Valerij Bardakov, Leonid Bokut, Andrei Vesnin
We study the conjugacy problem in cyclic extensions of free groups. It is shown that the conjugacy problem is solvable in split extensions of finitely generated free groups by virtually inner automorphisms. An algorithm for construction of the unique representative (the conjugacy normal form) for each conjugacy class is given.
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Triangle-free cyclic conjugacy class graph of a finite group
本文通过将元素替换为共轭类来推广增强幂图，并确定了该图无三角形的条件。
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.
In this paper, we set $η(G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We prove that if $G$ is a $p$-group of order $p^n$ and nilpotence class $l$, then $η(G)$ is bounded below by a linear function in $n/l$.
Conjugacy in a family of free-by-cyclic groups
本文研究了一族多项式增长的自由群与循环群半直积群 $H_m$ 的共轭问题，证明了其共轭子长度函数为线性，并给出了共轭问题及共轭搜索问题的多项式时间解。
In this paper, we set $η(G)$ to be the number of conjugacy classes of maximal cyclic subgroups of a finite group $G$. We compute $η(G)$ for all metacyclic $p$-groups. We show that if $G$ is a metacyclic $p$-group of order $p^n$ that is not dihedral, generalized quaternion, or semi-dihedral, then $η(G) \ge n-2$, and we determine when equality holds.
We show that the conjugacy problem is solvable in [finitely generated free]-by-cyclic groups, by using a result of O. Maslakova that one can algorithmically find generating sets for the fixed subgroups of free group automorphisms, and one of P. Brinkmann that one can determine whether two cyclic words in a free group are mapped to each other by some power of a given automorphism. The algorithm effectively computes a conjugating element, if it exists. We also solve the power conjugacy problem and give an algorithm to recognize if two given elements of a finitely generated free group are Reideme…
We give another proof of an observation of Thévenaz \cite{T1989} and present a fusion system version of it. Namely, for a saturated fusion system $\CF$ on a finite $p$-group $S$, we show that the number of the $\CF$-conjugacy classes of cyclic subgroups of $S$ is equal to the rank of certain square matrices of numbers of orbits, coming from characteristic bisets, the characteristic idempotent and finite groups realizing the fusion system $\CF$ as in our previous work \cite{P2010}.
Let G be a finite group and H a normal subgroup such that G/H is cyclic. Given a conjugacy class g^G of G we define its centralizing subgroup to be HC_G(g). Let K be such that H\le K\le G. We show that the G-conjugacy classes contained in K whose centralizing subgroup is K, are equally distributed between the cosets of H in K. The proof of this result is entirely elementary. As an application we find expressions for the number of conjugacy classes of K under its own action, in terms of quantities relating only to the action of G.
Products of nonprimary cyclic conjugacy classes in the general linear group
研究一般线性群中两个非主循环共轭类的乘积，证明其包含所有具有特定行列式的非标量矩阵。
Conjugacy classes of completely reducible cyclic subgroups of GL$(2, q)$
本文针对素数幂$q$及不被素数$p$整除的正整数$m$，确定了一般线性群GL$(2,q)$中阶为$m$的完全可约循环子群的共轭类数目。
In this paper, we set $η(G)$ to be the number of conjugacy classes of maximal cyclic subgroups of $G$. We consider $η$ and direct and semi-direct products. We characterize the normal subgroups $N$ so that $η(G/N) = η(G)$. We set $G^- = \{ g \in G \mid \langle g \rangle {\rm ~is~not ~maximal~cyclic} \}$. We show if $\langle G^- \rangle < G$, then $G/\langle G^- \rangle$ is either (1) an elementary abelian $p$-group for some prime $p$, (2) a Frobenius group whose Frobenius kernel is a $p$-group of exponent $p$ and a Frobenius complement has order $q$ for distinct primes $p$ and $q$, or (3) isomo…
The general {\bf surface group conjecture} asks whether a one-relator group where every subgroup of finite index is again one-relator and every subgroup of infinite index is free (property IF) is a surface group. We resolve several related conjectures given in [FKMRR]. First we obtain the Surface Group Conjecture B for cyclically pinched and conjugacy pinched one-relator groups. That is: if $G$ is a cyclically pinched one-relator group or conjugacy pinched one-relator group satisfying property IF then $G$ is free, a surface group or a solvable Baumslag-Solitar Group. Further combining results …
Free-by-cyclic groups are conjugacy separable
本文证明了所有有限生成的自由-循环群都是共轭可分离的，并展示了外自同构群的残余有限性。
Conjugacy in fibre products, distortion, and the geometry of cyclic subgroups
研究无挠双曲群中纤维积的共轭问题复杂性，通过建立不等式，将纤维积的共轭长度函数与\(Q\)中循环子群几何等联系起来，为扩展相关函数库提供工具。
In this paper we introduce a polynomial time algorithm that solves both the conjugacy decision and search problems in free abelian-by-infinite cyclic groups where the input is elements in normal form. We do this by adapting the work of Bogopolski, Martino, Maslakova, and Ventura in \cite{bogopolski2006conjugacy} and Bogopolski, Martino, and Ventura in \cite{bogopolski2010orbit}, to free abelian-by-infinite cyclic groups, and in certain cases apply a polynomial time algorithm for the orbit problem over $\Z^n$ by Kannan and Lipton.
On generalized (twisted) conjugacy separability of some extensions of groups
引入广义共轭可分性性质，证明群$G$在有限与循环扩张下扭曲共轭、Brinkmann共轭等可分性的等价与蕴含关系，并应用于自由群等群类。
Rivin conjectured that the conjugacy growth series of a hyperbolic group is rational if and only if the group is virtually cyclic. Ciobanu, Hermiller, Holt and Rees proved that the conjugacy growth series of a virtually cyclic group is rational. Here we present the proof confirming the other direction of the conjecture, by showing that the conjugacy growth series of a non-elementary hyperbolic group is transcendental. We also present and prove some variations of Rivin's conjecture for commensurability classes and primitive conjugacy classes. We then explore Rivin's conjecture for finitely gene…
Let W be a Coxeter group. In this paper, we establish that, up to going to some finite index normal subgroup W_0 of W, any two cyclically reduced expressions of conjugate elements of W_0 only differ by a sequence of braid relations and cyclic shifts. This thus provides a simple description of conjugacy classes in W_0. As a byproduct of our methods, we also obtain a characterisation of straight elements of W, namely of those elements w in W for which $\ell(w^n)=n\ell(w)$ for any integer n. In particular, we generalise previous characterisations of straight elements within the class of so-called…
The Conjugacy Ratio of Abelian-by-Cyclic Groups
研究了阿贝尔-循环群的共轭比，证明了该比为零除非群是虚拟阿贝尔群，并展示了非虚拟阿贝尔群的特殊情况。
We present a solution to the conjugacy decision problem and the conjugacy search problem in Garside groups, which is theoretically simpler than the usual one, with no loss of efficiency. This is done by replacing the well known cycling and decycling operations by a new one, called cyclic sliding, which appears to be a more natural choice. We give an analysis of the complexity of our algorithm in terms of fundamental operations with simple elements, so our analysis is valid for every Garside group. This paper intends to be self-contained, not requiring any previous knowledge of prior algorithms…
Let $n\geq 3$. In this paper we deal with the conjugacy problem in the Artin braid group quotient $B_n/[P_n,P_n]$. To solve it we use systems of equations over the integers arising from the action of $B_n/[P_n,P_n]$ over the abelianization of the pure Artin braid group $P_n/[P_n,P_n]$. Using this technique we also realize explicitly infinite virtually cyclic subgroups in $B_n/[P_n,P_n]$.
We characterize the group property of being with infinite conjugacy classes (or icc, i.e. in which all conjugacy classes beside 1 are infinite) for split extensions of groups.
We characterize the group property of being with infinite conjugacy classes (or icc, in which all conjugacy classes beside 1 are infinite) for extensions of some specific groups ; namely extensions of abelian, centerless, icc, or word hyperbolic groups.
We study the regularity of several languages derived from conjugacy classes in a finitely generated group G for a variety of examples including word hyperbolic, virtually abelian, Artin, and Garside groups. We also determine the rationality of the growth series of the shortlex conjugacy language in virtually cyclic groups, proving one direction of a conjecture of Rivin.
We calculate asymptotic estimates for the conjugacy growth function of finitely generated class 2 nilpotent groups whose derived subgroup is infinite cyclic, including the so-called higher Heisenberg groups. We prove that these asymptotics are stable when passing to commensurable groups, by understanding their twisted conjugacy growth. We also use these estimates to prove that, in certain cases, the conjugacy growth series cannot be a holonomic function.
We discuss time complexity of The Conjugacy Problem in HNN-extensions of groups, in particular, in Miller's groups. We show that for "almost all", in some explicit sense, elements, the Conjugacy Problem is decidable in cubic time. It is worth noting that the Conjugacy Problem in a Miller group may have be undecidable. Our results show that "hard" instances of the problem comprise a negligibly small part of the group.
The Conjugacy Problem for ascending HNN-extensions of free groups
该研究提出算法解决自由群上升HNN-扩张的共轭问题，同时给出相关算法处理自由群自同态动力学的特定问题，为群论相关问题的求解提供了算法支撑。
Let $H$ be an extension of a finite group $Q$ by a finite group $G$. Inspired by the results of duality theorems for étale gerbes on orbifolds, we describe the number of conjugacy classes of $H$ that maps to the same conjugacy class of $Q$. Furthermore, we prove a generalization of the orthogonality relation between characters of $G$.
Determining the length of short conjugators in a group can be considered as an effective version of the conjugacy problem. The conjugacy length function provides a measure for these lengths. We study the behaviour of conjugacy length functions under group extensions, introducing the twisted and restricted conjugacy length functions. We apply these results to show that certain abelian-by-cyclic groups have linear conjugacy length function and certain semidirect products $\Z^d \rtimes \Z^k$ have at most exponential (if $k >1$) or linear (if $k=1$) conjugacy length functions.
The conjugacy problem in cyclic extensions of one-ended hyperbolic groups
本研究针对无挠单端双曲群的自同构映射环面，通过典范JSJ树悬垂、Nielsen–Thurston约化细化及分块折叠得到余紧柱形分裂，将共轭问题归约为有理集判定，证明了这类循环扩张群的共轭问题可解。
Modelled on efficient algorithms for solving the conjugacy problem in hyperbolic groups, we define and study the permutation conjugacy length function. This function estimates the length of a short conjugator between words $u$ and $v$, up to taking cyclic permutations. This function might be bounded by a constant, even in the case when the standard conjugacy length function is unbounded. We give applications to the complexity of the conjugacy problem, estimating conjugacy growth rates, and languages. Our main result states that for a relatively hyperbolic group, the permutation conjugacy lengt…
The cyclic shift graph of a monoid is the graph whose vertices are the elements of the monoid and whose edges connect elements that are cyclic shift related. The Patience Sorting algorithm admits two generalizations to words, from which two kinds of monoids arise, the $\mathrm{rps}$ monoid and the $\mathrm{lps}$ (also known as Bell) monoid. Like other monoids arising from combinatorial objects such as the plactic and the sylvester, the connected components of the cyclic shift graph of the $\mathrm{rps}$ monoid consists of elements that have the same number of each of its composing symbols. In …
Knuth-Bendix algorithm and the conjugacy problems in monoids
本文针对幺半群共轭问题，扩展Knuth-Bendix经典重写理论，提出纳入循环共轭的重写算法方法。
A group G is a vGBS group if it admits a decomposition as a finite graph of groups with all edge and vertex groups finitely generated and free abelian. We prove that the multiple conjugacy problem is solvable between two n-tuples A and B of elements of G whenever the elements of A does not generate an elliptic subgroup. When the edge and vertex groups are infinite cyclic, i.e. G is a Generalized Baumslag-Solitar group, we prove that the multiple conjugacy problem is fully solvable.
We describe the conjugacy classes of the elements of the free product of two groups and their centralizers and, as a consequence, we correct the calculation of the cyclic and periodic cyclic homology of the group ring of the free product of two groups given in a previous paper.
Given a short exact sequence of groups with certain conditions, $1\to F\to G\to H\to 1$, we prove that $G$ has solvable conjugacy problem if and only if the corresponding action subgroup $A\leqslant Aut(F)$ is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form $\mathbb{Z}^2\rtimes F_m$, $F_2\rtimes F_m$, $F_n \rtimes \mathbb{Z}$, and $\mathbb{Z}^n \rtimes_A F_m$ with virtually solvable action group $A\leqslant GL_n(\mathbb{Z})$. Also, we give an easy way of constructing groups of the form $\mathbb{Z}^4\rtimes F_n$ and $F_3\rti…
We solve the twisted conjugacy problem on Thompson's group F. We also exhibit orbit undecidable subgroups of Aut(F), and give a proof that Aut(F) and Aut_+(F) are orbit decidable provided a certain conjecture on Thompson's group T is true. By using general criteria introduced by Bogopolski, Martino and Ventura in [5], we construct a family of free extensions of F where the conjugacy problem is unsolvable. As a byproduct of our techniques, we give a new proof of a result of Bleak-Fel'shtyn-Goncalves in [4] showing that F has property R_\infty, and which can be extended to show that Thompson's g…
Given a group $G$ and a family of subgroups $\mathcal{F}$, we consider its classifying space $E_{\mathcal F}G$ with respect to $\mathcal{F}$. When $\mathcal F = \mathcal{VC}yc$ is the family of virtually cyclic subgroups, Juan-Pineda and Leary conjectured that a group admits a finite model for this classifying space if and only if it is virtually cyclic. By establishing a connection to conjugacy growth we can show that this conjecture holds for linear groups. We investigate a similar question that was asked by Lück--Reich--Rognes--Varisco for the family of cyclic subgroups. Finally, we constru…
We characterize the group property of being with infinite conjugacy classes (or icc, i.e. infinite and of which all conjugacy classes except {1} are infinite) for groups which are defined by an extension of groups. We give characterizations for all different kinds of extension: direct product, semi-direct product, wreath products and general extension. We also give many particular results when the groups involved verify some additional hypothesis. The icc property is correlated to the Theory of Von Neumann algebras since a necessary and sufficient condition for the Von Neumann algebra of a gro…
The present paper records more details of the relationship between primitive elements and palindromes in F_2, the free group of rank two. We characterise the conjugacy classes of primitive elements which contain palindromes as those which contain cyclically reduced words of odd length. We identify large palindromic subwords of certain primitives in conjugacy classes which contain cyclically reduced words of even length. We show that under obvious conditions on exponent sums, pairs of palindromic primitives form palindromic bases for F_2. Further, we note that each cyclically reduced primitive …
Conjugacy classes of positive $3$-braids
本文研究正3辫的共轭类结构，提出直接且明确的描述方法，通过具体闭合形式确定任意正3辫的所有共轭。
Character formula for conjugacy classes in a coset
本文研究了有限群中共轭类与不可约特征标之间的关系，特别探讨了在循环群情况下陪集中的共轭类数量与特征标扩展的关系。
We characterize the group property of being with infinite conjugacy classes (or icc, i.e. \not= 1 and of which all conjugacy classes except 1 are infinite) for finite extensions of group.
This thesis deals with the conjugacy problem in groups and its twisted variants. We analyze recent results by Bogopolski, Martino, Maslakova and Ventura on the twisted conjugacy problem in free groups and its implication for the conjugacy problem in free-by-cyclic groups and some further group extensions. We also consider the doubly-twisted conjugacy problem in free groups. Staecker has developed an algorithm for deciding doubly-twisted conjugacy relations in the case where the involved homomorphisms satisfy a certain remnant inequality. We show how a similar condition affects the equalizer su…
We prove that one cannot algorithmically decide whether a finitely presented $\mathbb{Z}$-extension admits a finitely generated base group, and we use this fact to prove the undecidability of the BNS invariant. Furthermore, we show the equivalence between the isomorphism problem within the subclass of unique $\mathbb{Z}$-extensions, and the semi-conjugacy problem for deranged outer automorphisms.
In this paper we study probabilistic aspects such as (cyclic) subgroup commutativity degree and (cyclic) factorization number of ZM-groups. We show that these quantities can be computed using the sizes of the conjugacy classes of these groups. In the end, we point out a class of groups whose (cyclic) subgroup commutativity degree vanishes asymptotically.
Let $W$ be a Coxeter group. We provide a precise description of the conjugacy classes in $W$, in the spirit of Matsumoto's theorem. This extends to all Coxeter groups an important result on finite Coxeter groups by M. Geck and G. Pfeiffer from 1993. In particular, we describe the cyclically reduced elements of $W$, thereby proving a conjecture of A. Cohen from 1994.
Computing centralizers in [f.g. free]-by-cyclic groups
本文证明有限生成自由-循环群中元素的中心化子可计算，并由此得出共轭元集合可计算及带上下文无关约束的共轭问题可判定，最后提出若干开放问题。
We study groups $G$ where the $φ$-conjugacy class $[e]_φ=\{g^{-1}φ(g)~|~g\in G\}$ of the unit element is a subgroup of $G$ for every automorphism $φ$ of $G$. If $G$ has $n$ generators, then we prove that the $k$-th member of the lower central series has a finite verbal width bounded in terms of $n,k$. Moreover, we prove that if such group $G$ satisfies the descending chain condition for normal subgroups, then $G$ is nilpotent. Finally, if $G$ is a finite abelian-by-cyclic group, we construct a good upper bound of the nilpotency class of $G$.
We show that the conjugacy problem in a wreath product $A \wr B$ is uniform-$\mathsf{TC}^0$-Turing-reducible to the conjugacy problem in the factors $A$ and $B$ and the power problem in $B$. If $B$ is torsion free, the power problem for $B$ can be replaced by the slightly weaker cyclic submonoid membership problem for $B$. Moreover, if $A$ is abelian, the cyclic subgroup membership problem suffices, which itself is uniform-$\mathsf{AC}^0$-many-one-reducible to the conjugacy problem in $A \wr B$. Furthermore, under certain natural conditions, we give a uniform $\mathsf{TC}^0$ Turing reduction f…
We use wreath products to provide criteria for a group to be conjugacy separable or omnipotent. These criteria are in terms of virtual retractions onto cyclic subgroups. We give two applications: a straightforward topological proof of the theorem of Stebe that infinite-order elements of Fuchsian groups (of the first type) are conjugacy distinguished, and a proof that surface groups are omnipotent.
For groups of diffeomorphisms of $\T^2$ containing an Anosov diffeomorphism, we give a complete classification for polycyclic Abelian-by-Cyclic group actions on $\T^2$ up to both topological conjugacy and smooth conjugacy under mild assumptions. Along the way, we also prove a Tits alternative type theorem for some groups of diffeomorphisms of $\T^2$.
We study generalisations of conjugacy separability in restricted wreath products of groups. We provide an effective upper bound for $\mathcal{C}$-conjugacy separability of a wreath product $A \wr B$ in terms of the $\mathcal{C}$-conjugacy separability of $A$ and $B$, the growth of $\mathcal{C}$-cyclic subgroup separability of $B$, and the $\mathcal{C}$-residual girth of $B.$ As an application, we provide a characterisation of when $A \wr B$ is $p$-conjugacy separable. We use this characterisation to the provide for each prime $p$ an example of wreath products with infinite base group that are …
The conjugacy diameters of non-abelian finite $p$-groups with cyclic maximal subgroups
本文确定了半二面体2-群、广义四元数群和模p-群的共轭直径，推广了二面体群的已有结果。
Structure of conjugacy classes in Coxeter groups
本文通过循环移位和K-共轭构造结构共轭图，完整描述了任意Coxeter群中的共轭类结构，并证明该图连通，同时给出无限阶元素中心化子及自然分解等结果。
Products of strictly hyperbolic conjugacy classes in symplectic groups
本文研究了辛群中严格双曲共轭类的乘积问题，证明了两个循环严格双曲共轭类的乘积包含所有非标量元素，并验证了J.G.Thompson关于射影辛群共轭数猜想的特殊情况。
The subgroup pattern of a finite groups $G$ is the table of marks of $G$ together with a list of representatives of the conjugacy classes of subgroups of $G$. In this article we present an algorithm for the computation of the subgroup pattern of a cyclic extension of $G$ from the subgroup pattern of $G$. Repeated application of this algorithm yields an algorithm for the computation of the table of marks of a solvable group $G$, along a composition series of $G$.
Let $ϕ$ be an automorphism of a free group $F_n$ of rank $n$, and let $M_ϕ=F_n \rtimes_ϕ \mathbb{Z}$ be the corresponding mapping torus of $ϕ$. We study the group $Out(M_ϕ)$ under certain technical conditions on $ϕ$. Moreover, in the case of rank 2, we classify the cases when this group is finite or virtually cyclic, depending on the conjugacy class of the image of $ϕ$ in $GL_2(\mathbb{Z})$.
On the conjugacy problem for subdirect products of hyperbolic groups
研究超限群子直接积的共轭问题，证明其可解性取决于能否通过统一算法判断有限生成群中循环子群的成员资格。
All finite simple groups are determined with the property that every Galois orbit on conjugacy classes has size at most 4. From this we list all finite simple groups $G$ for which the normalized group of central units of the integral group ring ZG is an infinite cyclic group.
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