Sizes of spaces of triangulations of 4-manifolds and balanced presentations of the trivial group.
Let be any compact four-dimensional PL-manifold with or without boundary (e.g. the four-dimensional sphere or ball). Consider the space of all simplicial isomorphism classes of triangulations of endowed with the metric defined as follows: the distance between a pair of triangulations is the minimal number of bistellar transformations required to transform one of the triangulations into the other. Our main result is the existence of an absolute constant such that for every and all sufficiently large there exist more than triangulations of with at most simplices such that pairwise distances between them are greater than ( times).
This result follows from a similar result for the space of all balanced presentations of the trivial group. (“Balanced” means that the number of generators equals to the number of relations). This space is endowed with the metric defined as the minimal number of Tietze transformations between finite presentations. We prove a similar exponential lower bound for the number of balanced presentations of length with four generators that are pairwise -far from each other. If one does not fix the number of generators, then we establish a super-exponential lower bound for the number of balanced presentations of length that are -far from each other.
1. Main Results
In this paper we prove results about balanced presentations of the trivial group (Theorem B, Theorem C), triangulations of compact PL -manifolds (Theorem A), Riemannian metrics subject to some restrictions on some compact smooth -manifolds (Theorem A.1, Theorem A.2, Theorem A.1.1), and contractible -complexes (Theorem C.1). All of these theorems imply that the spaces of corresponding structures are large: we find an exponential, or in some cases super-exponential number of presentations (correspondingly triangulations, metrics, -complexes) which are extremely pairwise distant in some natural metrics. Exponentially many here means as a function of the length of presentations, number of simplices, etc.
The geometric theorems follow from the “group” theorems B and C, which are a development of results from [Lisa], [Bri15]. These papers contain independent and different constructions of infinite sequences of balanced presentations of the trivial group that are very distant from trivial presentations. Here we combine the techniques from [Lisa] and [Bri15] as well as some ideas of Collins [Col78] to produce an exponential and super-exponential number of pairwise distant presentations. (The growth is exponential in length, when the number of generators is fixed, and super-exponential , if it is arbitrary.) From a group-theoretic perspective the main technical novelty of the present paper is that here we are forced to treat balanced presentations of the trivial group that are very distant from trivial presentations as group-like objects, introduce concepts of homomorphisms and isomorphisms between these objects, and learn to prove that they are not isomorphic (when this is the case).
The “A” theorems are similar to results in dimension greater than obtained in [Nab95],[Nab96a],[Nab96a],[Nab96b],[Wei05] using different group-theoretic techniques. In these dimensions it was possible to obtain an even stronger analogue of Theorem A.2 (see [NW00],[NW03],[Wei05],[Nab10b]), without the topological assumption on the manifold and for a wider class of Riemannian metrics. It’s not clear if it is possible to implement the techniques of this paper to prove such a generalization. We will give the precise statements now.
Let be a PL-manifold. Here a triangulation of is a simplicial complex such that its geometric realization is PL-homeomorphic to . We do not distinguish between simplicially isomorphic triangulations and regard them as identical. Thus, the set of all triangulations of is discrete, and for each its subset that includes all triangulations with simplices is finite. It is easy to see that the cardinality of is at most for some . It is a major unsolved problem (cf. [Frö92], [Gro10], [Nab06]) if this cardinality can be majorized by an exponential function . The set can be endowed with a natural metric defined as the minimal number of bistellar transformations (a.k.a. Pachner moves) required to transform one of the two triangulations to the other. Recall that a bistellar transformation is a local operation on triangulations preserving their PL-homeomorphism class. These operations can be described as follows: One chooses a subcomplex of the triangulation formed by adjacent -dimensional simplices, , that is simplicially isomorphic to a subcomplex of the boundary of the standard -dimensional simplex. Then one removes from the triangulation and replaces it be the complementary subcomplex attaching it along the boundary of . Pachner proved that each pair of PL-homeomorphic finite simplicial complexes can be transformed one into the other by means of a finite sequence of bistellar transformations, so this distance is always finite.
In [Nab96b] it was proven that for each and each computable function there exists such that for each closed -dimensional manifold and each sufficiently large there exist at least distinct triangulations of with simplices such that all pairwise distances in between these triangulations are greater than . This result was shown to be also true for some closed -dimensional manifolds, namely those that can be represented as a connected sum of any closed -manifold and a certain number of copies of , where according to [Sta07] one can take . However, it is desirable to know whether or not this result holds for all closed four-dimensional PL-manifolds including . One motivation is a connection with Hartle-Hawking model of Quantum Gravity as well as other related models of Quantum Gravity such as Euclidean Simplicial Gravity (cf. [HH83], [ADJ97], [Nab06]). Another motivation is a natural desire to know if is as “large” as for more complicated -manifolds or the lack of topology somehow makes the spaces of triangulations smaller. The main result of our paper is that a slightly weaker version of this result holds for all closed -dimensional manifolds as well as compact -dimensional manifolds with boundary. To state it define functions by formulae
There exists such that for each compact -dimensional manifold (with or without boundary) and each positive integer for all sufficiently large there exist more than triangulations of with less than simplices such that the distance between each pair of these triangulations is at least .
Note that this result follows from its particular case when . Indeed, if one has distant triangulations of , once can form the connected sums of all these triangulations with a fixed triangulation of . Also, note that the same construction (exactly as in higher dimensions) implies the following Riemannian analog of Theorem A:
There exist positive constants , , , , such that for each closed -dimensional manifold and each non-negative integer for each sufficiently large there exist more than Riemannian metrics on such that 1) each of those metrics has sectional curvature between and , injectivity radius greater than , volume greater than but less than , and diameter less than ; 2) For each positive there is no sequence of jumps of length in the Gromov-Hausdorff metric that connects a pair of these metrics within the space of Riemannian metrics on with sectional curvature between and , volume and diameter . (If , the upper bound for the length of jumps becomes and the upper bound for the diameter is the same as in the condition 1.)
If the Euler characteristic of is non-zero and its sectional curvature satisfies , then the Gauss-Bonnet theorem implies that . Therefore, Theorem A.1 implies the existence of exponentially many connected components of the sublevel set of the diameter functional on the space of isometry classes of Riemannian metrics on with . Moreover, each pair of these components can merge only in a connected component of a sublevel set of diameter only for a much larger value of . A natural idea will be to look for minima of diameter on connected components of its sublevel sets, because the minimum of any continuous functional on a connected component of its sublevel set will be automatically a local minimum of this functional. In order to ensure the existence of the minimum of diameter on all connected components of its sublevel set we are going first to somewhat enlarge the considered space: Denote the closure of the space of Riemannian structures (i.e. isometry classes of Riemannian metrics) on with in the Gromov-Hausdorff topology by . Elements of are Alexandrov spaces with curvature bounded from both sides. They are isometry classes of -smooth metrics on , and the sectional curvature can be defined a.e. Consider the diameter as a functional on . Its minima in connected components of sublevel sets are also local minima on the whole space. Therefore, Theorem A.1 implies that the diameter has “many” local minima on , and at least some of these minima must be very deep. The number of these minima is at least exponential in a positive power of , and behaves as the exponential of our upper bound for the diameter. Therefore, denoting our upper bound for the diameter by , and observing that , we see that Theorem A.1 implies the following theorem:
There exists a positive constant such that for each closed -dimensional Riemannian manifold with non-zero Euler characteristic and each positive integer number the diameter regarded as a functional on has infinitely many distinct local minima such that 1) the sequence is an unbounded increasing sequence; 2) the number of such that is greater than ; 3) Each path or a sequence of sufficiently short jumps in that starts at and ends at a point with a smaller value of the diameter must pass through a point where the value of the diameter is greater than .
For the existence part of these results first appeared in [Nab96a] (Theorems 9, 11). Later [NW00] (see also [NW96]) the same and even stronger results were proven without the assumption that the Euler characteristic of does not vanish. (More importantly, the techniques of [NW00] can be applied to other Riemannian functionals, for example, to diameter regarded as a functional on the space of Alexandrov structures on with curvature .) The depths of local minima grow not only faster than any finite tower of exponentials but faster than any computable function. Also, it was proven in [NW00] that the distribution function for “deep” local minima of on grows at least exponentially. Shmuel Weinberger observed that the distribution function for the “deep” local minima of diameter is, in fact, doubly exponential ([Wei05], Theorem 1 on p. 128). For the corresponding existence results were proven in [LN].
Note that from Theorem A.1 almost immediately follows the following result with a somewhat nicer statement:
There exist positive constants and such that for each and each closed -dimensional Riemannian manifold with non-zero Euler characteristic for all sufficiently large there exists at least Riemannian metrics on such that Riemannian manifolds have volume , injectivity radius greater than and diameter less than with the following property: Let , and be any diffeomorphism. If we consider it as a map between Riemannian manifolds and , then either or will be greater than .
To see that Theorem A.1 follows from Theorem A.1.1, we can use the same Riemannian metrics on for both theorems. A well-known fact which is a part of all proofs of the Gromov-Cheeger compactness theorem is that two sufficiently Gromov-Hausdorff close Riemannian manifolds satisfying the conditions of Theorem A.1 (or Theorem A.1.1) are diffeomorphic. Moreover, the proofs yield concrete upper bounds for the the Lipschitz constants. Assuming that the condition 2) of Theorem A.1 does not hold we can multiply the upper bounds for the Lipschitz constants for the diffeomorphisms corresponding to small jumps in the condition 2 and obtain ”controlled” upper bounds for the composite diffeomorphisms, which contradicts Theorem A.1.1.
Theorems A, A.1, A.1.1 follow (see the last section) from the following theorem about balanced presentations. The length of a finite presentation of a group is defined as the sum of the lengths of all relators plus the number of generators.
There exists a constant such that for each and all sufficiently large there exist more than balanced presentations of the trivial group of length with four generators and four relators such that for each pair of these presentations one requires more than Tietze transformation in order to transform one of these presentations into the other.
If one does not restrict the number of generators and relators, then there is a superexponentially growing number of pairwise distant balanced presentations of the trivial group:
There exists a constant such that for each and all sufficiently large there exist more than balanced presentations of the trivial group of length such that for each pair of these presentations one requires more than Tietze transformation in order to transform one of these presentations into the other. Moreover, these balanced presentations can be chosen so that the length of each relation is equal to or , and each generator appears in at most relations.
If we take one of the balanced presentations in Theorems B or C, and consider its presentation complex, i.e. the -complex with one -cell, -cells corresponding to the generators of the presentation, and -cells corresponding to its relations, then we are going to obtain a family of contractible -complexes. In the situation of Theorem B these complexes will have -cells and -cells attached along words of length ; in the situation of Theorem C the number of -cells or -cells will be , and each -cell will be either a digon or a triangle. In the situation of Theorem B the number of these -complexes is at least , in the situation of Theorem C . We can endow each of these complexes by a path metric obtained by considering each -cell as a circle of length , and each -cell as the -disc with the length of circumference equal to the length of the corresponding relator. Also, we can subdivide these complexes into simplicial complexes with simplices by subdividing each -cell into triangles in an obvious way. Now given a pair of such contractible -complexes we can ask for specific Lipschitz maps (=homotopy equivalences ) , and Lipschitz homotopies between and the identity map of , and between and the identity map of such that the maximum of Lipschitz constants of is minimal possible. Theorems B and C imply that for all sufficiently large the Lipschitz constant of either , or at least one of the two homotopies must be greater than . (On the other hand and can be chosen as constant maps, so they do not need to have large Lipschitz constants.) The proof of this fact will not be presented here and is similar to the proofs of Theorems A and A.1 given below.
Alternatively, we can take the contractible simplicial -complexes constructed by triangulating presentation complexes of presentations from Theorem C, consider all possible subdivisions of , of , simplicial maps , define and , and, finally, consider all homotopies between and that are simplicial maps defined on some simplicial subdivisions of with values in for . (Observe, that, in general, are not simplicial maps from to , . However, we can subdivide each simplex of into simplices , where runs over all simplices of in (and, if and , then we might need to further subdivide quadrilaterals into pairs of triangles). After subdividing in a similar way, we will be able to regard as simplicial maps of the constructed subdivisions of into . Now the relative simplicial approximation theorem would imply the existence of some subdivisions of and simplicial homotopies between and .)
Now we can define the complexity of the quadruple of maps as the total number of -simplices in the subdivisions of , used to define . Define the witness complexity of homotopy equivalence of as the minimum of the complexity over all such quadruples of simplicial maps. Theorem C easily implies that:
For some and each for all sufficiently large there exist more than contractible -dimensional simplicial complexes with at most -simplices such that for each pair , (), the witness complexity of homotopy equivalence of and is at least .
Theorem C can be straightforwardly obtained from Theorem B for just by rewriting the finite presentations in an appropriate way. Theorem C.1 will be proven in the last section, and easily follows from our proof of Theorem C. Theorem A will be deduced from a modified version of Theorem C explained in section 6. One applies this theorem for such that in Theorem A satisfies . We realize the balanced finite presentations of the trivial group from Theorem C as “apparent” finite presentations of PL -spheres triangulated with simplices. More precisely, we start from the connected sum of several copies of (one copy for each generator) , realize relators by embedded circles and kill them by surgeries. Then we demonstrate that the resulting manifold can be triangulated in simplices. The “balanced” condition ensures that the resulting manifold will be a homology sphere. The triviality of the group implies that it is a homotopy sphere and, thus, by the celebrated theorem of M. Freedman homeomorphic to . But all our presentations obviously satisfy the Andrews-Curtis conjecture and, therefore, the resulting manifolds will be PL- (or smooth) spheres.
Of course, this construction can be also performed in all dimensions greater than four. However, for higher dimensions one has a much larger stock of suitable finite presentations of the trivial group (see [Nab96b]): One can start from a sequence of finite presentations used in the proof of S. Novikov theorem asserting the non-existence of the algorithm recognizing for each . There is no algorithm deciding which of these presentations are presentations of the trivial group, which means that the number of Tietze transformations required to transform presentations of the trivial group that appear in this sequence to the trivial presentation is not bounded by any computable function. On the other hand, all these presentations are presentations of groups with “obviously” trivial first and second homology groups. As a result, one can use the Kervaire construction ([Ker69]) to realize them as “apparent” finite presentations of -dimensional homology spheres which are diffeomorphic to (as the groups are trivial). However, there is no easy way to see that the homology spheres are diffeomorphic to , as otherwise we would be able to see that seed finite presentations are finite presentations of the trivial group. The starting sequence of finite presentations codes a halting problem for a fixed Turing machine, and it was noticed in [Nab96b] that one can use the concept of time-bounded Kolmogorov complexity and a theorem proven by Barzdin to choose this Turing machine so that it is possible to ensure the existence of not only some triangulations of far from the standard triangulation but the existence of an exponentially growing number of such triangulations (as in Theorem A).
The four-dimensional situation is different from the higher-dimensional case. J.P. Hausmann and S. Weinberger proved that the vanishing of the first two homology groups of a finitely presented group is no longer sufficient to realize this group as the fundamental group of a -dimensional homology sphere ([HW85]). (It was the main result of [Ker69] that for each is the necessary and sufficient condition for the existence of a smooth -dimensional homology sphere with fundamental group .) The only known general sufficient condition of realizability of a group given by a finite presentation as the fundamental group of a -dimensional homology sphere is that this finite presentation is balanced. The condition that the number of relators is equal to the number of generators is a very strong condition that seemingly precludes coding of Turing machines in such finite presentations. In fact, it is a famous unsolved problem whether or not there is an algorithm that decides if a group given by a balanced presentation is trivial. However, the example of the Baumslag-Gersten group that has Dehn function growing faster than a tower of exponentials of height suggests an idea of adding the second relation , where runs over words representing the trivial element in that have very large areas of their van Kampen diagrams. (Before going further recall that means , and observe that can also be written as , which shows that it can be obtained from the infinite cyclic group by performing two HNN-extensions with associated subgroups isomorphic to .) As a result, we obtain balanced presentations of the trivial group with two generators, such that a very large number of Tietze transformations is required to transform these presentations to the trivial presentation provided that one proceeds in the most obvious way (namely, first “proving” that using only the first relation, then concluding that from the second relation.) Thus, it is natural to conjecture that these balanced presentations of the trivial group will be very far from the trivial presentation and, therefore, can be used to obtain triangulations of and other -manifolds that are very far from the standard ones. The second author came with this idea in 1992 and discussed it during 90’s with many colleagues but was never able to prove this conjecture. (The conjecture was mentioned in [Nab06], that first appeared as 2001 IHES preprint IHES/M/01/35.) The difficulty is that Gersten’s proof of the fact that has a very rapidly growing Dehn function uses the fact that is non-trivial. In particular, Gersten considers the universal covering of the presentation complex of , and this cannot be done for presentations of the trivial group. On the other hand, it seems that it is difficult to apply methods based on considerations of van Kampen diagrams when one needs to deal with two relators rather than one.
This problem was solved by the first author in [Lisa], [Lisb]. The main new idea was use a modified version of the small cancellation theory over HNN-extensions. (It is easy to see that can be obtained from by performing two HNN-extensions.) In [Lisa] he used “long” words of very special form; recently he improved his approach so that it yields the conjecture for “most” words representing the trivial element including the most natural ones ([Lisb]). To get a flavour of the main idea of [Lisa] consider pseudogroups associated with finite presentations of a group where a word is regarded as trivial only if it can be represented as a product of not more than conjugates of relators, where is not too rapidly growing function of the length of . (This concept is similar to the notion of effective universal coverings introduced [Nab10a]). If we consider such “effective” pseudogroups produced from , the word will not represent the trivial element in a pseudogroup. Moreover, we can hope that for an appropriate choice of “effective” pseudogroups associated with with added relation satisfy a small cancellation condition and, therefore, are non-trivial as effective pseudogroups. This implies that corresponding balanced presentations of the trivial group are very far from the trivial presentation.
Unbeknown to the authors the problem of construction of balanced presentations of the trivial group that are far from the trivial presentation was also of interest to Martin Bridson (who also was not aware of the interest of the second author to this problem). His preprint [Bri15] with another solution appeared on arxiv two weeks after [Lisa] but, as we later learned, Bridson announced his solution in a series of talks in 2004-2006 and mentioned it in his ICM-2006 talk [Bri06]. We are going to describe his main ideas from our viewpoint involving the concept of effective pseudogroups. He also starts from . First, he considers a “fake” HNN-extension of with stable letter and new relation . As the resulting group is isomorphic to , but in the realm of effective pseudogroups is non-trivial, so from some intuitive viewpoint this can be regarded as an HNN-extension of effective pseudogroups. Then he takes two copies , of and considers the amalgamated free product of effective pseudogroups . Obviously, these presentations are the presentations of the trivial group but in the realm of effective pseudogroups this will be non-trivial pseudogroups. (Again, this is our interpretation of Bridson’s examples, not his. Also, at the moment there is no theory of effective pseudogroups. Yet this point of view suggests a possible simplification of Bridson’s construction, namely, “fake” amalgamated products of two copies of .)
The main idea of the proof of Theorem B (and, thus, all other results of the present paper) is the following. For each we produce exponentially many versions of the fake HNN-extension of . These versions involve variable words of length not exceeding , and words , , satisfying an “effective” small cancellation condition as in [Lisa], which gives even more control over . As the result, the finite presentations obtained from two copies of as in [Bri15] for two different will represent effective pseudo-groups that are not effectively isomorphic. This implies that they cannot be transformed one into the other by a “small” number of Tietze transformations. Thus, our construction is a hybrid of constructions in [Lisa] and [Bri15].
At the moment we do not have any infrastructure for a theory of effective pseudogroups (but hope to develop it in a subsequent paper). Therefore, the arguments involving the concept of effective pseudogroup should be regarded only as intuitive ideas useful for understanding of our actual proofs (that do not involve the concept of effective pseudogroups).
To give a flavour of our further arguments consider , where this time , denote two copies of a fixed word representing a non-trivial element of infinite order in the Baumslag-Gersten group , and is a variable word in . It is natural to conjecture that two such groups defined for different ’s are not isomorphic. To prove this fact one can investigate what can happen with the generators of under a possible isomorphism. One will be using the fact that here we are dealing with the amalgamated free products of two copies of , as well as the description of the outer automorphism group of found by A. Brunner in [Bru80]. Brunner proved that is isomorphic to the additive group of dyadic rationals. (A self-contained exposition of this result intended for geometers can be found in [Lisc].) In the present paper we will need to establish an effective version of Brunner’s theorem and, then, a non-existence of an “effective isomorphism” between “effective pseudogroups” and , which is an informal way of saying that and cannot be transformed one into the other by a “short” sequence of Tietze transformations.
Finally, note that it seems plausible that one can construct the desired presentations as , where runs over an exponentially large set of words in , but at the moment methods of [Lisa] seem insufficient to prove that these presentations are exponentially far from each other. But, if true, this would reduce the smallest possible number of generators in Theorem B from to .
In the next section we describe the construction of . In the third section we prove some quantitative results about Baumslag-Gersten group, necessary for the fourth section, where we essentially prove that any two different presentations from our construction are not isomorphic as effective pseudogroups. In the fifth section we prove Theorem B, in the sixth we prove Theorem C. In the last section we prove Theorems A, A.1, A.1.1.
2. Notation and the Construction
Denote by the tower of exponents of height , i.e. are recursively defined by , . Let . As usual, denotes , where can be words or group elements. Let be the length of the word . If represents the identity element, denote by the minimal number of 2-cells in a van Kampen diagram over the presentation with boundary cycle labeled by . For a presentation denote by its total length, i.e. the sum of lengths of the relators plus the number of generators.
Let , is called the Baumslag-Gersten group, an HNN-extension of the group (called the Baumslag-Solitar group).
We are going to use the same word of large area as in [Lisa].
Let be defined inductively as follows. Let
Here denotes , the commutator of and Suppose is defined, then let be the word obtained from by replacing subwords with . Finally, let .
We can make an estimate . In the terminology of [Lisa], is -reduced, i.e. any non-circular -band in a digram for is of length at least . This in particular implies that the area of is at least .
Before defining the presentations we prove a lemma.
Let be two different words in the alphabet , then in .
One can see that by choosing as the coset representative for and applying the theory of normal forms for an HNN-extension . Alternatively, one can see that in , where can be represented in the binary notation as follows. The length of the number is equal to . There is a digit for each in the word , the digit is placed in the -th position if there are letters to the right of that . The rest of the digits are . For example, , where . Clearly, this number is unique for a word . To finish the proof we notice that if , then because is the stable letter of , and therefore . Finally, if in , then in , because is an HNN-extension of . ∎
Let be an arbitrary word of length in , , where , appear only in non-negative degrees. Let , and define another copy of this presentation . Note, these are presentations of the infinite cyclic group. Let:
Replacing by , by , by , by we can reduce the number of generators to . Using the commutator notation the resulting presentations of the trivial group will look as follows:
Later we will use this construction for words of length and choose less than , and therefore the lengths of these presentations are . There are of them, and we will prove that they are very far from each other in the metric defined as the minimal number of Tietze transformations required to transform one presentation into the other. But for now we will treat the length of as a variable.
Results of [Bri15] imply that, in particular, . We will require finer results of the same type (see the next section) to prove that not only is large, but also that for different words of the type defined in the previous lemma the presentations are “far” from each other. The exponential number of such words as a function of length will give us the exponential number of such presentations. In order to prove Theorem B we will be using words of length and words with . Also, note that although given presentations of have generators and relators one can use four relations in order to eliminate generators (namely, ) as above and rewrite these presentations as balanced presentations with generators.
We introduce more notation. For denote by the minimal such that (equality in the free group), where neighbouring are from different factors of . For denote by the number of letters in . Similarly, for denote by the number of letters in , and for denote by the number of letters in .
3. Quantitative Results about the Baumslag-Gersten Group
As we mentioned before, this section contains technical lemmas similar in the spirit to Theorem B from [Bri15]. But we will prove finer results using “effective” small cancellation theory from [Lisa]. We will use these lemmas to obtain results about “effective” homomorphism in the next section.
Recall, that denotes the Baumslag-Solitar group, is the Baumslag-Gersten group, and were defined in the previous section.
Let be a non-empty word in that does not contain more than letter consecutively. Let be a word obtained from by replacing with , with , and with an element of , possibly different for any particular instance of , satisfying the following conditions. If is between two letters , then , if is between and , then , if is between and , then , and if is between two letters , then (not equal in ). Then if in G, .
Consider a van Kampen diagram for . Let us call letters on the boundary of this diagram “outer” if they come from . Recall that is the product of three commutators of the form
where in the other two commutators is replaced with and . First, notice that any -band in the diagram originating on an outer letter has to end on an outer letter. That follows from counting the number of letters and subtracting from that the number of letters between the two ends of a -band. This difference has to be equal to . Pick an outer letter and a -band corresponding to it. Pick another outer letter between the ends of this -band, there is the corresponding inner -band. Continue until we find a -band between neighbouring outer letters. We claim that this -band has length . The only possible neighbouring outer letters are (), (not a pinch), (coming from ), (coming from ), (coming from ), (coming from . The last four pairs are not pinches because of the requirements on . The claim and the lemma follow.
Assume , where are powers of or , and is as in Lemma 2.2. Then as words. Similarly, if for , then .
We look at cases. If one of the words is a power of then the other has to be the inverse of this power, and since , , which implies . If one of the words is a power of , then so is the other one and . We want to apply Lemma 3.1 and therefore want to check the conditions in its statement. Between and we can have . Between and we can have , , , , or . We need to check that if we conjugate any of them by we won’t get a power of . It is true for the second and the fifth element because . For the rest we will use the following fact, if , then and is even. Checking for the first element: . Similarly for the third: . And the fourth: . Therefore, the lemma follows from Lemma 3.1. ∎
Let , where is a word in , , where are or , are less than , and is as in Lemma 2.2. Then and . Furthermore, if , then through area less than .
When or is , the conclusion is clear. Suppose . Let be after all are replaced with . Similarly define . Then (one can see the proof of Theorem 2.3 from [Lisa] for a complete calculation). Consider an annular van Kampen diagram of area less than for this conjugation. The -bands on this diagram can not originate and end on the same boundary component (see the proof of Lemma 3.2). Therefore the number of letters is the same for both boundary components and .
Notice the decrease of the upper bound ( versus in Lemma 3.2) in the previous lemma. It is there for technical reasons, and we believe can be eliminated with some extra work. Similarly, the factor of in the next lemma is, probably, unnecessary.
Let be as in Lemma 2.2, then , where is a word in .
We can apply less than relations to convert to , obtaining a van Kampen diagram for of area less than . We want to apply our “effective” small cancellation theory from [Lisa] to this equality. We can view it as a van Kampen diagram with boundary over , because we have the equality in the free group: . Since the boundary of this diagram is also it can be viewed as a spherical van Kampen diagram over (see Figure 1), call it .
We want to apply Theorem 3.10 from [Lisa] to . This theorem says that if there is an HNN-extension ( in this case) and some added relations ( in this case) satisfying effective small cancellation conditions, then elements in which are not trivial in can not be effectively trivial in . In [Lisa] it was checked that the relator satisfies the effective metric condition and therefore condition . Similarly, satisfies . We don’t have the metric condition (our pieces can be exactly of length ), but it is not needed in the proof of that theorem. Secondly, our diagram is spherical, not planar, but we can just replace the Euler characteristic of the disk () with that of the sphere () in the proof of Theorem 3.10 from [Lisa] to get the same result. In the proof of that theorem the diagram is first made -reduced, i.e. -cells having large common boundaries are canceled out, then the contradiction is reached. We can not do that with spherical diagrams, because a spherical diagram might just disappear with the last cancellation not giving us a contradiction. Then Theorem 3.10 from [Lisa] implies that there is a cancellation in our diagram.
Cancellations happen when there are a lot of -bands between -cells. In [Lisa] it was shown that in this case these bands all have length , i.e. there is a pair of -cells that touch each other. Consider two cases. Case 1: there is a cancellation involving the first cell (conjugated by ) and case 2: a cancellation with the other cell. In case 1 we have two options: (see Figure 2 for the definition of ) is a loop on the diagram, or is. Note, loops in represent in because as groups. By assumption, , therefore , or . After canceling this pair of -cells we obtain a spherical diagram that has one pair of -cells. In these cells were connected by a curve spelling . After the cancellation the curve spells . Note, might not be equal to effectively. For example they might differ by one of the removed cells. In the two remaining cells have to cancel each other, therefore is a loop and thus . We know from before , and so . Case 2 is dealt with similarly.
Let be a non-empty freely reduced word in . We can think of as a word in . Then if , .
Suppose the area is less than . First we prove, that circular -bands are impossible. Find an innermost circular -band. Then we see that either a power of or a power of is in through area . This is impossible by Lemma 3.1. We checked the conditions required for an application of Lemma 3.1 in the proof of Lemma 3.2. Semi-circular -bands are impossible because . Therefore , which is impossible, a contradiction. ∎
Let be non-empty freely reduced words in . If (via area ) for some , then either is equal to a word in in via area or , for some .
Consider a diagram for . As in the proof of Lemma 3.5 there are no circular -bands, so the subdiagrams between -bands are over . Suppose there is a letter in (otherwise are powers of ). Then the -band originating on it has to go to for the same reasons explained in the proof of Lemma 3.5. The subdiagrams between such -bands are in , and we can use Lemma 3.2 to show that these -bands have length . That shows is equal to a word in in via area . ∎
4. Effective Isomorphisms
The goal of this section is to prove that there are no “effective” homomorphisms (with certain “effective injectivity” conditions) between and for . We are going to use the idea of Collins ([Col78]): Let be a homomorphism. Since the Baumslag-Solitar group () is an HNN extension of with the stable letter , elements in have the invariant -length (the number of , letters in a reduced form). Collins’ lemma implies that conjugation can not change -length, if the conjugated elements are cyclically -reduced. Since we have a conjugation , we see that -length of has to be , and that’s the first step in describing .
Now, is an HNN-extension of and therefore, by the same logic, -length of the image of is . This was used in [Bru80] to classify homomorphisms . We are going to extend this logic to , an “effective” HNN extension of , and finally an “effective” amalgamated free product of with itself (in free products with amalgamation the invariant length is the number of factors in the reduced form). To do this we need to “effectivise” Collins’ argument, which can be done by using van Kampen diagrams (see also a classification of maps using diagrams in [Lis15]). Then we will use our knowledge of the “effective” structure of to complete the proof. This effective structure is somewhat explicated by the technical lemmas of the previous section.
For a presentations , denote by the free group on the letters of the presentation.
For presentations , we say a group map is (we assume ) if for a letter , has length , and for a relator of , . If in addition (possibly ), then we say is .
One should think of an (L,N) map as an “effective” group homomorphism.
Let be words like in Lemma 2.2. Let , . In this section we will prove several lemmas about an effective homomorphism from to . We start with one and then simplify it until in the end we are able to prove that it exists only if . Each modification of the homomorphism either would be trivial on the level of pseudogroups, or a composition with a simple automorphism of : a conjugation or the transposition of the two factors of .
The next lemma is the first instant of the “effectivized” version of the Collins’ argument described in the beginning of this section. We also have to adapt this argument to free products with amalgamation instead of HNN extensions.
If is (where ), then there exists of type such that .
Diagrams over any free product with amalgamation consist of subdiagrams of cells from one or the other factor (regions), bounded by the mixed cells (the cells responsible for amalgamation). In the case of we have regions made up from cells and regions from bounded by the cells corresponding to . Therefore the pure regions are bounded by words in (or ) and pieces of the boundary of the diagram.
Consider a diagram for of area less than . Our goal is to find such that . Suppose , then . Consider the regions of the diagram in . In total they have twice as many boundary pieces on than on . Therefore there either exists a “no-hat” region with boundary on only (case 1), or a region with more than one boundary piece on (case 2), see Figure 3. In case 1 we have the equality of a no-hat piece of the boundary of the diagram to a word in , in case 2 we have the equality of a mixed piece (the part of the boundary between some two no-hat pieces) to a word in . In both cases we can find a word such that , the length of is less than , the length of is less than ( is a bound for the length of the words in or – the boundaries between pure regions), and . Thus, we define of type to be , for define . We have .
We can continue in this way defining