that contains only one point whose orbit in is bounded. Since the corresponding ray will then be by uniqueness, it will in fact be pointwise fixed by distality. Therefore, this will then finish the proof. -invariant
Suppose for a contradiction that contains distinct points with bounded orbits and denote their midpoint by In particular the orbit of . is also bounded. Let be the linear form corresponding to the compact base see ;Reference Bou81, II, §7, No. 3. By boundedness of the chosen orbits, there is such that , and , are bounded by for all We apply Lemma .13 to both pairs and with this Since . is locally convex, the lemma implies that there exists a continuous seminorm on and such that
We consider the convex weakly compact set corresponding to under translation by some Applying to . a denting theorem such as the lemma on p. 443 in Reference NA67, we find a continuous linear form and a constant such that the set is non-empty but of -diameter Denote by . the corresponding set which has the same , -diameter.
We choose a pre-image under the map and define by Notice that we have .
In particular, by minimality of in the class of closed invariant subcones containing bounded orbits, the bounded orbit cannot remain in the closed convex subcone of We can thus choose . with At least one of . or must also be negative; we assume We write . and recalling that , does not vanish on Now . belongs to and likewise for This implies . which is a contradiction since our choice guarantees , This completes the proof of Theorem .12.■
We now turn to Theorem 9. Consider an enumeration of and let be for -dominating We endow . with the weak-* topology induced by By our assumption on the action, the set . is equicontinuous (on we use throughout that equicontinuity implies in fact uniform equicontinuity for families of linear maps (Proposition 5 in );Reference Bou74, X, §2, No. 2). Therefore, this set is relatively compact in see Corollary 2 in ;Reference Bou81, III, §3, No. 4). The same holds for the convex balanced hull of since the convex balanced hull of an equicontinuous set is equicontinuous. Note that we argue by equicontinuity since, in general, the closed convex hull of a compact set need not be compact (in our case it would be true if , was supposed barrelled, but not in general). It follows that we can define an element by since the sequence of partial sums is weak-* Cauchy in this convex balanced hull.
Since is -dominating, is an order-unit for i.e. for any , there is with Thus, . is strictly positive on we define ; to be the intersection of with the affine hyperplane .
We claim that is weakly compact. Since is an order-unit, every is bounded on Thus . is weakly bounded. It is also weakly complete since is so, and therefore weak compactness follows; see e.g. Reference Cho69, 23.11.
At this point has been shown to have a weakly compact base so that it is locally compact in the weak topology. Since the representation is equicontinuous, all orbits are bounded and the action is distal. In other words, we can apply Theorem ,12 to finish the proof of Theorem 9.■
3. Integrals and cones
We begin the proof of Theorem 7 by showing that the fixed-point property for cones is, as expected, sufficiently general to be applied to cones of integrals in the sense of Definition 5. In other words, we establish 14:
Let be a non-zero bounded function on Consider the algebraic dual . endowed with the weak-* topology defined by The convex cone . of positive linear forms is proper since is spanned by positive elements. Moreover, is weakly complete (i.e. complete) in this topology (compare Reference Cho62 or recall that algebraic duals are weak-* complete Reference Bou81, II, §6, No. 7).
To justify that the is of locally bounded type, consider an element -action with The evaluation at . is a non-zero element and we claim that its is bounded in -orbit Indeed, a neighbourhood basis at . in is given by sets with and we have, as required, ; for all large enough scalars namely , .
For the cobounded type condition, observe that the topological dual of is itself Reference Rud91, 3.14, endowed with the finest locally convex topology (Reference Sch99, p. 56, Reference Bou81, II, §6, No. 1, Rem. 1)). Therefore the condition holds since by definition is by -dominated .
At this point we apply the fixed-point property and obtain a positive element in Upon renormalising, it remains only to show that . This follows from the fact that . is and -dominating is positive.■
We now prove the converse implication. Let be a weakly complete proper convex cone in a (locally convex, Hausdorff) topological vector space Suppose that we are given a . of locally bounded cobounded type. We can therefore choose a non-zero point -representation with bounded and a -orbit element -dominating To every . we associate a function on by This defines a linear positive map . from to the space of functions on with the pointwise order. Notice that this map is for the usual -equivariant by pre-composition; therefore its image is -actions by -dominated Moreover, since the orbit . is bounded, is a bounded function. In conclusion, we have obtained a positive linear map from -equivariant to .
Now let be an integral on normalised for Composing the above map . with we obtain a , positive linear map -invariant on with A priori . is an element of the algebraic dual of Suppose for a contradiction that . is not in under the canonical embedding By weak completeness, . is closed in for the weak-* topology induced by Therefore, the Hahn–Banach theorem implies that there is a continuous linear form . on with and for some Since . is a cone, we can suppose Moreover, by .Reference Rud91, 3.14, we have with Now . implies that must be positive since is positive and this contradiction shows that ; indeed contains a non-zero fixed point, namely .■
4. Translate properties
As we recalled in the introduction, the translate property for a bounded non-zero function on is equivalent to the existence of an invariant integral on However, the more important question would be whether there is an invariant integral on the larger space . For instance, when . is the characteristic function of a subset this stronger property is the one that is equivalent to the existence of an invariant finitely additive measure on , normalised for .
Unfortunately, to go from the translate property to this stronger one, it seems that it is necessary to know a priori that is amenable; see Problem on p. 18 in Reference Gre69.
All this is exposed in Reference Gre69, §1 and in Reference Ros74, but perhaps a small measure of confusion has persisted. (For instance, the amenability assumption is not used in Proposition 1.3.1 of Reference Gre69, though it is crucial in Theorem 1.3.2; in contrast, the amenability assumption is missing in Theorem 6.3.1 of Reference Sha04 for (2) finally, Corollary 1.2 of (3);Reference Ros74 is not a consequence, but rather a prerequisite, of the result preceding it.) We shall use notably a result of Moore Reference Moo13 to clarify the situation in Theorem 18 below.
We begin with a generalisation of the translate property.
Let us verify that the translate property is indeed a special case of the abstract translate property with We just need to check that for any non-zero bounded function . on there is a positive linear form , on with and which is bounded on the orbit Since . we can choose , with Then the evaluation at . is a linear form with the desired properties.
The same verification shows also that the abstract translate property contains the case of the translate properties for considered by Rosenblatt -setsReference Ros72 in the context of von Neumann’s questions on finitely additive measures.
The following consequence of Theorem 18 completes the proof of Theorem 7, except for point 5 which we defer to Section 9.
The proof of Theorem 18 given above uses only the fact that satisfies the translate property for characteristic functions of subsets of Therefore, we deduce the following variant of Corollary .19.
5. Smaller cones
In preparation for the proof of Theorem 8, we begin with two reduction statements.
The reason why the minimality statement is not completely obvious is that the union of bounded orbits is generally not closed.
A different notion of smallness is provided by the following result.
We cannot prove this simply by replacing with the closed convex cone spanned by a bounded because weakly separable spaces or cones are usually far from weakly first-countable -orbit,( provides examples). It also seems that one cannot conduct a limit argument by exhausting by subspaces of countable dimension. The reason is that subspaces of countable dimension cannot, in general, be full in the sense of ordered vector spaces—a fortiori not ideals in the Riesz space This deprives us from the likes of the compactness argument invoked in Proposition .21.
6. First stability properties
We begin to address the stability properties of the class of groups with the fixed-point property for cones; closure under quotients is obvious. Thanks to Theorem 7, the closure under subgroups and directed unions becomes easy. Indeed, for subgroups it suffices to show the following.
As for directed unions, we only need the following.
As we have already mentioned, group extensions do not preserve the fixed-point property for cones. However, extensions by and of finite groups are covered by the next two lemmas.
7. Central extensions
In order to prove that any central extension of a group with the fixed-point property for cones retains this property, we can restrict ourselves to the case of countable groups thanks to points 1 and 2 of Theorem 8. Indeed, a subgroup of a central extension is a central extension of a subgroup. Our strategy is to prove that if a group has a representation of locally bounded cobounded type on a weakly complete convex proper cone which is minimal in the sense of Proposition 21, then the centre of acts trivially on This then establishes the stability under central extensions. .
However, it will be crucial to know that we can in addition assume the weak topology of to have a countable neighbourhood basis, because this allows us to argue with extreme rays. Recall that even very familiar cones in separable Hilbert spaces need not have extreme rays (and their weak topology is not first-countable). This is the case of the self-dual cone in considered in Example 11.
In order to ensure the countable neighbourhood basis, we apply Proposition 21 to the cone provided by Proposition 22. We now choose an arbitrary element of the centre of and proceed to prove that fixes pointwise.
The set is a convex subcone of -invariant It is moreover closed. Indeed, if any net . with converges, then so does the net upon passing to a subnet because the sum map is proper; see Corollary 1 in Reference Bou81, II, §6, No. 8 together with Theorem 1(c) in Reference Bou71, I, §10, No. 2. This implies that the limit of belongs to Furthermore, . contains a non-zero bounded orbit because the map is Therefore, by minimality, -equivariant. .
It follows that every extreme ray of is set-wise preserved by As was already mentioned, there need not exist any extreme ray in general; however, the fact that . has a countable neighbourhood basis implies that is indeed the closed convex hull of its extreme rays; see Proposition 5 in Reference Bou81, II, §7, No. 2.
Thus, for on an extreme ray, there is a positive scalar such that On the other hand, for any . we define closed convex subcones -invariant by and A priori we cannot play the minimality of . against because we do not know whether these cones contain non-zero bounded But -orbits. is still a convex subcone and it is closed by Corollary 2 in -invariantReference Bou81, II, §6, No. 8. Since every extreme ray lies in at least one of or (according to whether or we conclude that ), is all of .
We now claim that for each one of the two cones , contains a non-zero bounded Indeed let -orbit. be a point of with bounded Then there exists -orbit. with Using again the properness of the map . we conclude readily that both , have bounded whence the claim since at least one of -orbits, is non-zero.
We can now invoke minimality again and deduce that for each one of the two cones , coincides with Upon possibly replacing . by we can assume that for , we have For all . we must have , since otherwise which we claim is impossible. Indeed, there is a non-zero point , with bounded hence in particular bounded -orbit, But -orbit. allows us to apply the properness of to and deduce that is bounded, contradicting .
In conclusion, and which shows that , acts trivially on was to be proved. —as■
8. Subexponential groups
Recall that a group is called subexponential if for every finite subset where , is the set of products of elements of -fold This holds in particular for all locally finite or abelian groups. .
The following property was first recorded by Jenkins, who proved moreover that it characterises subexponentiality Reference Jen76, Lemma 1.
Lemma 30 can of course also be reformulated with the inequality it suffices to switch left and right multiplication by replacing ; with and with In fact, the function constructed in the above proof satisfies both versions of this inequality simultaneously. .
The following proposition will complete the proof of the last remaining item in Theorem 8.
9. Reiter and ratio properties
This section regards approximation properties characterising the existence of integrals on and on It is a rather direct extension of the results known in the special case . of characteristic functions of subsets.
The following definition is a generalisation to functions of the ratio property introduced in Reference Ros72, §5 and Reference Ros73, §2 for subsets.
Next, still extending Reference Ros72, Reference Ros73, we consider a generalisation of the group property introduced by Dieudonné Reference Die60, p. 284 and now called the Reiter property.
Just as the Reiter property for groups characterises amenability Reference Rei68, Ch. 8, §6, we have:
In particular, this establishes the remaining equivalence 5 of Theorem 7.
It is apparent on the definitions that the property implies the -Reiter property; this is consistent with the following since the existence of integrals implies the translate property. -ratio
In conclusion, Theorem 7 shows that the fixed-point property for cones can be characterised both by the property and by the -ratio property, letting -Reiter range over all non-zero positive bounded functions on .
10. Further comments and questions
10.A. Topological groups
The definition of the fixed-point property for cones can be extended to topological groups simply by adding the requirement that the representation be (orbitally) continuous. For applications to integrals, the corresponding change is to consider those bounded functions that are right uniformly continuous. This topological generalisation can be pursued in two directions:
On the one hand, the case of locally compact groups. Here, most arguments can be adapted with suitable technical precautions. For instance, Jenkins’ criterion for subexponential groups (Lemma 1 in Reference Jen76) was stated and proved in this generality.
On the other hand, for more general topological groups, the corresponding equivalent characterisations are less flexible and we cannot prove the stability properties of Theorem 8 anymore. It turns out that such statements do indeed fail; this does not, of course, make the fixed-point property for cones any less intriguing for these groups.
The above example is analogous to the case of amenability of topological groups that are not locally compact; see Reference dlH82, p. 489.
10.B. Relativising
Given a subgroup of a group we say that , has the fixed-point property for cones conditionally with respect to if the following holds: for every of cobounded type on any weakly complete proper convex cone that admits a non-zero -representation point with bounded -fixed there is a non-zero -orbit, point. Thus, the condition of locally bounded type has been conditioned on -fixed .
It is straightforward to check that for normal subgroups this conditional property is equivalent to the fixed-point property for cones for the quotient group , In that sense, the conditional property is the analogue of the co-amenability studied notably by Eymard .Reference Eym72.
These conditional properties offer a non-trivial generalisation of the corresponding absolute fixed-point properties when we consider non-normal subgroups For instance, in the case of amenability, it was shown that co-amenability does not pass to subgroups in the natural sense ( .Reference MP03, Reference Pes03). In the present case, the situation is different:
This proposition is a consequence of the following one thanks to Remark 24.
The latter proposition is proved by the arguments of Section 3 without any change.
Contrariwise, the conditional property for cones lacks a feature of co-amenability, namely it is not transitive. This is due to the fact that the fixed-point property for cones is not preserved by group extensions.
The conditional property can be further extended to group actions, generalising the on -action In that setting, we simply replace the locally bounded type assumption by the following condition for a . on a set -action we assume that there is a non-zero : from -map to the cone whose image is bounded. The corresponding fixed-point property can again be characterised in terms of integrals on by the arguments of Section 3.
Finally, we point out that there is a dual way to relativise fixed-point properties to subgroups, which in the case at hand leads to the following. A subgroup has the fixed-point property for cones relatively to if for every of locally bounded type and of -representation type on any weakly complete proper convex cone, there is an -cobounded point. -fixed
In the classical case of amenability, this corresponds to relative amenability as studied in Reference CM14. This notion turns out to occur naturally in the study of Zimmer-amenability of non-singular actions on measure spaces, of group C*-algebras and boundaries Reference Oza14, as well as of invariant random subgroups Reference BDL16.
Just as in the case of relative amenability, this second relativisation may seem empty at first sight:
Indeed the property for is formally stronger and the converse follows from Remark 24.
However, in the topological context, the relative property is strictly weaker since it follows from the the fixed-point property for cones of the larger group thus, Example ;38 above is a case where these properties are not equivalent.
10.C. Review of some other properties
Conical fixed point properties have been considered earlier, but mostly, as far as we know, for the special case of cones with a compact base.
In that setting, Furstenberg Reference Fur65, §4 introduced the notion of Tychonoff groups by requiring that a ray be preserved (not necessarily pointwise fixed). Since the cone has a compact base this is equivalent to fixing a point in the projective action on the compact convex set , This property was further studied by Conze–Guivarc’h .Reference GC74 and Grigorchuk Reference Gri98. Albeit very interesting, the Tychonoff property is not relevant to the questions we investigated here (in particular the von Neumann–Tarski invariant measure problem). Indeed, already the infinite dihedral group—which is of polynomial growth—fails to be Tychonoff. This is witnessed e.g. by its representation as a matrix group generated by and acting on the positive quadrant of Conversely, but in the locally compact generality, connected groups of upper-triangular matrices are Tychonoff .Reference Fur65, p. 284, but fail to be supramenable unless they have polynomial growth Reference Jen73.
Still for cones with a compact base, Grigorchuk Reference Gri98, §5 considered the non-zero fixed-point property under the assumption that (all) orbits be bounded—this property follows from the fixed-point property introduced in the present article. He proved that Liouville groups enjoy that property for cones with compact bases, if we define “Liouville” by the triviality of the Poisson boundary of some finitely supported generating symmetric random walk. Liouville groups include all finitely generated subexponential groups by a result of Avez Reference Ave74, and we can reduce to the finitely generated case by a compactness argument. But Liouville groups also include some groups with free non-abelian subsemigroups, for instance the lamplighter (see e.g. Theorem 1.3 in Reference Kaĭ83); such groups have paradoxical subsets.
Jenkins has introduced in Reference Jen80 a non-zero fixed-point property for certain compact convex sets that are allowed to be more complicated than initial slices of cones with a compact base. He proved that his fixed-point property implies the existence of integrals (Proposition 5 in Reference Jen80). However, it does not hold for all subexponential groups (Remark on p. 370 in Reference Jen80) and therefore is too restrictive for our purposes.
Finally, another property introduced earlier by Jenkins Reference Jen76 is his “property F” (not to be confused with a different “property F” that he introduced earlier yet Reference Jen74). We recall his definition, which is related to our abstract translate property:
Suppose that has a linear representation on a vector space and that we are given and (the algebraic dual) such that the function is non-zero, bounded, and on Then there exists a net . of finite positive linear combinations of of -translates such that, for any given the net , converges to .
Jenkins proved Reference Jen76, Thm. 2 that subexponential groups enjoy this property F. We claim that in fact this property is equivalent to our fixed-point property for cones. It is rather straightforward to see that property F implies the abstract translate property. Now the latter implies the fixed-point property for cones by Theorem 7 above. It remains to see that the fixed-point property for cones implies property F. To this end, consider in the above notation, and endow its algebraic dual with the weak-* topology. Consider the cone of linear forms that are on the whole orbit Then an argument entirely similar to that of Section .3 implies the existence of a net as required by property F.
This equivalence also clarifies the picture with respect to another result of Jenkins: he proved Reference Jen76, Cor. 3 that subexponential groups, in addition to having property F, also admit integrals. It follows from the above equivalence (and again Theorem 7) that the existence of integrals is after all equivalent to property F.
10.D. Questions and problems
As we recalled in the introduction, amenability is equivalent to the fixed-point property for general convex compact sets in arbitrary (Hausdorff) locally convex spaces. However, when the group is countable, it suffices to consider Hilbert spaces (in their weak topology) instead of general locally convex spaces; this was recently proved in Reference GM17.
The corresponding representations would of course not be isometric in view of Theorem 9 (for countable groups). Also, the second question is not suited for general (uncountable) groups: for instance, a group with Bergman’s property Reference Ber06 would necessarily act equicontinuously (compare Reference GM17) and hence would have a non-zero fixed point by Theorem 12, although it might not even be amenable; an example is viewed as a group without topology.
It would be desirable to have a wider menagerie of examples of groups distinguishing the properties that we consider. First in line is the following:
In view of Theorem 8, such an example would in particular be provided by solving another problem:
We recall that the corresponding problem for supramenable groups is also open; in view of Reference KMR13, 3.4, the latter problem is equivalent to:
Comparing the last two problems, we record that we do not know if the implication from the fixed-point property for cones to supramenability can be reversed:
The classical Reiter property has equivalent variants (already introduced in Reference Die60, p. 284). Likewise, the group satisfies the property for all non-zero -ratio if and only if the same holds with the replaced by any -norm with -norm This is a straightforward verification where . , and , are to be replaced with appropriate powers. The same is true for the property using basic -Reiter inequalities.
The particular case of suggests the following:
We say that a unitary representation of a group on a Hilbert space has the operator property for a continuous operator -ratio of if for every finite set and every there is , with for all Likewise, the representation has the stronger operator . property if we can choose -Reiter with for all .
Returning closer to supramenability:
Acknowledgments
The author is grateful to Taka Ozawa and to the anonymous referee for helpul remarks on preliminary versions of this article.