Pub Date : 2023-10-19DOI: 10.46298/jgcc.2023.15.1.11315
Guba, Victor
This is a survey of our recent results on the amenability problem for Thompson's group $F$. They mostly concern esimating the density of finite subgraphs in Cayley graphs of $F$ for various systems of generators, and also equations in the group ring of $F$. We also discuss possible approaches to solve the problem in both directions.
{"title":"Amenability problem for Thompson's group $F$: state of the art","authors":"Guba, Victor","doi":"10.46298/jgcc.2023.15.1.11315","DOIUrl":"https://doi.org/10.46298/jgcc.2023.15.1.11315","url":null,"abstract":"This is a survey of our recent results on the amenability problem for Thompson's group $F$. They mostly concern esimating the density of finite subgraphs in Cayley graphs of $F$ for various systems of generators, and also equations in the group ring of $F$. We also discuss possible approaches to solve the problem in both directions.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135666569","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}
Pub Date : 2023-09-28DOI: 10.46298/jgcc.2023.15.1.11728
Michal Ferov, Mark Pengitore
In this article, we study the asymptotic behaviour of conjugacy separability for wreath products of abelian groups. We fully characterise the asymptotic class in the case of lamplighter groups and give exponential upper and lower bounds for generalised lamplighter groups. In the case where the base group is infinite, we give superexponential lower and upper bounds. We apply our results to obtain lower bounds for conjugacy depth functions of various wreath products of groups where the acting group is not abelian.
{"title":"Bounding conjugacy depth functions for wreath products of finitely generated abelian groups","authors":"Michal Ferov, Mark Pengitore","doi":"10.46298/jgcc.2023.15.1.11728","DOIUrl":"https://doi.org/10.46298/jgcc.2023.15.1.11728","url":null,"abstract":"In this article, we study the asymptotic behaviour of conjugacy separability for wreath products of abelian groups. We fully characterise the asymptotic class in the case of lamplighter groups and give exponential upper and lower bounds for generalised lamplighter groups. In the case where the base group is infinite, we give superexponential lower and upper bounds. We apply our results to obtain lower bounds for conjugacy depth functions of various wreath products of groups where the acting group is not abelian.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135387425","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}
Pub Date : 2023-08-28DOI: 10.46298/jgcc.2023..12200
A. Gaglione, D. Spellman
The Heisenberg group, here denoted $H$, is the group of all $3times 3$ upper unitriangular matrices with entries in the ring $mathbb{Z}$ of integers. A.G. Myasnikov posed the question of whether or not the universal theory of $H$, in the language of $H$, is axiomatized, when the models are restricted to $H$-groups, by the quasi-identities true in $H$ together with the assertion that the centralizers of noncentral elements be abelian. Based on earlier published partial results we here give a complete proof of a slightly stronger result.
{"title":"An axiomatization for the universal theory of the Heisenberg group","authors":"A. Gaglione, D. Spellman","doi":"10.46298/jgcc.2023..12200","DOIUrl":"https://doi.org/10.46298/jgcc.2023..12200","url":null,"abstract":"The Heisenberg group, here denoted $H$, is the group of all $3times 3$ upper\u0000unitriangular matrices with entries in the ring $mathbb{Z}$ of integers. A.G.\u0000Myasnikov posed the question of whether or not the universal theory of $H$, in\u0000the language of $H$, is axiomatized, when the models are restricted to\u0000$H$-groups, by the quasi-identities true in $H$ together with the assertion\u0000that the centralizers of noncentral elements be abelian. Based on earlier\u0000published partial results we here give a complete proof of a slightly stronger\u0000result.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89856287","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}
Pub Date : 2022-08-27DOI: 10.46298/jgcc.2023.14.2.10019
Lindsay Marjanski, Vincent Solon, Frank Zheng, Kathleen Zopff
In this paper, we study geodesic growth of numbered graph products; these are a generalization of right-angled Coxeter groups, defined as graph products of finite cyclic groups. We first define a graph-theoretic condition called link-regularity, as well as a natural equivalence amongst link-regular numbered graphs, and show that numbered graph products associated to link-regular numbered graphs must have the same geodesic growth series. Next, we derive a formula for the geodesic growth of right-angled Coxeter groups associated to link-regular graphs. Finally, we find a system of equations that can be used to solve for the geodesic growth of numbered graph products corresponding to link-regular numbered graphs that contain no triangles and have constant vertex numbering.
{"title":"Geodesic Growth of Numbered Graph Products","authors":"Lindsay Marjanski, Vincent Solon, Frank Zheng, Kathleen Zopff","doi":"10.46298/jgcc.2023.14.2.10019","DOIUrl":"https://doi.org/10.46298/jgcc.2023.14.2.10019","url":null,"abstract":"In this paper, we study geodesic growth of numbered graph products; these are\u0000a generalization of right-angled Coxeter groups, defined as graph products of\u0000finite cyclic groups. We first define a graph-theoretic condition called\u0000link-regularity, as well as a natural equivalence amongst link-regular numbered\u0000graphs, and show that numbered graph products associated to link-regular\u0000numbered graphs must have the same geodesic growth series. Next, we derive a\u0000formula for the geodesic growth of right-angled Coxeter groups associated to\u0000link-regular graphs. Finally, we find a system of equations that can be used to\u0000solve for the geodesic growth of numbered graph products corresponding to\u0000link-regular numbered graphs that contain no triangles and have constant vertex\u0000numbering.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"72 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83545543","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}
Pub Date : 2021-12-02DOI: 10.46298/jgcc.2021.13.2.8796
B. Fine, A. Gaglione, M. Kreuzer, G. Rosenberger, D. Spellman
In [FGRS1,FGRS2] the relationship between the universal and elementary theory of a group ring $R[G]$ and the corresponding universal and elementary theory of the associated group $G$ and ring $R$ was examined. Here we assume that $R$ is a commutative ring with identity $1 ne 0$. Of course, these are relative to an appropriate logical language $L_0,L_1,L_2$ for groups, rings and group rings respectively. Axiom systems for these were provided in [FGRS1]. In [FGRS1] it was proved that if $R[G]$ is elementarily equivalent to $S[H]$ with respect to $L_{2}$, then simultaneously the group $G$ is elementarily equivalent to the group $H$ with respect to $L_{0}$, and the ring $R$ is elementarily equivalent to the ring $S$ with respect to $L_{1}$. We then let $F$ be a rank $2$ free group and $mathbb{Z}$ be the ring of integers. Examining the universal theory of the free group ring ${mathbb Z}[F]$ the hazy conjecture was made that the universal sentences true in ${mathbb Z}[F]$ are precisely the universal sentences true in $F$ modified appropriately for group ring theory and the converse that the universal sentences true in $F$ are the universal sentences true in ${mathbb Z}[F]$ modified appropriately for group theory. In this paper we show this conjecture to be true in terms of axiom systems for ${mathbb Z}[F]$.
{"title":"The Axiomatics of Free Group Rings","authors":"B. Fine, A. Gaglione, M. Kreuzer, G. Rosenberger, D. Spellman","doi":"10.46298/jgcc.2021.13.2.8796","DOIUrl":"https://doi.org/10.46298/jgcc.2021.13.2.8796","url":null,"abstract":"In [FGRS1,FGRS2] the relationship between the universal and elementary theory\u0000of a group ring $R[G]$ and the corresponding universal and elementary theory of\u0000the associated group $G$ and ring $R$ was examined. Here we assume that $R$ is\u0000a commutative ring with identity $1 ne 0$. Of course, these are relative to an\u0000appropriate logical language $L_0,L_1,L_2$ for groups, rings and group rings\u0000respectively. Axiom systems for these were provided in [FGRS1]. In [FGRS1] it\u0000was proved that if $R[G]$ is elementarily equivalent to $S[H]$ with respect to\u0000$L_{2}$, then simultaneously the group $G$ is elementarily equivalent to the\u0000group $H$ with respect to $L_{0}$, and the ring $R$ is elementarily equivalent\u0000to the ring $S$ with respect to $L_{1}$. We then let $F$ be a rank $2$ free\u0000group and $mathbb{Z}$ be the ring of integers. Examining the universal theory\u0000of the free group ring ${mathbb Z}[F]$ the hazy conjecture was made that the\u0000universal sentences true in ${mathbb Z}[F]$ are precisely the universal\u0000sentences true in $F$ modified appropriately for group ring theory and the\u0000converse that the universal sentences true in $F$ are the universal sentences\u0000true in ${mathbb Z}[F]$ modified appropriately for group theory. In this paper\u0000we show this conjecture to be true in terms of axiom systems for ${mathbb\u0000Z}[F]$.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83076767","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}
Pub Date : 2021-06-22DOI: 10.46298/jgcc.2021.13.2.7617
A. Darbinyan, R. Grigorchuk, Asif Shaikh
For finitely generated subgroups $H$ of a free group $F_m$ of finite rank $m$, we study the language $L_H$ of reduced words that represent $H$ which is a regular language. Using the (extended) core of Schreier graph of $H$, we construct the minimal deterministic finite automaton that recognizes $L_H$. Then we characterize the f.g. subgroups $H$ for which $L_H$ is irreducible and for such groups explicitly construct ergodic automaton that recognizes $L_H$. This construction gives us an efficient way to compute the cogrowth series $L_H(z)$ of $H$ and entropy of $L_H$. Several examples illustrate the method and a comparison is made with the method of calculation of $L_H(z)$ based on the use of Nielsen system of generators of $H$.
{"title":"Finitely generated subgroups of free groups as formal languages and\u0000 their cogrowth","authors":"A. Darbinyan, R. Grigorchuk, Asif Shaikh","doi":"10.46298/jgcc.2021.13.2.7617","DOIUrl":"https://doi.org/10.46298/jgcc.2021.13.2.7617","url":null,"abstract":"For finitely generated subgroups $H$ of a free group $F_m$ of finite rank\u0000$m$, we study the language $L_H$ of reduced words that represent $H$ which is a\u0000regular language. Using the (extended) core of Schreier graph of $H$, we\u0000construct the minimal deterministic finite automaton that recognizes $L_H$.\u0000Then we characterize the f.g. subgroups $H$ for which $L_H$ is irreducible and\u0000for such groups explicitly construct ergodic automaton that recognizes $L_H$.\u0000This construction gives us an efficient way to compute the cogrowth series\u0000$L_H(z)$ of $H$ and entropy of $L_H$. Several examples illustrate the method\u0000and a comparison is made with the method of calculation of $L_H(z)$ based on\u0000the use of Nielsen system of generators of $H$.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74748874","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}
Pub Date : 2021-02-01DOI: 10.46298/jgcc.2021.13.1.7347
V. Yankovskiy
We find algebraic conditions on a group equivalent to the position of its Diophantine problem in the Chomsky Hierarchy. In particular, we prove that a finitely generated group has a context-free Diophantine problem if and only if it is finite.
{"title":"Groups with context-free Diophantine problem","authors":"V. Yankovskiy","doi":"10.46298/jgcc.2021.13.1.7347","DOIUrl":"https://doi.org/10.46298/jgcc.2021.13.1.7347","url":null,"abstract":"We find algebraic conditions on a group equivalent to the position of its\u0000Diophantine problem in the Chomsky Hierarchy. In particular, we prove that a\u0000finitely generated group has a context-free Diophantine problem if and only if\u0000it is finite.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86044712","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}
Pub Date : 2021-01-04DOI: 10.46298/jgcc.2021.13.2.7072
Jordan Sahattchieve
We generalize a result of Moon on the fibering of certain 3-manifolds over the circle. Our main theorem is the following: Let $M$ be a closed 3-manifold. Suppose that $G=pi_1(M)$ contains a finitely generated group $U$ of infinite index in $G$ which contains a non-trivial subnormal subgroup $Nneq mathbb{Z}$ of $G$, and suppose that $N$ has a composition series of length $n$ in which at least $n-1$ terms are finitely generated. Suppose that $N$ intersects nontrivially the fundamental groups of the splitting tori given by the Geometrization Theorem and that the intersections of $N$ with the fundamental groups of the geometric pieces are non-trivial and not isomorphic to $mathbb{Z}$. Then, $M$ has a finite cover which is a bundle over $mathbb{S}$ with fiber a compact surface $F$ such that $pi_1(F)$ and $U$ are commensurable.
{"title":"A fibering theorem for 3-manifolds","authors":"Jordan Sahattchieve","doi":"10.46298/jgcc.2021.13.2.7072","DOIUrl":"https://doi.org/10.46298/jgcc.2021.13.2.7072","url":null,"abstract":"We generalize a result of Moon on the fibering of certain 3-manifolds over\u0000the circle. Our main theorem is the following: Let $M$ be a closed 3-manifold.\u0000Suppose that $G=pi_1(M)$ contains a finitely generated group $U$ of infinite\u0000index in $G$ which contains a non-trivial subnormal subgroup $Nneq mathbb{Z}$\u0000of $G$, and suppose that $N$ has a composition series of length $n$ in which at\u0000least $n-1$ terms are finitely generated. Suppose that $N$ intersects\u0000nontrivially the fundamental groups of the splitting tori given by the\u0000Geometrization Theorem and that the intersections of $N$ with the fundamental\u0000groups of the geometric pieces are non-trivial and not isomorphic to\u0000$mathbb{Z}$. Then, $M$ has a finite cover which is a bundle over $mathbb{S}$\u0000with fiber a compact surface $F$ such that $pi_1(F)$ and $U$ are\u0000commensurable.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89686536","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}
Pub Date : 2020-09-22DOI: 10.46298/jgcc.2022.14.1.9776
A. Levine
We give an algorithm that decides whether a single equation in a group that is virtually a class $2$ nilpotent group with a virtually cyclic commutator subgroup, such as the Heisenberg group, admits a solution. This generalises the work of Duchin, Liang and Shapiro to finite extensions.
{"title":"Equations in virtually class 2 nilpotent groups","authors":"A. Levine","doi":"10.46298/jgcc.2022.14.1.9776","DOIUrl":"https://doi.org/10.46298/jgcc.2022.14.1.9776","url":null,"abstract":"We give an algorithm that decides whether a single equation in a group that\u0000is virtually a class $2$ nilpotent group with a virtually cyclic commutator\u0000subgroup, such as the Heisenberg group, admits a solution. This generalises the\u0000work of Duchin, Liang and Shapiro to finite extensions.","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"1635 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76281044","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}
Pub Date : 2020-05-11DOI: 10.46298/jgcc.2020.12.2.6649
Ansari Abdullah, A. Mahalanobis, V. Mallick
The elliptic curve discrete logarithm problem is considered a secure cryptographic primitive. The purpose of this paper is to propose a paradigm shift in attacking the elliptic curve discrete logarithm problem. In this paper, we will argue that initial minors are a viable way to solve this problem. This paper will present necessary algorithms for this attack. We have written a code to verify the conjecture of initial minors using Schur complements. We were able to solve the problem for groups of order up to $2^{50}$. Comment: 13 pages; revised for publication
{"title":"A new method for solving the elliptic curve discrete logarithm problem","authors":"Ansari Abdullah, A. Mahalanobis, V. Mallick","doi":"10.46298/jgcc.2020.12.2.6649","DOIUrl":"https://doi.org/10.46298/jgcc.2020.12.2.6649","url":null,"abstract":"The elliptic curve discrete logarithm problem is considered a secure\u0000cryptographic primitive. The purpose of this paper is to propose a paradigm\u0000shift in attacking the elliptic curve discrete logarithm problem. In this\u0000paper, we will argue that initial minors are a viable way to solve this\u0000problem. This paper will present necessary algorithms for this attack. We have\u0000written a code to verify the conjecture of initial minors using Schur\u0000complements. We were able to solve the problem for groups of order up to\u0000$2^{50}$.\u0000Comment: 13 pages; revised for publication","PeriodicalId":41862,"journal":{"name":"Groups Complexity Cryptology","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75605887","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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