Generalization of order separability for free products and omnipotence of free products of groups
aa r X i v : . [ m a t h . G R ] N ov Generalization of order separability for freeproducts and omnipotence of free products ofgroups.
Vladimir V. Yedynak
Faculty of Mechanics and Mathematics, Moscow State UniversityMoscow 119992, Leninskie gory, MSUedynak [email protected]
Abstract
It was proved that for any finite set of elements of a free product ofresidually finite groups such that no two of them belong to conjugatecyclic subgroups and each of them do not belong to a subgroup which isconjugate to a free factor there exists a homomorphism of the free productonto a finite group such that the order of the image of each fixed elementis an arbitrary multiple of a constant number.
Key words: free products, residual properties, omnipotence.
MSC:
Order separabilities are connected with the investigation of the correlation be-tween the orders of elements’ images after a homomorphism of a group onto afinite group. For example in [1] it was proved that for each elements u and v of a free group F such that u is conjugate to neither v nor v − there exists ahomomorphism of F onto a finite group such that the images of u and v havedifferent orders. In [6] it was proved that this property is inherited by freeproducts. This paper is devoted to the proof of the theorem that strengthensthe property of order separability for the class of free products of groups. Theorem.
Consider the group G = A ∗ B where the subgroups A and B are residually finite. Consider the elements u , . . . , u n such that u i ∈ G \{ ∪ g ∈ G ( g − Ag ∪ g − Bg ) } , u i , u j belongs to conjugate cyclic subgroups whenever i = j . Then there exists the natural number K such that for each orderedsequence l , . . . , l n of natural numbers there exists a homomorphism ϕ of G onto a finite group such that the order of ϕ ( u i ) is equal to Kl i The property under study in this work is closely connected with omnipotencewhich was investigated in [2], [3] where it was shown that free groups andfundamental groups of compact hyperbolic surfaces are omnipotent. Besides allfinite sets of independent elements whose orders are infinite in a Fuchsian groupof the first type also satisfy the property of omnipotence [4].Definition. The group G is called omnipotent if for each elements u , . . . , u n such that no two of them have conjugate nontrivial powers there exists a number K such that for each ordered sequence of natural numbers l , . . . , l n there existsa homomorphism ϕ of G onto a finite group such that the order of ϕ ( u i ) equals Kl i . 1he familiar property was also investigated in [7] where some sufficient con-ditions were found for n -order separability of free products. The group G is saidto be n -order separable if for a set S = { s , . . . , s n | s i = h − s ± j h, i = j } of n elements of G there exists a homomorphism of G onto a finite group mapping S onto a set whose elements have pairwise different orders.Notice that the theorem of this paper will enable to investigate the residualproperties of the fundamental group of graphs of groups whose vertex groupsare residually finite free products and edge groups are cyclic not belonging tosubgroups conjugate to free factors of vertex groups. We consider that for every graph there exists a mapping η from the set of edgesof this graph onto itself. For every edge e this mapping corresponds an edgewhich is inverse to e . Besides the following conditions are true: η ( η ( e )) = e foreach e , η is a bijection, for every edge e the beginning of e coincides with theend of the edge η ( e ).The graph is called oriented if from every pair of mutually inverse edges oneof them is fixed. The fixed edge is called positively oriented and the inverseedge is called negatively oriented.Let G be a free product of groups A and B . There exists a correspondencesuch that for every action of G on the set X at which both A and B act freelythere exists a graph Γ satisfying the following properties:1) for each c ∈ A ∪ B and for each vertex p of Γ there exists exactly oneedge labelled by c going into p and there exists exactly one edge labelled by theelement c which goes away from p .2) for every vertex p of Γ the maximal connected subgraph A ( p ) of Γ con-taining p whose positively oriented edges are laballed by the elements of A isthe Cayley graph of the group A with generators { A } ; we define analogicallythe subgraph B ( p ).3) we consider that for every edge e from the first item there exists the edgeinverse to e which does not bear a label; two edges with labels are not mutuallyinverse; edges with labels are positively oriented.Definition 1. We say that a graph is the free action graph of the group G = A ∗ B if it satisfies the properties 1), 2), 3).Note that if ϕ is the homomorphism of the group G such that ϕ A ∪ B is thebijection then the Cayley graph Cay ( ϕ ( G ); { ϕ ( A ) ∪ ϕ ( B ) } ) of the group ϕ ( G )with respect to the set of generators { ϕ ( A ) ∪ ϕ ( B ) } is the free action graph ofthe group G .Remark. In what follows appending a new edge with label to a free actiongraph we shall consider that it is positively oriented and the inverse edge wouldhave been appended. And if we delete an edge with label the inverse edge wouldhave been deleted.If e is the edge then α ( e ) , ω ( e ) are vertices which coincide with the beginningand the end of e correspondingly. 2f we have the free action graph Γ of the group G then there exists the actionof G on the set of vertices of Γ which is defined as follows. Let p be an arbitraryvertex of Γ. Then according to the definition of the free action graph for eachelement c from A ∪ B there exist edges e and f whose labels are equal to c suchthat α ( e ) = p, ω ( f ) = p . In this case the action of c on p is defined as follows: p ◦ c = ω ( e ) , p ◦ c − = α ( f ).Remark also that if we change the property 2) in the definition of the freeaction graph supposing that A ( p ) and B ( p ) are the Cayley graphs of the ho-momorphic images of the groups A and B correspondingly we also obtain thegraph such that there exists the action of the group G on the set of its vertices.Such a graph will be referred to as an action graph of the group G .Since there exists the action of G on the set of vertices of an action graph Γthere exists a homomorphism of G onto the group S n , where n is the cardinalnumber of the set of vertices of the graph Γ. Having a group G and its actiongraph Γ we shall denote this homomorphism as ϕ Γ .If e is the positively oriented edge of the action graph, then Lab( e ) is thelabel of e .Definition 2. Let u be a cyclically reduced element of the group G whichbelongs to neither A nor B and Γ is the action graph of G . Fix a vertex p of Γ.Then u -cycle in this action graph going from p is the cycle R = e . . . e n whichsatisfies the following properties:1) the path P is a closed path such that its beginning α ( P ) = p
2) consider u = u . . . u k where u i ∈ A ∪ B, u i , u i +1 as well as u , u k do notbelong to one free factor simultaneously; then k divides n and the edge e ik + j ispositively oriented and has a label u j , 1 j k (indices are modulo n )3) the cycle P is the minimal cycle which satisfies properties 1), 2).Definition 3. Suppose we have a path S = e · · · e n in the action graph. Thenthe label of this path is the element of the group which is equal to Q ni =1 Lab( e i ) ′ ,where Lab( e i ) ′ equals either the label of e i , if this edge is positively oriented,or Lab( e i ) ′ = Lab( η ( e i )) − otherwise. We shall denote the label of the path S as Lab( S ).Definition 4. Fix the graph Γ, p and q are vertices from Γ. Then we definethe distance between p and q as ρ ( p, q ) = min S l ( S ), where S is an arbitrarypath connecting p and q , l ( S ) is the number of edges in S .Notice that if a cycle S does not have l -near vertices then each subpath of S of length which less or equal than l is geodesic.Definition 5. Fix an arbitrary graph and a cycle S = e · · · e n in it. For everynonnegative integer number l we shall say that S does not have l -near vertices,if for every i, j, i = j, i, j n the distance between the vertices α ( e i ) , α ( e j )is greater or equal than min( l + 1 , | i − j | , n − | i − j | ).Definition 6. Suppose we have the u -cycle S . It is obvious that its labelequals the k -th power of u for some k . Then we say that the length of the u -cycle S is equal to k .Note that for the action graph Γ and cyclically reduced element u ∈ G \{ A ∪ B } the order of | ϕ Γ ( u ) | coincides with the less common multiple of lengths of3ll u -cycles in the graph Γ. Hence if there exists a u -cycle in the action graphwhose length equals t then | ϕ Γ ( u ) | is a multiple of t .Suppose u is an element of A ∗ B and u = u · · · u n is the irreducible formof u . Then the length of u is the number l ( u ) = n . The cyclic length l ′ ( u ) of anelement u is the length of the cyclically reduced element which is conjugate to u . Lemma 1.
Consider the group G = A ∗ B where A and B are finite, l and n are natural numbers, Q is a finite set of elements from G which are cyclicallyreduced and whose lengths are greater than 1. Then for each v ∈ Q there existsthe homomorphism ϕ of F onto a finite group such that for each u from Q the u -cycles in the Cayley graph Cay ( ϕ ( G ); { ϕ ( A ) ∪ ϕ ( B ) } ) of the group ϕ ( G ) donot have l -near vertices, | ϕ ( v ) | > n , and ϕ A ∪ B ∪ Q — injection. Proof.
For each q ∈ Q define the set L q which consists of the elements from G whose length is less or equal than l + 2 l ( q ) + 10 and which do not belong tothe subgroup generated by the element q . It is well known that free groupare subgroup separable [5]. Besides if a group is virtually subgroup separablethan it is subgroup separable (see [2] for example). Considering the abovethere exists the homomorphism ϕ q of the group G onto a finite group such that ϕ q ( L q ) ∩ h ϕ q ( q ) i is an empty set. There also exists the homomorphism ϕ ′ v of G onto a finite group such that ϕ ′ v ( v i ) = 1 where i = 1 , . . . , n and ϕ ′ v | A ∪ B ∪ Q is theinjection since virtually free groups are residually finite. The homomorphism ϕ : G → ( × h ∈ Q ϕ h ( G )) × ϕ ′ v ( G ) , ϕ : f Q h ∈ Q ( ϕ h ( f )) ϕ ′ v ( f ) is as required.Lemma 1 is proved.The following statement was proved in [2] Lemma 2.
Consider a group G and its elements g , . . . , g n possessing theproperty that for each j, j n, there exist constants K j, , . . . , K j,n suchthat for each natural m there exists a homomorphism ϕ j,m of G to a finite groupwith the condition that | ϕ j,m ( g k ) | = K j,k for all k = j and | ϕ j,m ( g j ) | = mK j,j .Then there exists the number K such that for each ordered sequence of naturalnumbers l , . . . , l n there exists the homomorphism ψ of G onto a finite groupsatisfying the property that | ψ ( g i ) | = Kl i . It follows from lemma 2 that that the theorem can be derived from the followingproposition.
Proposition.
Let G = A ∗ B be a free product of residually finite groups A and B , u, v , . . . , v n ∈ G . Elements u and v i do not belong to conjugatecyclic subgroups. Besides u does not belong to a subgroup which is conjugateto either A or B . Then there exist natural numbers L, K , . . . , K n such that4igure 1: The graph Γfor each natural i there exists a homomorphism ϕ of G onto a finite group suchthat | ϕ ( u ) | = Li, | ϕ ( v i ) | = K i , i n . Proof.
Since
A, B are residually finite we may consider that A and B are finite.Consider also that the elements u, v , . . . , v n of A ∗ B are cyclically reduced.Let us to define the following notation. Consider the action graph Γ of thegroup K ∗ L . Let S be the subset of Γ (e. g. vertex, edge, path, subgraphetc). Then having a set Γ , . . . , Γ n of copies of Γ we consider that S i denotesthe subset of Γ i corresponding to S in Γ.Put s = max b ∈{ u,v ,...,v n } l ( b ), and let k ′ be an arbitrary natural number suchthat k ′ l ( u ) > s . Put k = k ′ l ( u ). Denote by P the set of all nonunit elementswhose length is less or equal than 10 k . For Q = { u, v , . . . , v n } ∪ P accordingto lemma 1 there exists the homomorphism ϕ of G onto a finite group such that ϕ A ∪ B ∪ Q is the injection and for each s ∈ S which is cyclically reduced and whoselength is greater than 1 each s -cycle in the graph Γ = Cay ( ϕ ( G ); { ϕ ( A ) ∪ ϕ ( B )has no ( k + 4)-near vertices and | ϕ ( u ) | > k .Fix a natural number m > m copies of the graph Γ: Γ i , i m . In the graph Γ we fix a u -cycle S = e · · · e r . Without loss of generality we consider that Lab( e ) ∈ A . Put p i = α ( e i ), Lab ( e i ) = u i (see Figure 1). For each i, i m, we delete edgesincident to p i whose labels belong to A and delete also edges labelled by theelements of A whose begin or end points are p ik +2 . For each i we shall denotethe obtained graph as Γ ′ i . 5et ψ be the bijection between the subgraphs A ( p ) and A ( p k +1 ) which saveslabels of edges and ψ ( p ) = p k +1 .Fix an arbitrary edge e of Γ from the subgraph A ( p ) such that the cor-responding edge e i of Γ i was deleted. Let q = α ( e ) , r = ω ( e ). For each i, i m, if q = p we connect the vertices q i and ψ ( r ) i +1 by the newedge f i . If r = p we connect r i and ψ ( q ) i +1 by the edge f i . In both cases thelabel of f i coincides with Lab ( e ), besides if q = p then f i goes away from q i and if r = p then f i goes into r i .Now we need to complement the structure of obtained graph for to get theaction graph of the group A ∗ B . But it will not be the free action graph.For each i, i m, let us to add one new vertex n i to the subgraph Γ ′ i .Consider an arbitrary edge e from A ( p k +2 ) such that the corresponding edge e i was deleted from Γ i . Put q = α ( e ) , r = ω ( e ). If q = p k +2 then connect thevertices r i and n i by the edge g i . If r = p k +2 then the new edge g i connectsthe vertices q i and n i . Put Lab ( g i ) = Lab ( e ). The begin point of g i coincideswith either n i or q i .For each c ∈ A ∪ B and for each vertex p of the obtained graph which is notincident to an edge with label c add a loop with label c going from p .If we fix j then the union of the graph Γ ′ j and A ( p jk +1 ) , A ( p jk +2 ) , A ( p j ) isdenoted by ∆ j .We constructed the new graph ∆ which contains subgraphs Γ ′ j and ∆ j andis the action graph of the group G .In the graph ∆ the u -cycle S ′ going from the vertex p has the length ( | ϕ ( u ) | − k ′ ) m . From the properties of the homomorphism ϕ it follows that | ϕ ( u ) | > k = 10 k ′ l ( u ) > k ′ . Hence | ϕ ∆ ( u ) | > ( | ϕ ( u ) | − k ′ ) m > m .Let us to prove that for each i and for each v i -cycle T in the graph ∆all vertices of T belong to two subgraphs ∆ j , ∆ j +1 for some j . Supposethe contrary. That is we suppose that there exist pairwise different numbers j , j , j such that the vertices of T belong to all three subgraphs ∆ j , ∆ j , ∆ j .Note that different subgraphs ∆ k , ∆ k has the nonempty intersection if andonly if | k − k | = 1 and their intersection equals the subgraph A ( p l ) since m > l is equal to either k or k . So if ∆ j , ∆ j , ∆ j contain verticesof T there exists the number j such that the subgraphs ∆ j , ∆ j +1 , ∆ j +2 containthe vertices of T and there exists the path R which is the part of T and whichbelongs to ∆ j ∪ ∆ j +1 ∪ ∆ j +2 , R goes away from the vertex of ∆ j and goes intothe vertex of ∆ j +2 (indices are modulo m ).From the properties of R it follows that R contains its first and the last edges t j , r j correspondingly such that t j ∈ A ( p j ) , ω ( t j ) = p j +1 k +2 , r j ∈ A ( p j +11 ) , ω ( r j ) = p j +2 k +2 , and the rest edges of R are in ∆ j +1 .Because of our supposition that T goes from ∆ j +1 into ∆ j +2 it is possibleto deduce that R contains the subpath s · · · s l such that s · · · s l − belongsto Γ ′ j +1 and edges s , s l satisfy the following properties: α ( s ) ∈ A ( n j +1 ) ∪ B ( p j +1 k +2 ) , ω ( s l ) ∈ A ( p j +11 ) ∪ B ( p j +12 ).Denote the path e j +11 e j +12 · · · e j +1 k +1 as S u and s · · · s l − as S v i (see Figure2). Note that ρ ( α ( S u ) , α ( S v i )) , ρ ( ω ( S u ) , ω ( S v i )) ρ is6igure 2: The graph ∆taken with respect to Γ i ). Besides S v i is a part of some v i -cycle, S u is a partof the u -cycle S ′ . Since the elements u, v i of the group A ∗ B do not belongto conjugate cyclic subgroups and the length of the path S u is greater than10 s = 10 max z ∈{ u,v ,...,v n } ( l ( z )) the paths S v i and S u are different.7uppose that the length of the path S v i is less or equal than k +4 = l ( S u )+3.The paths S v i and S u and perhaps several edges whose number is less than 4compose the loop. Let g be the label of this loop. Then g is an element ofthe group G whose length is less or equal than 2 l ( S u ) + 6 = 2 k + 8 < k and ϕ ( g ) = 1. But this contradicts the condition on ϕ and the set Q . Thus the lengthof the path S v i is greater than k + 4 = l ( S u ) + 3. By the symmetry we may alsoassume that the length of the path T \ S v i is greater than k + 4: the structure ofthe part of T in ∆ j +2 is the same as in ∆ j +1 . But in this case ρ ( α ( S v i ) , ω ( S v i )) min( l ( S u ) + 3 , l ( S v i ) , l ( T \ S v i )) = min( k + 4 , l ( S v i ) , l ( T \ S v i )) = k + 4, since l ( S v i ) , l ( T ′ \ R ′ ) > k + 4. So the v i -cycle T containing S v i has ( k + 4)-nearvertices. This also contradicts the conditions on ϕ .Thus it is proved that for each i and for each v i -cycle T in the graph ∆there exists j, j m, such that all vertices of T are contained in ∆ j ∪ ∆ j +1 (indices are modulo m ). We deduce also that each u -cycle of ∆ which does notstart at p belongs to two subgraphs ∆ k , ∆ k +1 . This can be established bythe same way as it was shown that the analogical statement is true for v i -cycles.Now we shall denote the obtained graph ∆ for number m as ∆ ′ m . Considerthe set of graphs Λ m = ∆ ′ m , m = 1 , , .. .We shall show now that | ϕ Λ m ( v i ) | equals some constant number K i whichdoes not depend on m . Let R i,m be the set of lengths of all v i -cycles of Λ m . Thelocal structure of Λ m is the same: using the above notations and regarding thatΛ m is the union of ∆ , . . . , ∆ m it is obvious that the subgraphs ∆ k ∪ ∆ k +1 and∆ l ∪ ∆ l +1 are isomorphic and do not depend on m . Hence R i,m coincides withthe set of lengths of v i -cycles concentrated in ∆ ∪ ∆ and thereby R i,m = R i,t for all m, t . The same reasonings are true for all u -cycles of Λ m except for the u -cycle whose length is the multiple of 3 m so | ϕ Λ m ( u ) | = mK for some constant K which does not depend on m .Proposition is proved and therefore the theorem is also proved. Acknowledgements
I am grateful to Anton A. Klyachko, Ashot Minasyan, Denis Osin and HenryWilton for valuable conversations and for information about omnipotence.
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