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[EQUATION] since [EQUATION] and [MATH] Note that the proof above also shows that if [MATH] is the set of cancellations between [MATH] , then for any cancellation [MATH] of [MATH] we have |
[EQUATION] Now we have a tool to bound the length of a cancellation between two words in [MATH] from above. The next lemma bounds the length of a word in [MATH] from below. We will in the following call [MATH] |
positive if [MATH] consists only of positive powers of [MATH] . Similarly, we say [MATH] is negative if [MATH] consists only of negative powers of [MATH] . Note that if [MATH] is either positive or negative then we clearly have [MATH] |
Lemma 8.9 Let [MATH] and [MATH] . Then [EQUATION] Proof. It is enough to consider [MATH] Moreover, it is clear that if [MATH] , then |
[EQUATION] since there is no cancellation due to the fact that [MATH] is either positive or negative. For the remaining cases, we will begin with some observations. Assume [MATH] satisfy that [MATH] is reduced, [MATH] is positive and [MATH] is negative. Moreover, let [MATH] and [MATH] be such that [MATH] . Then we have... |
Now let [MATH] be neither positive or negative and let [MATH] . Fix [MATH] and [MATH] such that [MATH] and [MATH] , for all [MATH] |
Assume first that for all [MATH] we have [MATH] is neither positive nor negative. Then for each [MATH] , we may fix [MATH] together with positive [MATH] , and negative [MATH] , such that for each [MATH] there is a cyclically reduced cyclic conjugate of [MATH] and of [MATH] of the form |
[EQUATION] and [EQUATION] respectively. Then [MATH] and hence, by the observations above, we have [EQUATION] Next, assume that [MATH] is least such that [MATH] is either positive or negative. Then, as before, we may for each [MATH] chose [MATH] together with positive [MATH] and negative [MATH] such that for each [MATH]... |
[EQUATION] and [EQUATION] respectively. Then [MATH] and hence, by the observations above, we have [EQUATION] Finally, to finish the proof, note that for any [MATH] , we can chose [MATH] in the argument above. Thus, as |
[EQUATION] we obtain [EQUATION] as wanted. Note that [EQUATION] since [MATH] . This will be used all the time below to conclude that there is no bad cancellation in the various cases. |
We will now begin to argue that there is no bad cancellation between two words from [MATH] . So let [MATH] . The following decomposition will be useful. For [MATH] let |
[EQUATION] [EQUATION] and for [MATH] let [EQUATION] Then for [MATH] we have, up to cyclic permutation, that [EQUATION] and [EQUATION] |
is a reduced word whenever each factor is reduced, for all [MATH] . However, it is not necessarily cyclically reduced. If for some [MATH] we have [MATH] or [MATH] or [MATH] , then |
[EQUATION] is cyclically reduced. If [MATH] and [MATH] , then any possible reduction in the induced cycle of [MATH] is contained in [MATH] |
Claim 1: If [MATH] and [MATH] , then there is no bad cancellation between [MATH] and [MATH] Proof. It is easy to check that [MATH] satisfies the assumption of Lemma 8.8 , since [MATH] . Thus any cancellation, [MATH] , between [MATH] and [MATH] satisfies |
[EQUATION] So, by Lemma 8.9 , there cannot be any bad cancellation between [MATH] and [MATH] In the following, we let [EQUATION] |
Note that [MATH] Claim 2: If [MATH] with [MATH] and [MATH] , then there is no bad cancellation between [MATH] and [MATH] Proof. Note that either [MATH] or [MATH] . Assume without loss of generality that we are in the first case. Then the only powers of [MATH] occurring in [MATH] are [MATH] and [MATH] . Thus any cancell... |
[EQUATION] Moreover, by Lemma 8.9 , it holds that [EQUATION] since [MATH] . Thus there is no bad cancellation between [MATH] and [MATH] |
Now we will take care of the case where [MATH] are different, but none of them extends the other. Claim 3: If [MATH] with [MATH] [MATH] |
[EQUATION] then there is no bad cancellation between [MATH] and [MATH] Proof. First note that [MATH] will only contain [MATH] as powers of [MATH] , while [MATH] will only contain [MATH] as powers of [MATH] |
We now claim that for each [MATH] and [MATH] we have [MATH] contains the string [MATH] or [MATH] , for some [MATH] . Indeed [MATH] is of one of the forms [MATH] [MATH] [MATH] or [MATH] , for some [MATH] and [MATH] . By straightforward calculation the statement is clearly true for [MATH] or [MATH] . To see that the stat... |
Similarly, for each [MATH] and [MATH] , we have [MATH] contains the string [MATH] or [MATH] for some [MATH] Now let [MATH] satisfy that [MATH] and [MATH] . Then from the above it follows that any cancellation, [MATH] , between [MATH] and [MATH] is contained in either |
[EQUATION] [EQUATION] or in one of their inverses, for some [MATH] Therefore, by Lemma 8.8 , we have [EQUATION] So, by Lemma 8.9 , we have [MATH] |
Similarly any cancellation, [MATH] , between [MATH] and [MATH] is contained in either [EQUATION] [EQUATION] or in one of their inverses, for some [MATH] . So, by Lemma 8.8 and Lemma 8.9 , we also have [MATH] . Thus there cannot be any bad cancellation between [MATH] and [MATH] |
From Claim 1, Claim 2 and Claim 3 we may conclude that there is no bad cancellation between two words in [MATH] , i.e., that [MATH] satisfies the [MATH] cancellation property. |
We will now prove that there is no bad cancellation between a word from [MATH] and a word from [MATH] . To do so, let [MATH] [MATH] and |
[EQUATION] Then [MATH] . We will again consider fixed [MATH] . Let [MATH] be fixed such that [MATH] and [MATH] Claim 4: If [MATH] with [MATH] and [MATH] , then there is no bad cancellation between [MATH] and [MATH] |
Proof. This follows by the same arguments as the ones used in the beginning of the proof of Claim 2. Claim 5: If [MATH] [MATH] or [MATH] [MATH] . Then there is no bad cancellation between [MATH] and [MATH] |
Proof. This follows by arguments similar to those in the beginning of the proof of Claim 3. From Claim 4, we may assume that both [MATH] . Moreover, by Claim 5, there is no bad cancellation between [MATH] and [MATH] in the case [MATH] and [MATH] or in the case [MATH] and [MATH] . In the next three claims, we prove that... |
Claim 6: If [MATH] , then there is no bad cancellation between [MATH] and [MATH] Proof. Consider [MATH] for [MATH] and [MATH] . Either all the powers of [MATH] are positive or all the powers of [MATH] are negative. Below we have put these observations into a table. Here [MATH] and [MATH] refer to the sign of the occurr... |
[MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] It is easily seen that if the signs of the powers of [MATH] do not match, then there is no bad cancellation between [MATH] and [MATH] for the corresponding [MATH] . So assume that the sign of the powers of [MATH] in [MATH]... |
[EQUATION] [EQUATION] or in one of their inverses, for some [MATH] . Similarly, [MATH] is also contained in either [EQUATION] [EQUATION] |
or in one of their inverses for some [MATH] Thus, by 8.8 , we have [EQUATION] and hence, by 8.9 , there is no bad cancellation between [MATH] and [MATH] |
Claim 7: If [MATH] , then there is no bad cancellation between [MATH] and [MATH] Proof. First, consider the sign of the powers of [MATH] occurring in [MATH] and [MATH] for [MATH] |
[MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] It is easily seen that if the sign in two cells does not match, then there is no bad cancellation between [MATH] and [MATH] , for the corresponding [MATH] In case the signs of the powers of [MATH] are the same, one may use... |
Claim 8: If [MATH] , then there is no bad cancellation between [MATH] and [MATH] Proof. First, by considering the form of [MATH] and [MATH] for the words [MATH] , one finds that for all [MATH] , we have [MATH] and [MATH] will contain at most one occurrence of one of the strings |
[EQUATION] for some [MATH] . Moreover, for all [MATH] and [MATH] , we have that [MATH] and [MATH] contain at least one of the strings |
[EQUATION] for some [MATH] This is again straightforward to check, by considering the form of [MATH] and [MATH] Note that the above implies that [MATH] contains at most one of the strings |
[EQUATION] for some [MATH] . Moreover, for all [MATH] , we have that [MATH] contains at least one of the strings [EQUATION] for some [MATH] |
Conversely, the above also implies that [MATH] contains at most one of the strings [EQUATION] for some [MATH] . Moreover, for all [MATH] , we have that [MATH] contains at least one of the strings |
[EQUATION] for some [MATH] Thus, by making considerations and use of 8.8 and 8.9 , as in the proof of the earlier claims, we may conclude that there cannot be any bad cancellation between [MATH] and [MATH] |
Claim 9: If [MATH] and [MATH] , then there is no bad cancellation between [MATH] and [MATH] Proof. From earlier results it is enough to consider the case where [MATH] or [MATH] . Assume without loss of generality that we are in the first case. Then there exist [MATH] such that [MATH] and [MATH] , for some [MATH] . Then... |
[EQUATION] [EQUATION] or in one of their inverses. Therefore, by 8.8 we obtain [EQUATION] and hence, by 8.9 , we have [MATH] If [MATH] , then by 8.9 and the fact that [MATH] , we also have |
[EQUATION] So assume [MATH] . Then we have that each of [MATH] and [MATH] occurs at most once in [MATH] . But by the arguments in Claim 3, we have that for all [MATH] there is [MATH] such that [MATH] or [MATH] occur in [MATH] . Thus [MATH] or [MATH] occur in [MATH] for all [MATH] |
Therefore [MATH] is contained in either [EQUATION] [EQUATION] or in one of their inverses, for some [MATH] . Thus we obtain [EQUATION] |
So, by 8.9 , we have [MATH] , as well. We may therefore conclude that there is no bad cancellation between [MATH] and [MATH] Putting all the claims together we finally have a proof of the following theorem. |
Theorem 8.10 Let [MATH] for some [MATH] . Then there is a transversal [MATH] for the left cosets in [MATH] such that the set [EQUATION] |
satisfies the [MATH] cancellation property. Using this, and the criterion in 8.1 , it is now easy to construct continuum many non-atomic invariant random subgroups of [MATH] , which are [MATH] -invariant and weakly mixing with respect to the action of [MATH] |
Fix by 8.10 a transversal [MATH] for the left cosets in [MATH] such that the set [MATH] satisfies the [MATH] cancellation property. Moreover, fix an enumeration [MATH] such that [MATH] is the identity. Then we have |
[EQUATION] since it follows by 8.9 that [MATH] for all [MATH] . Thus, by Remark 8.2 , condition [MATH] in 8.1 is satisfied, and hence the proof of [MATH] in 8.1 provides a family as the one described above. |
There is also another consequence of 8.10 Let [MATH] denote the set of conjugacy classes of [MATH] and let [MATH] be the outer automorphism group of [MATH] . Now consider the natural action [MATH] . It is well known that there is a conjugacy class [MATH] such that [MATH] (see LS77 , page 45] ). Therefore for such [MATH... |
[EQUATION] With 8.10 we obtain the following strengthening of this result. Corollary 8.11 There exists a conjugacy class [MATH] such that |
[EQUATION] that is, [MATH] is disjoint from the (normal) subgroup [MATH] for [MATH] Proof. Let [MATH] be the conjugacy class of [MATH] for some [MATH] and assume towards contradiction that |
[EQUATION] Now choose by 8.10 a transversal [MATH] for the left cosets in [MATH] such that [MATH] satisfies the [MATH] cancellation property. Fix [MATH] such that [MATH] . Then we have |
[EQUATION] which contradicts that [MATH] for all [MATH] We do not know if the analogs of 8.10 and 8.11 hold for any [MATH] with [MATH] |
Department of Mathematics California Institute of Technology Pasadena, CA 91125 kechris@caltech.edu Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen vibquo@math.ku.dk |
# Source: arxiv 1806.08677 # Title: Bounded G-theory with fibred control # Sections: all # Downloaded: 2026-03-03T05:20:09.790540+00:00 |
Bounded -theory with fibred control Abstract. We use filtered modules over a Noetherian ring and fibred bounded control on homomorphisms to construct a new kind of controlled algebra with applications in geometric topology. The resulting theory can be thought of as a “pushout” of bounded [MATH] -theory with fibred cont... |
1. Introduction The purpose of this paper is to use filtered modules over a Noetherian ring with a fibred bounded control on homomorphisms to construct a bounded [MATH] -theory with fibred control. This theory can be thought of as a “pushout” of the bounded [MATH] -theory with fibred control constructed by the authors ... |
and the controlled [MATH] -theory constructed in Here is a summary of this situation: Throughout this paper, metric spaces such as [MATH] and [MATH] that appear in the square will be proper metric spaces in the sense that every closed bounded subspace is compact. The space or spectrum in the upper left corner represent... |
, reviewed here in the beginning of section 2.1 . This theory is defined for any ring of coefficients [MATH] . It is built out of free [MATH] -modules with generating sets parametrized over the metric space [MATH] . This allows to impose geometric control conditions on the homomorphisms [MATH] . The bounded control con... |
The spectrum in the upper right corner [MATH] is a generalization of this theory to the situation when the modules are parametrized by the product of two metric spaces [MATH] and [MATH] , and the control imposed on the homomorphisms is relaxed: it is essentially the bounded control across [MATH] but the bound is allowe... |
To describe the bottom row in the square and for the rest of the paper, we restrict to Noetherian rings [MATH] In place of parametrizations used to control homomorphisms between free modules, one can use filtrations of arbitrary [MATH] -modules by subsets of the metric space [MATH] and impose control conditions in term... |
for a single space [MATH] . The result was the bounded [MATH] -theory spectrum [MATH] . The definition involved promoting the setting from the additive structure for free modules in the definition of bounded [MATH] -theory to a specific non-split Quillen exact structure on a category of filtered [MATH] modules with mor... |
Now it is clear what the “pushout” [MATH] is supposed to mean. We want to look at the [MATH] -theory of a category built out of filtered modules over the product [MATH] where the morphisms have the fibred control condition of the type described for fibred [MATH] -theory. This time we are interested in very specific exc... |
Acknowledgement. We would like to thank the referee for excellent comments that improved the narrative and the precision of the paper. |
2. Elements of bounded -theory Bounded [MATH] -theory defined in is a variant of bounded [MATH] -theory of Pedersen and Weibel made applicable to more general, non-split exact structures. It was designed by the authors for a different purpose than the one in this paper. The old focus was on the equivariant theory in ad... |
in the form best fit for the fibred theory. 2.1. Basic definitions We start with a brief recollection of the bounded [MATH] -theory setup. The coefficients [MATH] for this theory can be an arbitrary associative ring. The bounded category |
[MATH] is the additive category of geometric [MATH] -modules whose objects are functions [MATH] which are locally finite assignments of free finitely generated [MATH] -modules [MATH] to points [MATH] of [MATH] The local finiteness condition requires precisely that for any bounded subset [MATH] the restriction of [MATH]... |
[EQUATION] with the property that the components [MATH] are zero for [MATH] for some fixed real number [MATH] The associated [MATH] -theory spectrum is denoted by [MATH] |
and is called the bounded K-theory of [MATH] 2.1.1 Remark We would like to remind the reader that the original paper of Pedersen and Weibel |
was already written in greater generality. If [MATH] is any additive category, it can be used as coefficients in this construction in place of finitely generated free [MATH] -modules using the same formulas as above. The outcome is the bounded category [MATH] which is again an additive category with the evident notion ... |
Pedersen and Weibel were more interested in the product situation and, in fact, repaired the non-uniform boundedness properties of morphisms in [MATH] by filtering the morphism sets. With that fix the category becomes isomorphic to [MATH] . We, instead, embraced the flexibility of this construction in |
with the idea of exploiting the additional deformations in [MATH] , the K-theory with fibred control , that the construction allows. This is the spectrum that shows up in Figure as [MATH] |
2.1.2 Notation For a subset [MATH] and a real number [MATH] [MATH] will stand for the metric [MATH] -enlargement [MATH] In this notation, the metric ball of radius [MATH] centered at [MATH] is [MATH] or simply [MATH] |
A variation of the basic construction of bounded [MATH] -theory is based on the following observation. For every object [MATH] and a subset [MATH] there is a free [MATH] -module [MATH] . In this context we say an element [MATH] is supported on a subset [MATH] if [MATH] Now the restriction from arbitrary [MATH] -linear ... |
In the rest of this section and the rest of the paper, we will restrict to the case of a Noetherian ring [MATH] Let [MATH] denote the power set of [MATH] partially ordered by inclusion and viewed as a category. If [MATH] is a left [MATH] -module, let [MATH] denote the family of all [MATH] -submodules of [MATH] partiall... |
2.1.3 Definition An [MATH] -filtered [MATH] -module is a module [MATH] together with a functor [MATH] from the power set of [MATH] to the family of [MATH] -submodules of [MATH] , both ordered by inclusion, such that the value on [MATH] is [MATH] It will be most convenient to think of [MATH] as the functor above and use... |
reduced if [MATH] An [MATH] -homomorphism [MATH] of [MATH] -filtered modules is boundedly controlled if there is a fixed number [MATH] such that the image [MATH] is a submodule of [MATH] |
for all subsets [MATH] of [MATH] The objects of the category [MATH] are the reduced [MATH] -filtered [MATH] -modules, and the morphisms are the boundedly controlled homomorphisms. |
The category [MATH] we constructed is clearly an additive category, but the more interesting structure for developing its [MATH] -theory is a certain Quillen exact structure. For a good modern exposition of exact categories we refer to Keller |
; there is also a leisurely review of relevant basic theory in , section 2] Let us recall some standard terms. If a category has kernels and cokernels for all morphisms, it is called preabelian . If, in addition, the canonical map [MATH] for each morphism [MATH] is monic and epic but not necessarily invertible, we will... |
semi-abelian , cf. 16 , pages 167-168] and . It is abelian if it is also balanced in the sense that the canonical map is an isomorphism. Recall also that a category is called cocomplete if it contains colimits of arbitrary small diagrams, cf. Mac Lane 13 , chapter V] |
2.1.4 Remark If [MATH] is unbounded, [MATH] is not a balanced category and therefore not an abelian category. For an explicit description of a boundedly controlled morphism in |
[MATH] which is an isomorphism of left [MATH] -modules but whose inverse is not boundedly controlled, we refer to 14 , Example 1.5] |
It turns out that the kernels and cokernels in [MATH] can be characterized using an additional property a boundedly controlled morphism may or may not have. |
2.1.5 Definition A morphism [MATH] in [MATH] is called boundedly bicontrolled if there exists a number [MATH] such that in addition to inclusions of submodules |
[EQUATION] there are inclusions [EQUATION] for all subsets [MATH] In this case we will say that [MATH] has filtration degree [MATH] and write [MATH] |
2.1.6 Definition We define the admissible monomorphisms in [MATH] be the boundedly bicontrolled homomorphisms [MATH] such that the map [MATH] is a monomorphism. We define the admissible epimorphisms be the boundedly bicontrolled homomorphisms [MATH] such that [MATH] is an epimorphism. |
Let the class [MATH] of exact sequences consist of the sequences [EQUATION] where [MATH] is an admissible monomorphism, [MATH] is an admissible epimorphism, and [MATH] |
There are numerous examples of semi-abelian categories that are classical or have appeared recently in analysis and algebra that are listed in section 4 of |
or in . For us [MATH] is of major interest. 2.1.7 Theorem [MATH] is a cocomplete semi-abelian category. The class of exact sequences [MATH] gives an exact structure on [MATH] |
Proof. This fact is contained in Proposition 2.6 and Theorem 2.13 of We want to recall the explicit construction of kernels and cokernels [MATH] for future reference. For a boundedly controlled morphism |
[MATH] , the kernel of [MATH] in [MATH] has the [MATH] -filtration [MATH] where [MATH] This gives a kernel of [MATH] in [MATH] Similarly, let [MATH] be the [MATH] -filtration of the image of |
[MATH] in [MATH] by [MATH] . Let [MATH] be the cokernel of [MATH] in [MATH] , which is the quotient [MATH] Then there is a filtration [MATH] of [MATH] defined by [MATH] where the maps between quotients are induced from the structure maps of [MATH] The resulting epimorphism [MATH] |
gives a boundedly bicontrolled morphism of filtration [MATH] . Lemma 2.5 of verifies the universal properties of the cokernel [MATH] |
Now suppose [MATH] is an arbitrary cocomplete semi-abelian category. All of the definitions and proofs presented so far can be interpreted verbatim to describe a bounded category [MATH] if instead of [MATH] -modules one uses objects from [MATH] . We should point out that the exposition in |
is constructed in lesser generality with [MATH] assumed to be abelian. It is important for the fibred version to relax abelian to semi-abelian. However, |
can still serve as a good reference because throughout that paper only semi-abelian properties of [MATH] are used. In particular we have this conclusion. |
2.1.8 Theorem For any cocomplete semi-abelian category [MATH] [MATH] is a cocomplete semi-abelian category. Of course, the category of [MATH] -modules [MATH] is a cocomplete abelian category and so can serve as a basic example of cocomplete semi-abelian coefficients [MATH] . In this case [MATH] is precisely [MATH] |
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