Abstract. A cylinder is the set of infinite words with fixed prefix u. A double-cylinder is “the same” for bi-infinite words. We show that for every word u and any automorphism of the free group F the image is a finite union of cylinders. The analogous statement is true for double cylinders. We give (a) an algorithm, and (b) a precise formula which allows one to determine this finite union of cylinders.
{"title":"Cylinders, multi-cylinders and the induced action of Aut(Fn)","authors":"Fedaa Ibrahim","doi":"10.1515/gcc-2012-0017","DOIUrl":"https://doi.org/10.1515/gcc-2012-0017","url":null,"abstract":"Abstract. A cylinder is the set of infinite words with fixed prefix u. A double-cylinder is “the same” for bi-infinite words. We show that for every word u and any automorphism of the free group F the image is a finite union of cylinders. The analogous statement is true for double cylinders. We give (a) an algorithm, and (b) a precise formula which allows one to determine this finite union of cylinders.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"78 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114129260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract. Cyclic words are equivalence classes of cyclic permutations of ordinary words. When a group is given by a rewriting relation, a rewriting system on cyclic words is induced, which is used to construct algorithms to find minimal length elements of conjugacy classes in the group. These techniques are applied to the universal groups of Stallings pregroups and in particular to free products with amalgamation, HNN-extensions and virtually free groups, to yield simple and intuitive algorithms and proofs of conjugacy criteria.
{"title":"Cyclic rewriting and conjugacy problems","authors":"V. Diekert, A. Duncan, A. Myasnikov","doi":"10.1515/gcc-2012-0020","DOIUrl":"https://doi.org/10.1515/gcc-2012-0020","url":null,"abstract":"Abstract. Cyclic words are equivalence classes of cyclic permutations of ordinary words. When a group is given by a rewriting relation, a rewriting system on cyclic words is induced, which is used to construct algorithms to find minimal length elements of conjugacy classes in the group. These techniques are applied to the universal groups of Stallings pregroups and in particular to free products with amalgamation, HNN-extensions and virtually free groups, to yield simple and intuitive algorithms and proofs of conjugacy criteria.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134060220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract. The objective of this work is to survey several digital signatures proposed in the last decade using non-commutative groups and rings and propose a digital signature using non-commutative groups and analyze its security.
{"title":"Non-commutative digital signatures","authors":"Delaram Kahrobaei, Charalambos Koupparis","doi":"10.1515/gcc-2012-0019","DOIUrl":"https://doi.org/10.1515/gcc-2012-0019","url":null,"abstract":"Abstract. The objective of this work is to survey several digital signatures proposed in the last decade using non-commutative groups and rings and propose a digital signature using non-commutative groups and analyze its security.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"84 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131435497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract. We investigate the cogrowth and distribution of geodesics in R. Thompson's group F.
摘要研究了R. Thompson’s群中测地线的共生长和分布。
{"title":"On the cogrowth of Thompson's group F","authors":"M. Elder, A. Rechnitzer, T. Wong","doi":"10.1515/gcc-2012-0018","DOIUrl":"https://doi.org/10.1515/gcc-2012-0018","url":null,"abstract":"Abstract. We investigate the cogrowth and distribution of geodesics in R. Thompson's group F.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"96 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124965837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let X be a locally finite tree and let G = Aut(X). Then G is naturally a locally compact group. A discrete subgroup Γ ≤ G is called an X-lattice, or a tree lattice if Γ has finite covolume in G. The lattice Γ is encoded in a graph of finite groups of finite volume. We describe several methods for constructing a pair of X-lattices (Γ′, Γ) with Γ ≤ Γ′, starting from ‘edge-indexed graphs’ (A′, i′) and (A, i) which correspond to the edge-indexed quotient graphs of their (common) universal covering tree by Γ′ and Γ respectively. We determine when finite sheeted topological coverings of edge-indexed graphs give rise to a pair of lattice subgroups (Γ, Γ′) with an inclusion Γ ≤ Γ′. We describe when a ‘full graph of subgroups’ and a ‘subgraph of subgroups’ constructed from the graph of groups encoding a lattice Γ′ gives rise to a lattice subgroup Γ and an inclusion Γ ≤ Γ′. We show that a nonuniform X-lattice Γ contains an infinite chain of subgroups Λ1 ≤ Λ2 ≤ Λ3 ≤ ⋯ where each Λ k is a uniform Xk -lattice and Xk is a subtree of X. Our techniques, which are a combination of topological graph theory, covering theory for graphs of groups, and covering theory for edge-indexed graphs, have no analog in classical covering theory. We obtain a local necessary condition for extending coverings of edgeindexed graphs to covering morphisms of graphs of groups with abelian groupings. This gives rise to a combinatorial method for constructing lattice inclusions Γ ≤ Γ′ ≤ H ≤ G with abelian vertex stabilizers inside a closed and hence locally compact subgroup H of G. We give examples of lattice pairs Γ ≤ Γ′ when H is a simple algebraic group of K-rank 1 over a nonarchimedean local field K and a rank 2 locally compact complete Kac–Moody group over a finite field. We also construct an infinite descending chain of lattices ⋯ ≤ Γ2 ≤ Γ1 ≤ Γ ≤ H≤ G with abelian vertex stabilizers.
{"title":"Tree lattice subgroups","authors":"Lisa Carbone, Leigh Cobbs, G. Rosenberg","doi":"10.1515/gcc.2011.001","DOIUrl":"https://doi.org/10.1515/gcc.2011.001","url":null,"abstract":"Abstract Let X be a locally finite tree and let G = Aut(X). Then G is naturally a locally compact group. A discrete subgroup Γ ≤ G is called an X-lattice, or a tree lattice if Γ has finite covolume in G. The lattice Γ is encoded in a graph of finite groups of finite volume. We describe several methods for constructing a pair of X-lattices (Γ′, Γ) with Γ ≤ Γ′, starting from ‘edge-indexed graphs’ (A′, i′) and (A, i) which correspond to the edge-indexed quotient graphs of their (common) universal covering tree by Γ′ and Γ respectively. We determine when finite sheeted topological coverings of edge-indexed graphs give rise to a pair of lattice subgroups (Γ, Γ′) with an inclusion Γ ≤ Γ′. We describe when a ‘full graph of subgroups’ and a ‘subgraph of subgroups’ constructed from the graph of groups encoding a lattice Γ′ gives rise to a lattice subgroup Γ and an inclusion Γ ≤ Γ′. We show that a nonuniform X-lattice Γ contains an infinite chain of subgroups Λ1 ≤ Λ2 ≤ Λ3 ≤ ⋯ where each Λ k is a uniform Xk -lattice and Xk is a subtree of X. Our techniques, which are a combination of topological graph theory, covering theory for graphs of groups, and covering theory for edge-indexed graphs, have no analog in classical covering theory. We obtain a local necessary condition for extending coverings of edgeindexed graphs to covering morphisms of graphs of groups with abelian groupings. This gives rise to a combinatorial method for constructing lattice inclusions Γ ≤ Γ′ ≤ H ≤ G with abelian vertex stabilizers inside a closed and hence locally compact subgroup H of G. We give examples of lattice pairs Γ ≤ Γ′ when H is a simple algebraic group of K-rank 1 over a nonarchimedean local field K and a rank 2 locally compact complete Kac–Moody group over a finite field. We also construct an infinite descending chain of lattices ⋯ ≤ Γ2 ≤ Γ1 ≤ Γ ≤ H≤ G with abelian vertex stabilizers.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126278900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let g, p ∈ [0↑∞[, the set of non-negative integers. Let A g,p denote the group consisting of all those automorphisms of the free group on t [1↑p] ∪ x [1↑g] ∪ y [1↑g] which fix the element ∏ j∈[p↓1] tj ∏ i∈[1↑g][xi, yi ] and permute the set of conjugacy classes {[tj ] : j ∈ [1↑p]}. Labruère and Paris, building on work of Artin, Magnus, Dehn, Nielsen, Lickorish, Zieschang, Birman, Humphries, and others, showed that A g,p is generated by what is called the ADLH set. We use methods of Zieschang and McCool to give a self-contained, algebraic proof of this result. (Labruère and Paris also gave defining relations for the ADLH set in A g,p ; we do not know an algebraic proof of this for g ⩾ 2.) Consider an orientable surface S g,p of genus g with p punctures, with (g, p) ≠ (0, 0), (0, 1). The algebraic mapping-class group of S g,p , denoted , is defined as the group of all those outer automorphisms of 〈t [1↑p] ∪ x [1↑g] ∪ y [1↑g] | ∏ j∈[p↓1] tj ∏ i∈[1↑g][xi, yi ]〉 which permute the set of conjugacy classes . It now follows from a result of Nielsen that is generated by the image of the ADLH set together with a reflection. This gives a new way of seeing that equals the (topological) mapping-class group of S g,p , along lines suggested by Magnus, Karrass, and Solitar in 1966.
{"title":"The Zieschang–McCool method for generating algebraic mapping-class groups","authors":"Lluís Bacardit, Warren Dicks","doi":"10.1515/GCC.2011.007","DOIUrl":"https://doi.org/10.1515/GCC.2011.007","url":null,"abstract":"Abstract Let g, p ∈ [0↑∞[, the set of non-negative integers. Let A g,p denote the group consisting of all those automorphisms of the free group on t [1↑p] ∪ x [1↑g] ∪ y [1↑g] which fix the element ∏ j∈[p↓1] tj ∏ i∈[1↑g][xi, yi ] and permute the set of conjugacy classes {[tj ] : j ∈ [1↑p]}. Labruère and Paris, building on work of Artin, Magnus, Dehn, Nielsen, Lickorish, Zieschang, Birman, Humphries, and others, showed that A g,p is generated by what is called the ADLH set. We use methods of Zieschang and McCool to give a self-contained, algebraic proof of this result. (Labruère and Paris also gave defining relations for the ADLH set in A g,p ; we do not know an algebraic proof of this for g ⩾ 2.) Consider an orientable surface S g,p of genus g with p punctures, with (g, p) ≠ (0, 0), (0, 1). The algebraic mapping-class group of S g,p , denoted , is defined as the group of all those outer automorphisms of 〈t [1↑p] ∪ x [1↑g] ∪ y [1↑g] | ∏ j∈[p↓1] tj ∏ i∈[1↑g][xi, yi ]〉 which permute the set of conjugacy classes . It now follows from a result of Nielsen that is generated by the image of the ADLH set together with a reflection. This gives a new way of seeing that equals the (topological) mapping-class group of S g,p , along lines suggested by Magnus, Karrass, and Solitar in 1966.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125063086","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract. In 2003 Cohn and Umans introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets of a group satisfying the triple product property (TPP) as a means to bound the exponent of matrix multiplication. We present two new characterizations of the TPP, which are used for theoretical considerations and for TPP test algorithms. We describe the algorithms for all known TPP tests and present the runtime differences between their GAP implementations. We prove that the search for non-trivial-sized TPP triples of subgroups of a given group can be restricted to the set of its non-normal subgroups, and apply this, together with other preconditions, to describe brute-force search algorithms for largest-sized TPP triples of subgroups and subsets. In addition we present the results of the subset brute-force search for all groups of order up to 32 and selected results of the subgroup brute-force search for 2-groups, and . Our results for the groups and suggest tentative answers to certain questions posed by Cohn and Umans.
{"title":"Search and test algorithms for triple product property triples","authors":"Ivo Hedtke, Sandeep Murthy","doi":"10.1515/gcc-2012-0006","DOIUrl":"https://doi.org/10.1515/gcc-2012-0006","url":null,"abstract":"Abstract. In 2003 Cohn and Umans introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets of a group satisfying the triple product property (TPP) as a means to bound the exponent of matrix multiplication. We present two new characterizations of the TPP, which are used for theoretical considerations and for TPP test algorithms. We describe the algorithms for all known TPP tests and present the runtime differences between their GAP implementations. We prove that the search for non-trivial-sized TPP triples of subgroups of a given group can be restricted to the set of its non-normal subgroups, and apply this, together with other preconditions, to describe brute-force search algorithms for largest-sized TPP triples of subgroups and subsets. In addition we present the results of the subset brute-force search for all groups of order up to 32 and selected results of the subgroup brute-force search for 2-groups, and . Our results for the groups and suggest tentative answers to certain questions posed by Cohn and Umans.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115053171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Bowen defined the growth rate of an endomorphism of a finitely generated group and related it to the entropy of a map ƒ : M ↦ M on a compact manifold. In this note we study the purely group theoretic aspects of the growth rate of an endomorphism of a finitely generated group. We show that it is finite and bounded by the maximum length of the image of a generator. An equivalent formulation is given that ties the growth rate of an endomorphism to an increasing chain of subgroups. We then consider the relationship between growth rate of an endomorphism on a whole group and the growth rate restricted to a subgroup or considered on a quotient. We use these results to compute the growth rates on direct and semidirect products. We then calculate the growth rate of endomorphisms on several different classes of groups including abelian and nilpotent.
Bowen定义了有限生成群的自同态的增长率，并将其与紧流形上映射f: M × M的熵联系起来。本文研究了有限生成群的自同态增长率的纯群论问题。我们证明了它是有限的，并以一个生成器图像的最大长度为界。给出了将自同态的增长率与子群的递增链联系起来的等价公式。然后，我们考虑了整群上的自同态的增长率与子群上的增长率或商上的增长率之间的关系。我们用这些结果来计算直接产品和半直接产品的增长率。然后我们计算了包括阿贝尔群和幂零群在内的几个不同类别群上的自同态的增长率。
{"title":"Growth rate of an endomorphism of a group","authors":"K. Falconer, B. Fine, Delaram Kahrobaei","doi":"10.1515/gcc.2011.011","DOIUrl":"https://doi.org/10.1515/gcc.2011.011","url":null,"abstract":"Abstract Bowen defined the growth rate of an endomorphism of a finitely generated group and related it to the entropy of a map ƒ : M ↦ M on a compact manifold. In this note we study the purely group theoretic aspects of the growth rate of an endomorphism of a finitely generated group. We show that it is finite and bounded by the maximum length of the image of a generator. An equivalent formulation is given that ties the growth rate of an endomorphism to an increasing chain of subgroups. We then consider the relationship between growth rate of an endomorphism on a whole group and the growth rate restricted to a subgroup or considered on a quotient. We use these results to compute the growth rates on direct and semidirect products. We then calculate the growth rate of endomorphisms on several different classes of groups including abelian and nilpotent.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"186 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2011-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124748743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We prove that the complexity of the conjugacy problems for wreath products and for free solvable groups is decidable in polynomial time. For the wreath product AwrB, we must assume the decidability in polynomial time of the conjugacy problems for A and B and of the power problem in B. Using this result and properties of the Magnus embedding, we show that the conjugacy and conjugacy search problems in free solvable groups are computable in polynomial time.
{"title":"Polynomial time conjugacy in wreath products and free solvable groups","authors":"S. Vassileva","doi":"10.1515/gcc.2011.005","DOIUrl":"https://doi.org/10.1515/gcc.2011.005","url":null,"abstract":"Abstract We prove that the complexity of the conjugacy problems for wreath products and for free solvable groups is decidable in polynomial time. For the wreath product AwrB, we must assume the decidability in polynomial time of the conjugacy problems for A and B and of the power problem in B. Using this result and properties of the Magnus embedding, we show that the conjugacy and conjugacy search problems in free solvable groups are computable in polynomial time.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125765678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract. For every odd prime and every integer , there is a Heisenberg group of order that has pairwise nonisomorphic quotients of order . Yet, these quotients are virtually indistinguishable. They have isomorphic character tables, every conjugacy class of a non-central element has the same size, and every element has order at most . They are also directly and centrally indecomposable and of the same indecomposability type. Nevertheless, there is a polynomial-time algorithm to test for isomorphisms between these groups.
{"title":"Isomorphism in expanding families of indistinguishable groups","authors":"M. Lewis, James B. Wilson","doi":"10.1515/gcc-2012-0008","DOIUrl":"https://doi.org/10.1515/gcc-2012-0008","url":null,"abstract":"Abstract. For every odd prime and every integer , there is a Heisenberg group of order that has pairwise nonisomorphic quotients of order . Yet, these quotients are virtually indistinguishable. They have isomorphic character tables, every conjugacy class of a non-central element has the same size, and every element has order at most . They are also directly and centrally indecomposable and of the same indecomposability type. Nevertheless, there is a polynomial-time algorithm to test for isomorphisms between these groups.","PeriodicalId":119576,"journal":{"name":"Groups Complex. Cryptol.","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133687597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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