aa r X i v : . [ m a t h . G R ] S e p A DESCRIPTION OF
Aut(dV n ) AND
Out(dV n ) USINGTRANSDUCERS
LUKE ELLIOTT
Contents
1. Introduction 12. Preliminaries 23. Generalizing the transducers of Grigorchuk, Nekrashevich, andSushchanskii 34. Generalizing the synchronizing homeomorphisms of Bleak, Cameron,Maissel, Navas, and Olukoya 65. A closer look at the groups d O n, Abstract
The groups dV n are an infinite family of groups, first introduced by C. Mart´ınez-P´erez, F. Matucci and B. E. A. Nucinkis, which includes both the Higman-Thompsongroups V n (= 1 V n ) and the Brin-Thompson groups nV (= nV ). A description of thegroups Aut( G n,r ) (including the groups G n, = V n ) has previously been given by C.Bleak, P. Cameron, Y. Maissel, A. Navas, and F. Olukoya. Their description uses thetransducer representations of homeomorphisms of Cantor space introduced a paper of R.I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskii, together with a theorem of M.Rubin. We generalise the transducers of the latter paper and make use of these transducersto give a description of Aut( dV n ) which extends the description of Aut(1 V n ) given in theformer paper. We make use of this description to show that Out( dV ) ∼ = Out( V ) ≀ S d , andmore generally give a natural embedding of Out( dV n ) into Out( G n,n − ) ≀ S d . Introduction
In Matthew G. Brin’s 2004 paper [5], he introduces the family of simple groups dV , which serve as d -dimensional analogues to Thompson’s group V . The presentpaper is concerned with finding a “nice” way to represent the automorphism groupsof the groups dV . To do this we follow a similar path to that in [2] but via a morecategory theoretic perspective. This enables us to prove a conjecture made byNathan Barker in 2012: Theorem 1.1.
For all d ≥ , we have Out( dV ) ∼ = Out( V ) ≀ S d (using the standardaction of S d on d points). We view the transducers of [8] as a category in their own right, and then identifysubcategories of transducers which are more appropriate for representing homeo-morphisms of “ n -dimensional” Cantor Spaces. Similarly to [2], we then employ Rubin’s Theorem [15] to represent Aut( dV ) by transducers. We also extend thedescription of Out( V ) given in [2] to Out( dV ).From this perspective we are able to represent the automorphisms of the encom-passing family of groups dV n , first introduced in the paper [11] of Mart´ınez-P´erez,Matucci, and Nucinkis. We also describe the outer automorphisms of these groupswith transducers and give the following theorem extending the one given for dV : Theorem 1.2.
For all d ≥ and n ≥ we have Out( dV n ) ∼ = (cid:8) T ∈ Out( G n,n − ) d (cid:12)(cid:12) Q i The outer automorphisms of Thompson groups have a history in the literature.In [4, 7] Brin and Fernando Guzm´an study the automorphisms of F and T typegroups. As previously motioned, the authors of [2] gave a means of describingOut( G n,r ) with transducers, in particular the way the groups Out(1 V n ) are viewedin this paper is theirs. More recently Feyishayo Olukoya has used transducer basedmethods to study the outer automorphisms of the groups T n,r in [12].The family of groups dV have also been extensively studied in the literature. In[5] it is proved that the groups dV are all infinite, simple, and finitely generated.In [6] Brin goes on to give an explicit finite presentation for 2 V with 8 generatorsand 70 relations. The paper [3] of Collin Bleak and Daniel Lanoue uses Rubin’sTheorem to show that dV and nV are non-isomorphic for d = n .In [9], Johanna Hennig and Francesco Matucci show that in general dV can befinitely presented with 2 d + 4 generators and 10 d + 10 d + 10. More recently in [14],Martyn Quick builds much smaller presentations for dV , using only 2 generators aswell as 2 d + 3 d + 13 relations.It’s shown in [2] that Aut( G n,r ) embeds in the rational group R of finite trans-ducers as defined in [8]. In [1] it is shown that there is a natural topologicalconjugacy embedding 2 V into R as well (which can be naturally generalised to dV ). It is therefore natural to ask if this conjugacy sends Aut( dV ) to a subgroupof R . In this paper we give examples to demonstrate that this fails for all d ≥ k -monoids” (see [10]). Proposition 3.9 of [10]suggests that the methods of this document are likely only compatible with groupscorresponding to the k-monoids which are finite products of finite rank free monoids.AcknowledgementsI would like to thank my supervisor Collin Bleak for reading a draft of this paper,and giving a lot of helpful advice as to how to better present the results within it.I would also like to thank Jim Belk for recommending that I look into theautomorphisms of 2 V .2. Preliminaries We will compose functions from left to right and we will always index from 0.For n ∈ N , we will use the notations X n and n to denote the set { , , . . . , n − } .We use the former when thinking of this set as an alphabet, and the latter whenthinking of an initial segment of N . DESCRIPTION OF Aut(dV n ) AND Out(dV n ) USING TRANSDUCERS 3 We denote the free monoid of all finite words over a finite alphabet X by X ∗ .That is X ∗ := [ n ∈ N X n . This notably includes the empty word which we denote by ε . We also follow thestandard convention of identifying a letter with a word of length 1. If w ∈ X ∗ ,then we define | w | to be the length of w as a word (which is actually the same asits cardinality). If X is an alphabet, then the set X ∗ ∪ X ω is naturally partiallyordered by: x ≤ y if and only if x is a “prefix” of y. We also extend this partial order to the sets ( X ∗ ∪ X ω ) d . Definition 2.1 We will denote all projection morphisms by π , π , . . . in all cate-gories with products (the specific morphism used will be determined by the context).If x ∈ ( X ∗ n ) d , y ∈ ( X ∗ n ∪ X ωn ) d and x ≤ y , then we define y − x to be the unique z ∈ ( X ∗ n ∪ X ωn ) d such that xz = y (using coordinate-wise concatenation). Definition 2.2 If n ≥ 2, the we define C n := X ωn to be the usual Cantor spacewith the product topology. Moreover if w ∈ ( X ∗ n ) d then we define w C dn := (cid:8) x ∈ C dn (cid:12)(cid:12) w ≤ x (cid:9) . Note that these sets are clopen, and the collection of all such sets is a basis for C dn .Such basic open sets will be referred to as cones . Definition 2.3 If X is a topological space, then we denote the homeomorphismgroup of X by H ( X ).We can now give the definition of the groups dV n which will we will use throughoutthe paper. Definition 2.4 Suppose that F , F are finite subsets of ( X ∗ n ) d , such that (cid:8) f C dn (cid:12)(cid:12) f ∈ F (cid:9) and (cid:8) f C dn (cid:12)(cid:12) f ∈ F (cid:9) are partitions of C dn , and φ : F → F is a bijection. We call such sets F , F completeprefix codes for C dn . We then define the prefix exchange map f φ : C dn → C dn by:If w ∈ F and x ∈ w C dn , then( x ) f φ = ( wφ )( x − w ) . Such prefix exchange maps are always homeomorphisms and the set of all suchmaps under composition forms the group dV n (or just dV if n = 2). Remark 2.5. There is a complete prefix code for C dn of size m if and only if m ∈ (1 + ( n − N ) . This is because all complete prefix codes can be obtained bystarting with the trivial prefix code of size , and repeatedly splitting cones into n smaller cones. Generalizing the transducers of Grigorchuk, Nekrashevich, andSushchanskii A transducer, as introduced by Grigorchuk, Nekrashevich, and Sushchanskii(which we shorten to GNS) in [8], can be thought of as a way of assigning eachletter of an alphabet, a transformation of a “state set”, together with a word towrite for each state. These are then extended to all words in the input alphabet LUKE ELLIOTT via the universal property of the free monoid. With reading elements of a monoidin mind, the following is a natural generalisation of their transducer definition. Definition 3.1 We say that T := ( Q T , D T , R T , π T , λ T ) is a transducer if:(1) Q T is a set (called the set of states),(2) D T is a semigroup (called the domain semigroup),(3) R T is a semigroup (called the range semigroup),(4) π T : Q T × D T → Q T is a (right) action of D T on the set Q T (called thetransition function),(5) λ T : Q T × D T → R T is a function with the property that for all q ∈ Q T , s, t ∈ D T we have( q, st ) λ T = ( q, s ) λ T (( q, s ) π T , t ) λ T (called the output function).We will often refer to the domain semigroup and range semigroup of a transduceras simply its domain and range.Note that a one state transducer is equivalent to a semigroup homomorphism. Definition 3.2 Let A, B be transducers. We say that φ is a transducer homo-morphism from A to B (written φ : A → B ), if φ is a 3-tuple ( φ Q , φ D , φ R ) with thefollowing properties:(1) φ R : R A → R B is a semigroup homomorphism,(2) φ D : D A → D B is a semigroup homomorphism,(3) φ Q : Q A → Q B is a function, such that for all q ∈ Q A and s ∈ D T we have( q, s ) π A φ Q = ( qφ Q , sφ D ) π B and ( q, s ) λ A φ R = ( qφ Q , sφ D ) λ B . If furthermore, the maps φ D , φ R are the identity maps, then we say that φ is strong . Remark 3.3. If transducer homomorphisms are composed component-wise, thentransducers become a category when given transducer homomorphisms, or strongtransducer homomorphisms. Definition 3.4 A transducer homomorphism φ , is called a quotient map if eachof φ Q , φ D , φ R is surjective.We say that transducers A and B are isomorphic (denoted A ∼ = B ) if they areisomorphic in the category of transducers and transducer homomorphisms. Simi-larly, we say that A and B are strongly isomorphic (denoted A ∼ = S B ) if they areisomorphic in the category of transducers and strong transducer homomorphisms.The next definition gives us a means of minimizing our transducers which coin-cide with the GNS notion of combining equivalent states. Definition 3.5 If T is a transducer with R T cancellative, then we define its minimal transducer M T to be ( Q T / ∼ M T , D T , R T , π M T , λ M T ) where ∼ M T , π M T and λ M T are defined by:(1) ∼ M T is the equivalence relation (cid:8) ( p, q ) ∈ Q T (cid:12)(cid:12) ( p, s ) λ T = ( q, s ) λ T for all s ∈ D T (cid:9) , (2) if q ∈ Q T , s ∈ D T then ([ q ] ∼ MT , s ) π M T = [( q, s ) π T ] ∼ MT ,(3) if q ∈ Q T , s ∈ D T then ([ q ] ∼ MT , s ) λ M T = [( q, s ) λ T ] ∼ MT .It is routine to verify that this is well-defined, the natural strong quotient candidate q T : T → M T , with ( p ) q T Q = [ p ] ∼ MT is a strong quotient map. Moreover all strongquotient maps with domain T are left divisors of q T . DESCRIPTION OF Aut(dV n ) AND Out(dV n ) USING TRANSDUCERS 5 Definition 3.6 If T is a transducer, and we restrict Q T , D T , and R T to setswhich are (together) closed under the transition and output functions, then weobtain another transducer. We call such a transducer a subtransducer of T . Definition 3.7 If d ∈ N and n ∈ N \{ , } , then we define a ( d, n )-transducer tobe a transducer T with ( X ∗ n ) d as its domain and range, and such that the transitionfunction is a monoid action.If T is a transducer, q ∈ Q T , and w ∈ D T , then will view the maps ( q, · ) π T , and( q, · ) λ T as reading w though a path in T from q , ending at the state ( q, w ) π T , andwriting ( q, w ) λ T along the way (similarly to GNS transducers). Note that unlikeGNS transducers, there isn’t always a “best” way of splitting up this path intominimal steps.If T is a ( d, n )-transducer then (like GNS transducers) we can naturally extendthis idea to “infinite words”, which in this case means elements of ( X ωn ) d , by readingarbitrarily long prefixes of an element and taking the limit of the elements written. Definition 3.8 If T is a ( d, n )-transducer and q ∈ Q T , then we define f T,q :( X ωn ) d → ( X ωn ) d ∪ ( X ∗ n ) d to be the map which maps w ∈ ( X ωn ) d to the word writtenwhen w is read in T from the state q .Note that if A is a ( d, n )-transducer, q ∈ Q A and φ : A → B is a strong transducerhomomorphism, then f A,q = f B, ( q ) φ Q . In particular this is true of the homomor-phism q A . Definition 3.9 Similarly to GNS we say that a ( d, n )-transducer T is degenerate if there are any finite elements in the image of f T,q for any q ∈ Q T .We will often use the following fact without comment: Remark 3.10. If T is a non-degenerate ( d, n ) -transducer and q ∈ Q T , then forall m ∈ N there is k ∈ N such reading an element of ( X kn ) d always writes a wordwhose length is at least m in every coordinate. There are 2 important ways by which we combine our transducers, there is“composition” as was done in GNS, and taking products in the categorical sense. Definition 3.11 If A and B are transducers, such that the range of A is containedin the domain of B , then we define their composite by AB = ( Q AB , D AB , R AB , π AB , λ AB ) . Where(1) Q AB := Q A × Q B , D AB := D A , R AB := R B ,(2) (( a, b ) , s ) π AB = (( a, s ) π A , ( b, ( a, s ) λ A ) π B ),(3) (( a, b ) , s ) λ A,B = ( b, ( a, s ) λ A ) λ B .As was the case in GNS, this definition is constructed so that whenever A, B are non-degenerate ( d, n )-transducers, and ( p, q ) ∈ A × B , we obtain f A,p f B,q = f AB, ( p,q ) . Definition 3.12 If ( A ) i ∈ I are transducers, then we define Q i ∈ I A i := P where Q P := Y i ∈ I Q A i , D P := Y i ∈ I D A i , R P := Y i ∈ I R A i , and for all ( p i ) i ∈ I ∈ P Q and ( s i ) i ∈ I ∈ D P we have(( p i ) i ∈ I , ( s i ) i ∈ I ) π P = (( p i , s i ) π A i ) i ∈ I , (( p i ) i ∈ I , ( s i ) i ∈ I ) λ P = (( p i , s i ) λ A i ) i ∈ I . LUKE ELLIOTT For i ∈ I we then define π i : P → A i to be the transducer homomorphism( π i , π i , π i ). One can verify that this is a product in the category theoretic sense(using transducer homomorphisms but not strong transducer homomorphisms).The following definition gives us, for each homeomorphism h of C dn , a transducer M h representing it. From the definition, one can see that this transducer hasno inaccessible states, has complete response and has no distinct but equivalentstates. So in particular when d = 1, the transducer M h is the minimal transducerrepresenting h as described by GNS. Definition 3.13 If h ∈ H ( C dn ), then we define T h to be the ( d, n )-transducer with(1) Q T h := ( X ∗ n ) d ,(2) ( s, t ) π T h = st ,(3) (( s, t ) λ T h ) π i is b − s , where b is the longest common prefix of the words inthe set (( st C dn ) h ) π i and s is the longest common prefix of the words in theset (( s C dn ) h ) π i .(As h is a homeomorphism, the set ( st C dn ) h is always open and thus ( s, t ) λ T h isalways an element of ( X ∗ n ) d .) Moreover, as was the case in GNS, if q = 1 ( X ∗ n ) d then f T h ,q = h . We also define M h := M T h . Remark 3.14. If h ∈ H ( C dn ) and q ∈ Q M h , then f M h ,q is injective with clopenimage. The proof of the following theorem is analogous to the proof of the analogoustheorem in GNS, (the above construction deals with the homeomorphism case). Theorem 3.15. A function h : C dn → C dn is continuous if and only if there is anon-degenerate ( d, n ) -transducer T and q ∈ Q T such that h = f T,q . Generalizing the synchronizing homeomorphisms of Bleak,Cameron, Maissel, Navas, and Olukoya As was the case in [2] when analyzing Aut( G n,r ), we now want to restrict to thetransducers which give us the automorphisms we want. We thus extend the notionof synchronization given there. Definition 4.1 We say that a ( d, n )-transducer T is synchonizing at level k if forall q , q ∈ Q T and w ∈ ( X ∗ n ) d with min( (cid:8) | w π i | (cid:12)(cid:12) i ∈ n (cid:9) ) ≥ k , we have ( q , w ) π T =( q , w ) π T . We say that T is synchronizing if it is synchronizing at any level. The synchronizing length of a synchronizing transducer T ismin( (cid:8) k ∈ N (cid:12)(cid:12) T is synchronizing at level k (cid:9) ) . In this case we define the function s T := (cid:8) ( w, q ) ∈ ( X ∗ n ) d × Q T (cid:12)(cid:12) for all p ∈ Q T we have ( w, p ) π T = q (cid:9) . So s T is basically π T restricted to the part of it’s domain when the input state isnot needed. The image of s T , denoted Core( T ), is called the core of T .It is useful to think of the core of a synchronising ( d, n )-transducer as the placereached when a sufficient amount of information has been read in each coordi-nate. In particular, if a word is read from any core state of a synchronizing ( d, n )-transducer T then you stay in the core, thus Core( T ) is a (synchronizing) subtrans-ducer of T (when given the restrictions of the transition and output functions of T ). DESCRIPTION OF Aut(dV n ) AND Out(dV n ) USING TRANSDUCERS 7 Start Core... . . . (1 , ε ) / ( ε, , ε ) / ( ε, ε, / ( ε, ε )( ε, / ( ε, ε ) (1 , ε ) / ( ε, , ε ) / ( ε, 01) ( ε, / ( ε, ε )( ε, / ( ε, ε )(1 , ε ) / (1 , ε )(0 , ε ) / (0 , ε )( ε, / ( ε, 1) ( ε, / ( ε, , ε ) / ( ε, , ε ) / ( ε, ε, / ( ε, ε )( ε, / ( ε, ε ) Figure 1. A minimal transducer with ( X ∗ ) as domain and range.(This represents the baker’s map in 2 V ).If q ∈ Q T , then Core( T ) = ( { q } × ( X kn ) d ) s T (where k is the synchronizing lengthof T ), so Core( T ) is always finite.The following proposition is routine to verify, and shows that our transducerframework describes dV n in a manner analogous to the way in which the transducersof GNS describe V n . Proposition 4.2. If h ∈ H ( C dn ) , then h ∈ dV n if and only if M h is a synchronizingtransducer whose core consists of a single “identity” state. Unlike for V n , the transducers for elements of dV n can sometimes be infinite.For example the transducer representing the baker’s map of 2V is infinite (as canbe seen in Figure 1) as it can’t write anything until something is read in the firstcoordinate.We will now introduce the monoids d S n, , ^ d O n, , d B n, and d O n, which gener-alise the monoids S n, , g O n, , B n, and O n, of [2]. Definition 4.3 We say an element f ∈ H ( C dn ) is synchronizing if M f is synchro-nizing. We define d S n, to be the set of synchronizing elements of H ( C dn ). LUKE ELLIOTT Remark 4.4. If A, B are synchronizing, non-degenerate transducers then so istheir composite AB . This works as you can synchronize the first coordinate usingthe synchronizing property of A , and once the first coordinate is in the finite non-degenerate core of A , one can read enough so that the output of A synchronizes B as well. Corollary 4.5. If f, g ∈ d S n, then Core( M f M g ) is a subtransducer of the com-posite transducer Core( M f ) Core( M g ) . Corollary 4.6. The set d S n, is always a monoid.Proof. It follows from Remark 4.4 that, if f, g ∈ d S n, , then M f M g is synchonizing.We will now essentially minimise M f M g in the GNS fashion and obtain M fg .Let q f := (1 ( X ∗ n ) d ) q T f and q g := (1 ( X ∗ n ) d ) q T g . We have f g = f M f ,q f f M g ,q g = f M f M g , ( q f ,q g ) . If we then define A to be the transducer with the same states, domain, rangeand transition function as M f M g but with ( q, w ) λ A = b − s where b is the longestcommon prefix of the set ( w C dn ) f M f M g ,q and s is the longest common prefix of theset ( C dn ) f M f M g ,q (it follows from Remark 3.14 that b and s are finite). Let A ′ bethe subtransducer of A consisting of the states that are accessible from ( q f , q g ) (theimage of (( q f , q g ) , · ) π A ).It follows from the definition that A ′ is a strong quotient of the transducer T fg .Thus M fg is a strong quotient of A ′ . As A has the same transitions as M f M g , itfollows tat A is synchronizing. Moreover, since A ′ is a subtransducer of A , we getthat A ′ is synchronizing and thus M fg is also synchronizing (as a strong quotientof A ′ ). (cid:3) Corollary 4.7. The set ^ d O n, := (cid:8) [Core( M f )] ∼ = S (cid:12)(cid:12) f ∈ d S n, (cid:9) naturally forms amonoid, which is a quotient of d S n, .Proof. We define [Core( M f )] ∼ = S [Core( M g )] ∼ = S = [Core( M fg )] ∼ = S . This is well defined as the strong isomorphism type Core( M fg ) can be found by re-moving incomplete response from Core( M f ) Core( M g ), combining equivalent statesand passing to the core (in the same manner as the proof of Corollary 4.6). (cid:3) Definition 4.8 We define d B n, to be the group of units of d S n, , and d O n, := (cid:8) [Core( M f )] ∼ = S (cid:12)(cid:12) f ∈ d B n, (cid:9) . Lemma 4.9. The map f [Core( M f )] ∼ = S is a surjective group homomorphismfrom d B n, to d O n, with kernel dV n .Proof. This map is a homomorphism by the definition of multiplication in ^ d O n, ,it is surjective by the definition of d O n, and thus as d B n, is a group, d O n, is also.The identity of d O n, is the image of the identity map, and is thus the single state“identity” transducer. From Remark 4.2, we get that dV n is the kernel. (cid:3) We have now introduced the monoids we need. We now begin showing that d B n, coincides with the normalizer of dV n in H ( C dn ) (the case with d = 1 was done in[2]). DESCRIPTION OF Aut(dV n ) AND Out(dV n ) USING TRANSDUCERS 9 Lemma 4.10. Let h ∈ N H ( C dn ) ( dV n ) and s, t ∈ ( X ∗ n ) d \{ ( X ∗ n ) d } . Let q h := (1 ( X ∗ n ) d ) q T h .There exists K h,s,t ∈ N such that for all a ∈ (cid:8) w ∈ ( X ∗ n ) d (cid:12)(cid:12) min( (cid:8) | w π i | (cid:12)(cid:12) i ∈ d (cid:9) ) ≥ K h,s,t (cid:9) we have ( q , sa ) π M h = ( q , ta ) π M h .Proof. For all x ∈ ( X ∗ n ) d , let q x := ( q h , x ) π M h . Let f ∈ dV n be such that f replacesthe prefix s with the prefix t . By the choice of h , there is some g ∈ nV such that h − f h = g and so f h = hg .Let q f := (1 ( X ∗ n ) d ) q T f and q g := (1 ( X ∗ n ) d ) q T g . It follows that f M f M h , ( q f ,q h ) = f h = hg = f M h M g , ( q h ,q g ) . Let I f , I g be the core states of M f and M g respectively(which don’t do anything). Note that(( q f , q h ) , s ) π M f M h = ( I f , q t ) , (( q h , q g ) , s ) π M h M g = ( q s , ( q g , ( q h , s ) λ M h ) π M g ) . Let K ∈ N be such that for all w ∈ ( X ∗ n ) d with min( (cid:8) | w π i | (cid:12)(cid:12) i ∈ d (cid:9) ) ≥ K , we havemin( (cid:8) | (( q h , s ) π M h , w ) λ M h π i | (cid:12)(cid:12) i ∈ d (cid:9) ) is at least the synchronizing length of M g .Let a ∈ ( X ∗ n ) d be arbitrary such that min( (cid:8) | w π i | (cid:12)(cid:12) i ∈ d (cid:9) ≥ K . We have(( q f , q h ) , sa ) π M f M h = ( I f , q ta ) , (( q h , q g ) , sa ) π M h M g = ( q sa , I g ) . Thus, for all v ∈ ( X ωn ) d we have(( q f , q h ) , sa ) λ M f M h ( v ) f M h ,q ta = (( q f , q h ) , sa ) λ M f M h ( v ) f M f M h , ( I f ,q ta ) = ( sav ) f h = ( sav ) hg = (( q h , q g ) , sa ) λ M h M g ( v ) f M h M g , ( q sa ,I g ) = (( q h , q g ) , sa ) λ M h M g ( v ) f M h ,q sa It follows that (( q f , q h ) , sa ) λ M f M h and (( q h , q g ) , sa ) λ M h M g are comparable ineach coordinate.If (( q f , q h ) , sa ) λ M f M h and (( q h , q g ) , sa ) λ M h M g differed in any coordinate it wouldfollow that either the map f M h ,q ta or f M h ,q sa has its image contained in a propercone. This is impossible as M h by definition has no incomplete response. Thus(( q f , q h ) , sa ) λ M f M h = (( q h , q g ) , sa ) λ M h M g .From the equality(( q f , q h ) , sa ) λ M f M h ( v ) f M h ,q ta = (( q h , q g ) , sa ) λ M h M g ( v ) f M h ,q sa it follows that f M h ,q ta = f M h ,q sa . As M h is minimal it follows that q ta = q sa asrequired. (cid:3) Lemma 4.11. The group N H ( C dn ) ( dV n ) is contained in d B n, .Proof. The proof of this is essentially the same as Corollary 6.17 of [2]. The ideais as follows. We need only show containment in d S n, because N H ( C dn ) ( dV n ) is agroup with the same identity as d S n, . Thus we need only show the synchronizingcondition. So it suffices to show that, for any h ∈ N H ( C dn ) ( dV n ), there is a K ∈ N such that the state reached by reading any word from (1 ( X ∗ n ) d ) q T h is determinedby the last K letters of the word (in every coordinate). We do this by collapsingany giving input word from the front by repeated applications of Lemma 4.10 using s with size 1 and t with size 2 (where size means the sum of the lengths of thecoordinates). (cid:3) We now recall the theorem of Rubin which connects our arguments to automor-phism groups: Theorem 4.12 (Rubin’s Theorem [15]) . Let G be a group of homeomorphisms ofa perfect, locally compact, Hausdorff topological space X . For U ⊆ X let G U := { g ∈ G : ( x ) g = x for all x ∈ X \ U } . Suppose further that for all x ∈ X and U aneighbourhood of x , we have ( x ) G U is somewhere dense. If φ : G → G is a groupisomorphism then there is a ψ φ ∈ H ( X ) such that ( g ) φ = φ − φ gψ φ for all g ∈ G . In [2], it is shown that Rubin’s theorem allows us to naturally embed Aut( G n,r )into H ( C n,r ). This same argument also applies to dV n , and in fact to any groupwith an action satisfying the hypothesis of Rubin’s theorem. Corollary 4.13. The groups Aut( dV n ) and N H ( C dn ) ( dV n ) are isomorphic. Theorem 4.14. The groups Aut( dV n ) and d B n, are isomorphic.Proof. We have Aut( dV n ) ∼ = N H ( C dn ) ( dV n ) by Corollary 4.13, and we have N H ( C dn ) ( dV n ) ⊆ d B n, ⊆ N H ( C dn ) ( dV n )by Lemma 4.11 and Lemma 4.9. (cid:3) Corollary 4.15. The groups Out( dV n ) and d O n, are isomorphic.Proof. This follows from Theorem 4.14 and Lemma 4.9. (cid:3) Corollary 4.16. For d, m ∈ N \{ } and n ∈ N \{ , } the group Aut( dV n ) m embedsin the group Aut(( md ) V n ) .Proof. If ( d B n ) m acts on ( C dn ) m ∼ = C dmn in the natural fashion, then these homeo-morphisms are contained in the group ( md ) B n . (cid:3) Corollary 4.17. The group Aut( dV n ) is countably infinite.Proof. The group dV n is countably infinite and the group d O n, is countable (itconsists of the isomorphism classes of finite things). (cid:3) A closer look at the groups d O n, We now want to pin down what the core transducers representing d O n, looklike. We’re going to end up with a semidirect product, so we’ll deal with the actingpart of the product first. Before that we introduce a notation which we will userepeatedly throughout this section. Definition 5.1 We define F d,n,S := (cid:8) w ∈ ( X ∗ n ) d (cid:12)(cid:12) w π i = ε for all i ∈ d \ S (cid:9) That is F d,n,S is the submonoid of ( X ∗ n ) d , consisting of those elements only allowedto be non-trivial in the coordinates in S . Lemma 5.2. Let T be a transducer representing an element of ^ d O n, . If i ∈ d then there is a unique ( i ) ψ T ∈ d , such that for all q ∈ Q T and l ∈ F d,n, { i } , we have ( q, l ) λ T ∈ F d,n, { ( i ) ψ T } . DESCRIPTION OF Aut(dV n ) AND Out(dV n ) USING TRANSDUCERS 11 Proof. We start by showing the existence of ( i ) ψ T . First note that for all q ∈ Q T ,the map f T,q is necessarily injective (Remark 3.14). Suppose for a contradictionthat there is i ∈ d , l , l ∈ F d,n, { i } , q , q ∈ Q T and α, β ∈ d such that α = β , | ( q , l ) λ T π α | > | ( q , l ) λ T π β | > 0. We may assume without loss of generalitythat α = 0 and β = 1.For all j ∈ d \{ , } let q j ∈ Q T , i j ∈ d and l j ∈ F d,n, { i j } be such that | ( q j , l j ) λ T π j | > T is non-degenerate and the F setsgenerate ( X ∗ n ) d ). Also let i = i = i .It is now the case that if we read a word in coordinate i j , it’s possible to writein coordinate j (if we’re in the correct state). Moreover i = i , so there is acoordinate b such that we can write words in any coordinate without reading fromcoordinate b . We will use this observation to contradict injectivity.For each state q j we choose some w j ∈ ( q j ) s − T . If we read w j l j from anywherewe will write non-trivially into the coordinate j . Consider w j as w j,b w ′ j where w j,b ∈ F d,n, { b } and w ′ j ∈ F d,n,d \{ b } .Consider the elements s m := w ,b w ,b . . . w d − ,b ( w ′ l ) m ( w ′ l ) m . . . ( w ′ d − l d − ) m ∈ ( X ∗ n ) d . As the w j,b type elements commute will all other kind of elements in the productdefining of s m , by commuting the words so that w j,b ( w ′ j l j ) j is a part of the productdefining s m , it follows that for any q ∈ Q T we have | ( q, s m ) λ T π j | ≥ m for all j .Thus all elements of ( X ωn ) d which have all the s m as prefixes have the same imageunder f T,q . This is a contradiction as there are infinitely many such elements and f T,q is injective.It remains to show the uniqueness of ( i ) ψ T . The only way ( i ) ψ T could be non-unique is if T never writes anything when reading from coordinate i . In this caseit follows from the existence of the other ( j ) ψ T , that there is some coordinate intowhich T never writes, which is impossible as T is non-degenerate. (cid:3) Definition 5.3 If T is a transducer representing an element of ^ d O n, , then wedefine ψ T : d → d to be the map which was shown to be well-defined in Lemma 5.2. Theorem 5.4. The group d O n, is isomorphic to d K n, ⋊ S d , where S d is using itsstandard action of on d points and d K n, = (cid:8) [ T ] ∼ = S ∈ d O n, (cid:12)(cid:12) ψ T = id (cid:9) .Proof. Note that the map [ T ] ∼ = S → ψ T is a monoid homomorphism to the fulltransformation monoid on d points. As d O n, is a group, it follows that ψ T isalways a permutation for [ T ] ∼ = ∈ d O n, . To see that the map is onto the symmetricgroup and the extension splits, note that for any f ∈ S d the map( p , p , . . . , p d − ) h −→ ( p (0) f , p (1) f , . . . , p ( d − f )is an element of d B n, . Moreover [Core( M h )] ∼ = S maps to f under the homomor-phism. (cid:3) We next need to understand the group d K n, . To this end, we recall the groups O n,n − of [2]. These are groups of synchronizing core (1 , n )-transducers, which areisomorphic to the outer automorphism groups of G n,n − . In [2], it was shown that O n,n − contains O n,j for all j , and that a (1 , n )-transducer represents an element of O n,n − if and only if it is minimal (in the senseof GNS), synchronizing, it is its own core, all it’s states are injective, all its stateshave clopen image and it’s invertible. In particular 1 O n, = O n, is a subgroup of O n,n − . Theorem 5.5. If [ T ] ∼ = S ∈ d K n, then there are T , T , . . . T d − ∈ O n,n − such that T ∼ = S Y i ∈ d T i . Proof. For each i ∈ d, let ∼ i := (cid:8) ( p, q ) ∈ Q T (cid:12)(cid:12) there is w ∈ F d,n, { i } with ( p, w ) π T = q (cid:9) . One can check that each ∼ i is an equivalence relation. If q ∈ Q T , i ∈ d , we restrictthe domain and range of T to F d,n, { i } and restrict the state set of T to [ q ] ∼ i , thenwe obtain a subtransducer S q,i of T .Moreover for all w ∈ F d,n,d \{ i } , one can check that the map q ( q, w ) π T is astrong transducer isomorphism from S q,i to S ( q,w ) π T ,i .We will show that S q,i ∈ O n,n − (if we make the natural identification between F d,n, { i } and X ∗ n ). It suffices to check that the conditions given in [2] are satisfied.The transducer S q,i has no inaccessible states by the definition of ∼ i , it has completeresponse because T has complete response, it’s synchonizing and it’s own corebecause T is and it has injective state functions because T does (Remark 3.14). ByRemark 3.14, each state function f T,q of T has clopen image. As the image of astate function f S q,i ,p of S q,i is a projection of the image of the corresponding statefunction f T,p of T (because ψ T is well-defined), it follows that this image of f T,p is compact and open (hence clopen). It remains to show that S q,i has no distinctbut equivalent states. As T has no distinct but equivalent states, it suffices to showthat if p , p are equivalent states in S q,i , then they are equivalent in T . Let j ∈ d ,and w ∈ F d,n, { j } . It suffices to show that ( p , w ) λ T = ( p , w ) λ T . If j = i thenthis follows by the assumption on p , p . Otherwise let s ∈ F d,n, { i } be such that( p , s ) π T = p . Then( p , w ) λ T = ( p , ws ) λ T π j = ( p , sw ) λ T π j = (( p , s ) λ T ( p , w ) λ T ) π j = ( p , w ) λ T as required. So we can conclude that S q,i ∈ O n,n − (if we make the natural identi-fication between F d,n, { i } and X ∗ n ).For each i ∈ d, let S q,i be isomorphic to T i ∈ O n,n − via an isomorphism whichuses π i as the domain and range isomorphisms (recall that q has no affect on theisomorphism type). Moreover let φ q,i : S q,i → T i be the unique such transducerisomorphism (this is unique as the image of any state is determined by any of itssynchronizing words). Let φ i : T → T i be the transducer homomorphism with φ iD = φ iT = π i , φ iQ = [ q ∈ Q T φ q,iQ . We then define φ : T → Q i ∈ d T i to be the unique transducer homomorphismsuch that for all i ∈ d we have φ π i = φ i . We need to show that φ is a strong isomorphism. We have φ D = φ R = id bydefinition, so we need only show that φ Q is a bijection. It must be surjective as DESCRIPTION OF Aut(dV n ) AND Out(dV n ) USING TRANSDUCERS 13 it’s target is synchronizing, and so each state is the image of the state reached in T by reading any of it’s synchronizing words. Injectivity is also immediate as T wasassumed to be minimal and hence has no proper strong quotients. (cid:3) It is routine to check that the multiplication in d K n, is also compatible with themultiplication in O n,n − , so we can make the following definition: Definition 5.6 We define an embedding α : d K n, → O dn,n − by Y i ∈ d ([ T ] ∼ = S ) α π i ∼ = S T. Corollary 5.7. The group Out( dV n ) embeds in the group O n,n − ≀ S d . Proof. This follows from Definition 5.6(Theorem 5.5), Theorem 5.4 and Corol-lary 4.15. (cid:3) We now have a connection between d O n, and O n,n − . To pin this down preciselywe recall the map sig : O n,n − → ( Z / ( n − Z , × ) of [13] Definition 7.6.This group homomorphism takes an O n,n − transducer to the unique element( m +( n − Z ) ∈ Z / ( n − Z , such that when a cone in read through the transducer,the transducer writes m disjoint cones. We can naturally use this map to define anew homomorphism from O dn,n − . Definition 5.8 We define the homomorphism sig d : O dn,n − → ( Z / ( n − Z , × )by ( T )sig d = Y i ∈ d (( T ) π i )sig . Moreover we observe that this definition functions as one might expect: Lemma 5.9. If T ∈ O dn,n − , then ( T ) sig d the unique element ( m + ( n − Z ) of Z / ( n − Z , such that m disjoint cones are written when a cone in read through Q i ∈ d ( T π i ) .Proof. This is well-defined by Remark 2.5. The result follows from the observationsthat a ( d, n ) cone is the same as a product of d (1 , n ) cones, and the set of wordswritable from a state in a product of transducers is the product of the sets of wordswhich can be written in each coordinate. (cid:3) In Proposition 7.7 in [13] and Theorem 9.5 of [2], the signature map has beenused to identify the groups O n,r inside of O n,n − . We can now do the same with d K n, via a similar argument. Lemma 5.10. If T ∈ ( O n,n − ) d , then P := Q i ∈ d T π i is strongly isomorphic to atransducer representing an element of d K n, if and only if T ∈ ker( sig d ) .Proof. ( ⇒ ) : Suppose that f ∈ B n, is such that the transducer P is stronglyisomorphic to Core( M f ). If U is clopen in C dn , then let count( U ) be the smallestnumber of cones in a decomposition of U into cones. Let k be the synchronizing length of M f . It follows that1 + ( n − Z = count(( C dn ) f ) + ( n − Z = X w ∈ ( X kn ) d count(( C dn ) f M f , ( w ) s Mf ) + ( n − Z = X w ∈ ( X kn ) d ( T )sig d + ( n − Z = n kd ( T )sig d + ( n − Z = ( T )sig d + ( n − Z . The result follows.( ⇐ ) : Let q ∈ Q P be arbitrary. Then let S a ∈ A a C dn be a decomposition ofimg( f P,q ) into disjoint cones. We have that | A | ∈ n − Z . Let k ∈ N begreater that | A | , and such that for all w ∈ ( X kn ) d and i ∈ d , we have | ( q, w ) λ P π i | ≥ max {| a π j | : a ∈ A, j ∈ d } . For all w ∈ ( X kn ) d let a w ∈ A be such that a w is a prefixof ( q, w ) λ P . For all a ∈ A we now have that { (( q, w ) λ P − a w ) img( f P, ( q,w ) π T ) : w ∈ ( X kn ) d has a w = a } is a partition of C dn .As | A | ∈ n − N , there is a complete prefix code B of size | A | . Let φ : A → B be a bijection. It follows that { ( a w ) φ (( q, w ) λ P − a w ) img( f P, ( q,w ) π P ) : w ∈ ( X kn ) d } is a partition of C dn . We now define an element f ∈ H ( C dn ) as follows:If w ∈ ( X kn ) d and x ∈ C dn then( wx ) f = ( a w ) φ (( q, w ) λ P − a w )( x ) f P, ( q,w ) π P . It is routine to verify that M f is synchronizing and has core strongly isomorphic to P . Thus f ∈ d S n, , and P represents an element of ^ d O n, . As ker(sig d ) is a group, T − ∈ ker(sig d ) and so by the same argument P ′ := Q i ∈ d ( T π i ) − also representsand element of ^ d O n, . Thus P is d O n, . As P is a product of (1 , n )-transducers wealso have ψ P = id , so the result follows. (cid:3) We now have all the tools to prove Theorem 1.2 from the introduction, (thestatement is a bit simpler now that we’ve defined sig d ). Theorem 5.11. For all d ≥ and n ≥ we have Out( dV n ) ∼ = ker( sig d ) ⋊ S d , where the action of S d is the standard permutation of coordinates.Proof. By Corollary 4.15 we have Out( dV n ) ∼ = d O n, . Thus the result follows fromTheorem 5.4 and Lemma 5.10. (cid:3) Corollary 5.12. For all d ≥ we have Out( dV ) ∼ = Out( V ) ≀ S d (using the standardaction of S d on d points).Proof. This follows from the previous theorem together with the observation that( Z / (2 − Z , × ) is the trivial group. (cid:3) DESCRIPTION OF Aut(dV n ) AND Out(dV n ) USING TRANSDUCERS 15 q q / / / 01 0 /ε / / / / / / Figure 2. Two transducers with domain and range X ∗ Rationality and Representations Unfortunately, unlike with B , , representing elements of 2 B , with transducerscan sometimes result in a transducer which has infinitely many states. In Figure 1we see that the baker’s map when represented by a transducer, in the way describedin this paper, has infinitely many states.However, if we want our pictures to be finite, then we can consider the submonoidof ( X ∗ n ) d consisting of those elements w ∈ ( X ∗ n ) d such that | w π i | is the same for all i . As elements of d B n, are synchronizing, it follows that if we restrict the domainof a minimal transducer representing an element of d B n, to this submonoid, thenwe will only need finitely many states to represent it. In Figure 3 we see the baker’smap represented in this fashion. The main problem with this representation is thatcomposing functions represented with these transducers is much harder (due to thefact that the transducers have distinct domains and ranges).Figure 2 displays two elements of B , , where the q are the initial states. Thesetransducers are particularly nice as they are finite. The first one acts by swappingthe strings “0” and “00” wherever it sees them, and the second one is the identity.These transducers are special in that they are each equal to their own cores. Thus,these pictures also represent elements of O , if we ignore the choice of initial state.In Figure 4 we see the categorical product of the transducers in Figure 2, whichrepresents an element of 2 B , (and also an element of 2 O , ).In [2], it is shown that B , consists of rational homeomorphims, and in [1]Theorem 5.2 it is shown that 2 V naturally embeds in R via conjugation by ahomeomorphism between C and C . The map of [1] Theorem 5.2 acts by con-verting a pair of words ( x x . . . , y y . . . ) to a word ( x , y )( x , y ) . . . over thealphabet { (0 , , (0 , , (1 , , (1 , } . It is natural to ask if the same map gives anembedding of 2 B , into R . However it is routine to verify that if we conjugate the q Core(1 , / ( ε, , / ( ε, 01) (1 , / ( ε, , / ( ε, 00) (0 , / (0 , , / (0 , , / (1 , 0) (1 , / (1 , Figure 3. The subtransducer of the transducer in Figure 1 with { w ∈ ( { , } ∗ ) : | w π | = | w π |} as domain and no longer accessi-ble states removed. q (1 , ε ) / (1 , ε ) (0 , ε ) / (0 , ε )( ε, / ( ε, 1) ( ε, / ( ε, , ε ) / (01 , ε )(0 , ε ) / ( ε, ε ) ( ε, / ( ε, ε, / ( ε, , ε ) / (1 , ε )(0 , ε ) / (00 , ε ) ( ε, / ( ε, 1) ( ε, / ( ε, , ε ) / (1 , ε )(0 , ε ) / (0 , ε ) ( ε, / ( ε, ε, / ( ε, Figure 4. The transducer obtained by taking the categoricalproduct of the transducers in Figure 2homeomorphism defined by the transducer of Figure 4, then the resulting map isnot rational. This happens because this transducer is a product of a transducerwhich never resizes words, with a transducer which does this arbitrarily amounts DESCRIPTION OF Aut(dV n ) AND Out(dV n ) USING TRANSDUCERS 17 (for example by reading the words (01) n ). So it seems that there is no good wayto describe the groups Aut(dV n ) using only finite transducers. References [1] James Belk and Collin Bleak. Some undecidability results for asynchronous transducersand the Brin-Thompson group 2 V . Transactions of the American Mathematical Society ,369(5):31573172, Dec 2016.[2] Collin Bleak, Peter Cameron, Yonah Maissel, Andrs Navas, and Feyishayo Olukoya. Thefurther chameleon groups of Richard Thompson and Graham Higman: Automorphisms viadynamics for the Higman groups G n,r , 2016.[3] Collin Bleak and Daniel Lanoue. A family of non-isomorphism results. Geometriae Dedicata ,146(1):21–26, Jun 2010.[4] Matthew G. Brin. The chameleon groups of richard j. thompson: Automorphisms and dy-namics. Publications mathmatiques de lIHS , 84(1):533, Dec 1996.[5] Matthew G. Brin. Higher dimensional Thompson groups. Geometriae Dedicata , 108(1):163–192, Oct 2004.[6] Matthew G. Brin. Presentations of higher dimensional Thompson groups. Journal of Algebra ,284(2):520558, Feb 2005.[7] Matthew G. Brin and Fernando Guzmn. Automorphisms of generalized thompson groups. Journal of Algebra , 203(1):285348, May 1998.[8] R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanski˘ı. Automata, dynamical systems,and groups. Proc. Steklov Inst. Math , 231:128–203, 2000.[9] Johanna Hennig and Francesco Matucci. Presentations for the higher dimensional Thomp-son’s groups nV, 2011.[10] Mark V Lawson and Alina Vdovina. Higher dimensional generalizations of the thompsongroups, 2019.[11] Conchita Mart´ınez-P´erez, Francesco Matucci, and Brita EA Nucinkis. Cohomological finite-ness conditions and centralisers in generalisations of thompsons group v. In Forum Mathe-maticum , volume 28, pages 909–921. De Gruyter, 2016.[12] Feyishayo Olukoya. Automorphisms of the generalised Thompson’s group T n,r . In Prepara-tion, 2018.[13] Feyishayo Olukoya. Automorphisms of the generalised Thompson’s group T n,r , 2019.[14] Martyn Quick. Permutation-based presentations for Brin’s higher-dimensional Thompsongroups nV , 2019.[15] Matatyahu Rubin. Locally moving groups and reconstruction problems. In Ordered groupsand infinite permutation groups , volume 354 of