aa r X i v : . [ m a t h . G R ] J u l A FAMILY OF NON-ISOMORPHISM RESULTS
COLLIN BLEAK AND DANIEL LANOUE
Abstract.
We give a short argument showing that if m , n ∈{ , , . . . } ∪ { ω } , then the groups mV and nV are not isomorphic.This answers a question of Brin. Introduction
In the paper [2], Brin introduces groups nV , where n ∈ { , , . . . } ∪{ ω } . These groups can be thought of as “higher dimensional” analoguesof Thompson’s group V . In particular, while V = 1 V is a group ofhomeomorphisms of the standard, deleted-middle-thirds Cantor set C ,the group nV can be thought of as a group of homeomorphisms of C n .The groups nV are all simple (see [4]) and finitely presented ([1].Also, they each contain copies of every finite group. Because of theseproperties, it is non-trivial to detect whether these groups are pairwiseisomorphic through the use of standard algebraic machinery. On theother hand, by Rubin’s Theorem, much of the structure of these groupsis encoded in their actions as groups of homeomorphisms. Thus, theauthors of this note turned to dynamical/topological approaches.In the seminal article [2], Brin shows that the group 2 V is not iso-morphic to V . He also asks (in his talk [4]) whether mV and nV canbe isomorphic if m = n . In this paper, we answer that question.As Brin demonstrates in [2], the groups V and 2 V act on C and C in such a way that Rubin’s Theorem (see [5]) applies; any isomorphism φ : V → V would induce a unique homeomorphism ψ : C → C , sothat for v ∈ V , φ ( v ) = ψ ◦ v ◦ ψ − . As C and C are homeomorphic,Rubin’s Theorem does not directly demonstrate that the groups V and2 V are non-isomorphic. However, in [2], Brin in finds an element in 2 V which acts with periodic orbits of length k for arbitrarily large integers k . As he also demonstrates that V does not contain such an element,he is able to deduce that 1 V = V is not isomorphic with 2 V .Although we also employ Rubin’s Theorem, our approach is differ-ent from that used by Brin above (although it owes somewhat to Brin’swork in [3], where he classifies the automorphism groups of R. Thomp-son’s groups F and T ). Our chief result is stated below. Date : October 24, 2018.1991
Mathematics Subject Classification.
Key words and phrases.
Higher Dimensional R. Thompson Groups nV, germs,Rubin’s Theorem.
Theorem 1.
Let m , n ∈ { , , . . . } ∪ { ω } , with m = n , then mV isnot isomorphic to nV . Background Requirements
In this paper, most of the work is in setting up the correct per-spective. Once that is achieved, the proof of Theorem 1 is almost atriviality.Throughout this section, we will provide the background and set thestage for that proof.2.1.
A group of germs.
Suppose X is a topological space, and H isa subgroup of the full group of homeomorphisms of X . For any x ∈ X ,define the set F ix ( H,x ) = { h ∈ H | h ( x ) = x } .We place an equivalence relation on F ix ( H,x ) . Say that f , g ∈ F ix ( H,x ) are equivalent if there is a neighborhood U of x so that f | U = g | U . In this case we write f ∼ x g and we denote the equivalence classof any element h ∈ F ix ( H,x ) by [ h ] x . We further denote the set ofequivalence classes that result by G ( H,x ) . This last set is known as theset of germs of H at x which fix x .The following is standard. It is also given explicitly as a portion ofLemma 3.4 in [3]. Lemma 1.
Given a topological space X , a point x ∈ X , and a groupof homeomorphisms H of X , the set G ( H,x ) forms a group under theoperation [ f ] x ∗ [ g ] x = [ f g ] x . C n , n -rectangles, and the groups nV . There is a well-knowncorrespondence between finite binary strings, and subsets of the Can-tor set C . We interpret a string as inductive choices of halves. Forinstance, the string “011” would means take the left half of the Can-tor set (throwing out the right half), then take the right half of whatremains, and finally pass to the right half of that. The diagram belowillustrates this notation.PSfrag replacements 011If we pass to the limit and consider infinite binary strings, we obtainthe standard bijection from 2 N to the Cantor set. That is, we abusenotation by having an element s of the Cantor set correspond to a map s : N → { , } . In this notation, we would denote s (0) by s , so that, FAMILY OF NON-ISOMORPHISM RESULTS 3 by an abuse of notation, we can write s as an infinite string, that is s = s s . . . . In this case, we can define a prefix of s as any finitesubstring beginning with s . We will call an infinite binary string w an infinite tail of s if s = P w for some finite prefix P of s .Elements s, t ∈ N are near to each other when they have long com-mon prefixes. This induces the standard topology of the Cantor set.In this description, a point z of the Cantor set will be rational if andonly if it corresponds to an infinite string of the form P w = P ww . . . ,where P is a prefix string and w is some non-empty finite string.We will now fix n as a positive integer. Let us first define a specialclass of subsets of C n . R is an n -rectangle in C n if there is a collectionof finite binary strings P , P , . . . , P n − so that R = (cid:8) z ∈ C n | z = ( x , x , x , . . . , x n − ) and x i = P i z i where z i ∈ N (cid:9) . In this case, by an abuse of notation, we will say R = ( P , P , . . . , P n − ).We now define a special class of maps, n -rectangle maps, as follows.Suppose D = ( P , P , . . . , P n − ) and R = ( Q , Q , . . . , Q n − ) are n -rectangles. We define the n -rectangle map τ ( D,R ) : D → R by therule z z ′ where the i ’th coordinate P i z i of z determines the i ’thcoordinate Q i z i of z ′ . We may also refer to these as prefix maps.We note that a prefix map τ ( D,R ) (as above) scales dimension i inaccord with the length of P i and the length of Q i . For instance, if P i = 1 and Q i = 011, then in dimension i , the map will take a halfof the Cantor set and map it affinely over an eighth of the Cantor set.In particular, any such map will have a scaling factor of 2 n for n ∈ Z .(In terms of metric scaling, perceiving the Cantor set as the standarddeleted-middle-thirds subset of the interval, the scaling factors wouldof course be 3 n . We will use the 2 n point of view in the remainder ofthis note.)We can now define a pattern to be a partition of C n into a finitecollection of n -rectangles. An element f of nV now corresponds to ahomeomorphism from C n to C n for which there is an integer k , a do-main pattern D , and a range pattern R , each pattern with k rectangles,so that f can be realized as the union of k disjoint n -rectangle maps,each carrying a n -rectangle of D to a n -rectangle of R . It should benoted that two different pairs of partitions can correspond to the samemap. The diagram below demonstrates a typical such map in 2 V . COLLIN BLEAK AND DANIEL LANOUE
PSfrag replacements 11 22 33In the case of ωV , the ω -rectangles are only allowed to restrict thedomain in finitely many dimensions. That is, R = ( P , P , P , . . . ) is an ω rectangle if and only if only finitely many of the P i are non-emptystrings. Thus, ωV can be thought of as a direct union of the nV groupsfor finite n .2.3. Rubin’s Theorem and some groups of germs.
In order tostate Theorem 2 below, we need to give a further definition. If X is atopological space and and F is a subgroup of the group of homeomor-phisms of X , then we will say that F is locally dense if and only if forany x ∈ X and open neighborhood U of x the set (cid:8) f ( x ) | f ∈ F, f | ( X − U ) = 1 | ( X − U ) (cid:9) has closure containing an open set.The following is the statement of Rubin’s Theorem, as given by Brinas Theorem 2 in [2]. It is a modification of Rubin’s statement Theorem3.1 in [5], which statement appears to contain a minor technical error. Theorem 2 (Rubin) . Let X and Y be locally compact, Hausdorff topo-logical spaces without isolated points, let H( X ) and H( Y ) be the auto-morphism groups of X and Y, respectively, and let G ⊆ H( X ) and H ⊆ H( Y ) be subgroups. If G and H are isomorphic and are both locallydense, then for each isomorphism φ : G → H there is a unique home-morphism ψ : X → Y so that for each g ∈ G , we have φ ( g ) = ψgψ − . If we combine Rubin’s Theorem with our previous work on the groupof germs, we get a lemma which appears to be a very mild extensionof Lemma 3.4 from [3].
Lemma 2.
Suppose X and Y are locally compact, Hausdorff topolog-ical spaces without isolated points, and that G and H are respectivelysubgroups of the homeomorphism groups of X and Y , so that G and H are both locally dense. Suppose further that φ : G → H is an isomor-phism, and that ψ is the homeomorphism induced by Rubin’s Theorem.If x ∈ X and y ∈ Y so that ψ ( x ) = y , then ψ induces an isomorphism ψ : G ( G,x ) → G ( H,y ) . FAMILY OF NON-ISOMORPHISM RESULTS 5
Proof.
This is a straightforward exercise in calculation.For [ v ] x ∈ G ( G,x ) , define ψ ([ v ] x ) = [ ψvψ − ] y .We first show that ψ is well defined. Let f ∼ x g with f, g ∈ F ix ( G,x ) ,so that f | U = g | U with U an open neighborhood of x . Then N = ψ ( U )is an open neighborhood of y = ψ ( x ) since ψ is a homeomorphism.For z ∈ N , ψ − ( z ) ∈ U , so ψf ψ − ( z ) = ψgψ − ( z ). Thus [ ψf ψ − ] y =[ ψgψ − ] y .Now we show that ψ is a homomorphism. We have ψ ([ f ] x [ g ] x ) = ψ ([ f g ] x ) = [ ψf gψ − ] y = [ ψf ψ − ψgψ − ] y = [ ψf ψ − ] y [ ψgψ − ] y = ψ ([ f ] x ) ψ ([ g ] x ) . Finally, we show that ψ is a bijection. Since ψ − conjugates H to G , we also have an induced map ψ − . Now, direct calculation as aboveshows that ψ − ◦ ψ and ψ ◦ ψ − are the identity maps on G ( G,x ) and G ( H,y ) respectively. (cid:3) Conclusion
We are now in a position to prove Theorem 1. We describe ourresults using notation for some n < ω and generally leave to the readerthe extension to n = ω .3.1. Some germs of nV . Let us calculate the group G ( nV,x ) , for x ∈ C n .For x = ( x , . . . , x n ) ∈ C n , let | x | denote the cardinality of the set { i : x i rational } .We now observe the following. Lemma 3.
For x ∈ C n , G ( nV,x ) ∼ = Z | x | .Proof. Let x = ( x , x , . . . , x n − ), with | x | = k and assume withoutmeaningful loss of generality that x , ..x k − are rational. In particular,let us write x i = A i w i for each index i with 0 ≤ i ≤ k −
1, where A i is the shortest prefix so that x i can be written in this fashion, andwhere w i is the shortest word so that an infinite tail of x i is of the form w i . Suppose f ∈ nV with f ( x ) = x , and let j be the minimal integerso that f admits a decomposition as a union of j disjoint n -rectanglemaps, { f i : D i → R i | ≤ i ≤ j } , with f = f ∪ f . . . ∪ f j .Assume further that a is the index so that x ∈ D a . By our earlierassumptions, x ∈ R a as well. We can now restrict our attention to τ ( D a ,Q a ) . We have that there are finite strings P ( a,i ) and Q ( a,i ) so that R a = ( P ( a, , P ( a, , . . . , P ( a,n − ) and Q a = ( Q ( a, , Q ( a, , . . . , Q ( a,n − ).Since x is fixed by f , we must have that for each index m ≥ k , P ( a,m ) = Q ( a,m ) . Also, we see that for each index m < k , P ( a,m ) = A m ( w ) s m and Q ( a,m ) = A m ( w ) t m , for non-negative integers s m and t m . COLLIN BLEAK AND DANIEL LANOUE
Define for f , the tuple ( s − t , s − t , . . . , s k − − t k − ) ∈ Z k . We willcall this association ˆ σ x : F ix ( nV,x ) → Z k .If g , h ∈ F ix ( nV,x ) , with [ g ] x = [ h ] x , then ˆ σ x ( g ) = ˆ σ x ( h ), as otherwisethe scaling of the two maps in some dimension could not be the samein some small neighborhood of x .In particular, the map ˆ σ x induces a set map σ x : G ( nV,x ) → Z k . Thereader may now confirm that the map σ x is in fact an isomorphism ofgroups. (cid:3) Germ isomorphisms.
For any integer n , C n is a locally com-pact, Hausdorff topological space without isolated points (recall that C n is homeomorphic to C for any integer n ). Given any x ∈ C n withopen neighborhood U , and y ∈ U , there is a map in nV which swaps avery small n -rectangle around x with a very small n -rectangle around y , and is otherwise the identity. In particular, the group nV acts lo-cally densely on C n . A similar argument works for ωV , where our n -rectangles are given with long prefixes for the first N coordinates forsome N , and are empty afterwords. Thus, Rubin’s Theorem appliesfor nV for any index n in { , , . . . } ∪ { ω } .We are finally in position to prove our main theorem.Suppose m and n are valid indices, with m < n , and suppose φ : nV → mV is an isomorphism, with ψ : C n → C m the homeo-morphism induced by Rubin’s Theorem. Let x be a point in C n withall n coordinates rational, and let y = ψ ( x ). The map ψ induces anisomorphism ψ : G ( nV,x ) → G ( mV,y ) . But observe, the rank of G ( nV,x ) is n , while the rank of G ( mV,y ) must be less than or equal to m . Inparticular, no such isomorphism ψ can exist. References
1. Bleak, C., Hennig, J., Matucci, J.:
Presentations for the Groups nV , In Prepa-ration (2008).2. Brin, M.:
Higher Dimensional Thompson Groups , Geom. Dedicata (2004)163-192.3. Brin, M.:
The chameleon groups of Richard J. Thompson: automorphisms anddynamics , Inst. Hautes tudes Sci. Publ. Math. No. 84 (1996), 5-334. Brin, M.:
Cousins of Thompson’s groups, III (Higher dimensional groups) ,”Thompson’s Groups: New Developments and Interfaces”, CIRM, Marseille,France. June 5th, 2008.5. Rubin, M.:
Locally Moving Groups and Reconstruction Problems, OrderedGroups and Infinite Permutation Groups , Kluwer Acad. Publ., Dordrecht, 1996,pp. 121–157.
FAMILY OF NON-ISOMORPHISM RESULTS 7
Department of Mathematics, University of Nebraska, Lincoln, NE68588-0130, USA,
E-mail address : [email protected] Department of Mathematics, Northeastern University, Boston, MA02115, USA,
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