Mitra Modi, Mathew Seedhom, Alexander Ushakov
We prove that the conjugacy problem in the first Grigorchuck group $Γ$ can be solved in linear time. Furthermore, the problem to decide if a list of elements $w_1,\ldots,w_k\inΓ$ contains a pair of conjugate elements can be solved in linear time. We also show that a conjugator for a pair of conjugate element $u,v\inΓ$ can be found in polynomial time.
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In this paper we prove that the Conjugacy Problem in the Grigorchuk group $Γ$ has polynomial time complexity.
Spinal groups and multi-GGS groups are both generalisations of the well-known Grigorchuk-Gupta-Sidki (GGS-)groups. Here we give a necessary condition for spinal groups to be conjugate, and we establish a necessary and sufficient condition for multi-GGS groups to be conjugate. We also introduce a natural common generalisation of both classes, which we call polyspinal groups. Our results enable us to give a negative answer to a question of Bartholdi, Grigorchuk and Sunik, on whether every finitely generated branch group is isomorphic to a weakly branch spinal group.
The conjugacy growth function counts the number of distinct conjugacy classes in a ball of radius $n$. We give a lower bound for the conjugacy growth of certain branch groups, among them the Grigorchuk group. This bound is a function of intermediate growth. We further proof that certain branch groups have the property that every element can be expressed as a product of uniformly boundedly many conjugates of the generators. We call this property bounded conjugacy width. We also show how bounded conjugacy width relates to other algebraic properties of groups and apply these results to study the …
A branch group with unsolvable conjugacy problem
本文通过将任意有限生成剩余有限群保持共轭关系地嵌入有限生成分支群，构造出字问题可解但共轭问题不可解的分支群，回答了相关公开问题。
We suggest several automaton groups as key-agreement platforms for Anshl-Anshel-Goldfeld metascheme, they include Grigorchuk and universal Grigorchuk groups, Hanoi 3-Towers group, Basilica group and a subgroup of the affine group with the unsolvable conjugacy problem
$\mathcal{C}$-Hereditarily conjugacy separable groups and wreath products
该研究给出限制 wreath 积为 $\boldsymbol{\textit{C}}$-遗传共轭可分群的充要条件，证明 Grigorchuk 群是 2-遗传共轭可分群，构造了无穷多类非多循环遗传共轭可分群。
Contracting Self-similar Groups in Group-Based Cryptography
提出使用自相似压缩群作为基于同时共轭搜索问题(SCSP)的密码方案平台，利用其核肖像正规形式抵抗现有攻击，并分析了Grigorchuk群等实例的计算安全性。
We prove a formula connecting the number of unipotent conjugacy classes in a maximal parabolic subgroup of a finite general linear group with the numbers of unipotent conjugacy classes in various parabolic subgroups in smaller dimensions. We generalise this formula and deduce a number of corollaries; in particular, we express the number of conjugacy classes of unitriangular matrices over a finite field in terms of the numbers of unipotent conjugacy classes in maximal parabolic subgroups over the same field. We show how the numbers of unipotent conjugacy classes in parabolic subgroups of small …
We investigate the conjugacy growth of finitely generated linear groups. We show that finitely generated non-virtually-solvable subgroups of GL_d have uniform exponential conjugacy growth and in fact that the number of distinct polynomials arising as characteristic polynomials of the elements of the ball of radius n for the word metric has exponential growth rate bounded away from 0 in terms of the dimension d only.
In this paper we study twisted conjugacy classes and the $R_{\infty}$-property for classical linear groups. In particular, we prove that the general linear group ${\rm GL}_n(K)$ and the special linear group ${\rm SL}_n(K)$ possess $R_{\infty}$-property, if $n\geq3$ and $K$ is an infinite integral domain with trivial group of automorphisms (Theorem 1), or $K$ is an integral domain, which has a zero characteristic and for which ${\rm Aut}(K)$ is torsion (Theorem 2).
Let $k$ be an algebraically closed field, $G$ a linear algebraic group over $k$ and $φ\in Aut(G)$, the group of all algebraic group automorphisms of $G$. Two elements $x, y$ of $G$ are said to be $φ$-twisted conjugate if $y=gxφ(g)^{-1}$ for some $g\in G$. In this paper we prove that for a connected non-solvable linear algebraic group $G$ over $k$, the number of its $φ$-twisted conjugacy classes is infinite for every $φ\in Aut(G)$.
A note on conjugacy problem for finite Sylow subgroups of infinite linear groups
本文证明了在存在至少一个有限Sylow 2-子群的条件下，伪有限$\mathfrak{M}_c$群中Sylow 2-子群的共轭性，并通过Platonov的反例说明伪有限性假设不可或缺。
Subracks and second homology of the conjugacy classes of finite projective special linear groups of degree two
基于Dickson子群定理描述PSL(2,q)共轭类子rack，确定其极小非交换子rack，给出完全群共轭类关联群与Schur乘子关系的一般结果，进而明确q>3时PSL(2,q)共轭类关联群及二阶同调。
Guba and Sapir asked if the simultaneous conjugacy problem was solvable in Diagram Groups or, at least, for Thompson's group F. We give a solution to the latter question using elementary techniques which rely purely on the description of F as the group of piecewise linear orientation-preserving homeomorphisms of the unit interval. The techniques we develop extend the ones used by Brin and Squier allowing us to compute roots and centralizers as well. Moreover, these techniques can be generalized to solve the same question in larger groups of piecewise-linear homeomorphisms.
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.
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。
If $u$ and $v$ are two conjugate elements of a hyperbolic group then the length of a shortest conjugating element for $u$ and $v$ can be bounded by a linear function of the sum of their lengths, as was proved by Lysenok. Bridson and Haefliger showed that in a hyperbolic group the conjugacy problem can be solved in polynomial time. We extend these results to relatively hyperbolic groups. In particular, we show that both the conjugacy problem and the conjugacy search problem can be solved in polynomial time in a relatively hyperbolic group, whenever the corresponding problem can be solved in pol…
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。
We complete the program begun by Brin and Squier of characterising conjugacy in Thompson's group $F$ using the standard action of $F$ as a group of piecewise linear homeomorphisms of the unit interval.
Let $G$ be a linear algebraic group over an algebraically closed field $k$ and $\mathrm{Aut}_{\mathrm{alg}}(G)$ the group of all algebraic group automorphisms of $G$. For every $φ\in \mathrm{Aut}_{\mathrm{alg}}(G)$ let $\mathcal{R}(φ)$ denote the set of all orbits of the $φ$-twisted conjugacy action of $G$ on itself (given by $(g,x)\mapsto gxφ(g^{-1})$, for all $g,x\in G$). We say that $G$ has the algebraic $R_\infty$-property if $\mathcal{R}(φ)$ is infinite for every $φ\in \mathrm{Aut}_{\mathrm{alg}}(G)$. In \citep{bb} we have shown that this property is satisfied by every connected non-solva…
There have been several attempts to extend the notion of conjugacy from groups to monoids. The aim of this paper is study the decidability and independence of conjugacy problems for three of these notions (which we will denote by $\sim_p$, $\sim_o$, and $\sim_c$) in certain classes of finitely presented monoids. We will show that in the class of polycyclic monoids, $p$-conjugacy is "almost" transitive, $\sim_c$ is strictly included in $\sim_p$, and the $p$- and $c$-conjugacy problems are decidable with linear compexity. For other classes of monoids, the situation is more complicated. We show t…
Let $S$ be a semigroup. The elements $a,b\in S$ are called primarily conjugate if $a=xy$ and $b=yx$ for certain $x,y\in S$. The relation of conjugacy is defined as the transitive closure of the relation of primary conjugacy. In the case when $S$ is a monoid, denote by $G$ the group of units of $S$. Then the relation of $G$-conjugacy is defined by $a\sim_G b \iff a=g^{-1}bg$ for certain $g\in G$. We establish the structure of conjugacy classes for regular epigroups (i.e. semigroups such that some power of each element lies in a subgroup). As a corollary we obtain a criterion of conjugacy in ter…
Linear representations, crystallographic quotients, and twisted conjugacy of virtual Artin groups
本文研究了虚拟Artin群的线性表示、晶体商与扭曲共轭，构造了其线性表示，证明了球面Coxeter群对应的商群为晶体群，分类了挠元素，并证明直角虚拟Artin群具有$R_\infty$性质。
We give a unified solution to the conjugacy problem for Thompson's groups F, T, and V. The solution uses strand diagrams, which are similar in spirit to braids and generalize tree-pair diagrams for elements of Thompson's groups. Strand diagrams are closely related to piecewise-linear functions for elements of Thompson's groups, and we use this correspondence to investigate the dynamics of elements of F. Though many of the results in this paper are known, our approach is new, and it yields elegant proofs of several old results.
In this paper, we classify conjugacy classes of centralizers of irreducible subgroups in $PSL(n,\mathbb{C})$ using alternate modules a.k.a. finite abelian groups with an alternate bilinear form. When $n$ is squarefree, we prove that these conjugacy classes are classified by their isomorphism classes. More generally, we define a finite graph related to this classification whose combinatorial properties are expected to help us describe the stratification of the singular (orbifold) locus in some character varieties.
The character degree product and the conjugacy length product for finite general linear groups
本文针对 K. Harada 关于有限群不可约特征标次数乘积整除共轭类长度乘积的猜想，在有限一般线性群与有限酉群上完成了验证。
A {\it $k$-involution} is an involution with a fixed point set of codimension $k$. The conjugacy class of such an involution, denoted $S_k$, generates $\text{Möb}(n)$-the the group of isometries of hyperbolic $n$-space-if $k$ is odd, and its orientation preserving subgroup if $k$ is even. In this paper, we supply effective lower and upper bounds for the $S_k$ word length of $\text{Möb}(n)$ if $k$ is odd, and the $S_k$ word length of $\text{Möb}^+(n)$, if $k$ is even. As a consequence, for a fixed codimension $k$ the length of $\text{Möb}^{+}(n)$ with respect to $S_k$, $k$ even, grows linearly …
Let q be a power of a prime and n a positive integer. Let P(q) be a parabolic subgroup of the finite general linear group GL(n,q). We show that the number of P(q)-conjugacy classes in GL(n,q) is, as a function of q, a polynomial in q with integer coefficients. This answers a question of J. Alperin.
We classify the real and strongly real conjugacy classes in $GL_n(q)$, $SL_n(q)$, $PGL_n(q)$, $PSL_n(q)$, and all quasi-simple covers of $PSL_n(q)$. In each case we give a formula for the number of real, and the number of strongly real, conjugacy classes.
Uniform Upper Bounds for Conjugacy Separability Growth in Nilpotent Groups
针对幂零群的共轭可分性增长函数，本文给出次数至多线性于幂零类、二次于Hirsch长度的显式多项式上界，并证明二步幂零情形下该界最优，同时纠正了文献错误。
Implementation: The conjugacy problem in right-angled Artin groups
本文用Python实现了Crisp等人2009年提出的线性时间算法，用于解决直角Artin群中的共轭问题，不仅能判定两元素是否共轭，还能在共轭时返回具体的共轭元素。
We investigate Levi subgroups of a connected reductive algebraic group G, over a ground field K. We parametrize their conjugacy classes in terms of sets of simple roots and we prove that two Levi K-subgroups of G are rationally conjugate if and only if they are geometrically conjugate. These results are generalized to arbitrary connected linear algebraic K-groups. In that setting the appropriate analogue of a Levi subgroup is derived from the notion of a pseudo-parabolic subgroup.
We say that two elements of a group or semigroup are $\Bbbk$-linear conjugates if their images under any linear representation over $\Bbbk$ are conjugate matrices. In this paper we characterize $\Bbbk$-linear conjugacy for finite semigroups (and, in particular, for finite groups) over an arbitrary field $\Bbbk$.
In this article, we propose a geometric framework dedicated to the study of van Kampen diagrams of graph products of groups. As an application, we find information on the word and the conjugacy problems. The main new result of the paper deals with the computation of conjugacy length functions. More precisely, if $Γ$ is a finite graph and $\mathcal{G}= \{ G_u \mid u \in V(Γ) \}$ a collection of finitely generated groups indexed by the vertices of $Γ$, then $$\max\limits_{u \in V(Γ)} \mathrm{CLF}_{G_u}(n) \leq \mathrm{CLF}_{Γ\mathcal{G}}(n) \leq (D+1) \cdot n + \max\limits_{Δ\subset Γ\ \text{com…
We compute conjugacy classes in maximal parabolic subgroups of the general linear group. This computation proceeds by reducing to a ``matrix problem''. Such problems involve finding normal forms for matrices under a specified set of row and column operations. We solve the relevant matrix problem in small dimensional cases. This gives us all conjugacy classes in maximal parabolic subgroups over a perfect field when one of the two blocks has dimension less than 6. In particular, this includes every maximal parabolic subgroup of GL_n(k) for n < 12 and k a perfect field. If our field is finite of …
Stability, bounded generation and strong boundedness
本文通过经典代数K理论的稳定性考量，为整体域整数环上特殊线性群和辛群的广义共轭直径提供秩线性界，确定了其增长率。
Given an automorphism $ϕ:Γ\to Γ$, one has an action of $Γ$ on itself by $ϕ$-twisted conjugacy, namely, $g.x=gxϕ(g^{-1})$. The orbits of this action are called $ϕ$-twisted conjugacy classes. One says that $Γ$ has the $R_\infty$-property if there are infinitely many $ϕ$-twisted conjugacy classes for every automorphism $ϕ$ of $Γ$. In this paper we show that SL$(n,\mathbb{Z})$ and its congruence subgroups have the $R_\infty$-property. Further we show that any (countable) abelian extension of $Γ$ has the $R_\infty$-property where $Γ$ is a torsion free non-elementary hyperbolic group, or SL$(n,\math…
Let $p$ be a prime and let $\mathbb{C}$ be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of $\mathrm{GL}(p,\mathbb{C})$ up to conjugacy. That is, we give a complete and irredundant list of $\mathrm{GL}(p,\mathbb{C})$-conjugacy class representatives as generating sets of monomial matrices. Copious structural information about non-solvable finite irreducible monomial subgroups of $\mathrm{GL}(p,\mathbb{C})$ is also proved, enabling a classification of all such groups bar one family. We explain the obstacles in that exceptional case. For $p\leq 3$, w…
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…
Using generating functions, we enumerate regular semisimple conjugacy classes in the finite classical groups. For the general linear, unitary, and symplectic groups this gives a different approach to known results; for the special orthogonal groups the results are new.
In this note we investigate the conjugacy problem in lamplighter groups with particular interest in the role of their geometry. In particular we show that the conjugacy length function is linear.
Let $K$ be a field and $n\geq 3$. Let $E_n(K)\leq H\leq GL_n(K)$ be an intermediate group and $C$ a noncentral $H$-class. Define $m(C)$ as the minimal positive integer $m$ such that $\exists i_1,\dots,i_m\in\{\pm 1\}$ such that the product $C^{i_1}\dots C^{i_m}$ contains all nontrivial elementary transvections. In this article we obtain a sharp upper bound for $m(C)$. Moreover, we determine $m(C)$ for any noncentral $H$-class $C$ under the assumption that $K$ is algebraically closed or $n=3$ or $n=\infty$.
We study the word and conjugacy problems in lacunary hyperbolic groups (briefly, LHG). In particular, we describe a necessary and sufficient condition for decidability of the word problem in LHG. Then, based on the graded small-cancellation theory of Olshanskii, we develop a general framework which allows us to construct lacunary hyperbolic groups with word and conjugacy problems highly controllable and flexible both in terms of computability and computational complexity. As an application, we show that for any recursively enumerable subset $\mathcal{L} \subseteq \mathcal{A}^*$, where $\mathca…
In the paper we study twisted conjugacy classes and isogredience classes for automorphisms of reductive linear algebraic groups. We show that reductive linear algebraic groups over some fields of zero characteristic possess the $R_\infty$ and $S_\infty$ properties.
The conjugacy problem for a finitely generated group $G$ is the two-variable problem of deciding for an arbitrary pair $(u,v)$ of elements of $G$, whether or not $u$ is conjugate to $v$ in $G$. We construct examples of finitely generated, computably presented groups such that for every element $u_0$ of $G$, the problem of deciding if an arbitrary element is conjugate to $u_0$ is decidable in quadratic time but the worst-case complexity of the global conjugacy problem is arbitrary: it can be any c.e. Turing degree , can exactly mirror the Time Hierarchy Theorem, or can be $\mathcal{NP}$-complet…
Let $V$ be a vector space over the field of order $2$. We investigate subgroups of the linear group $GL(V)$ which are generated by a conjugacy class $D$ of elements of order $3$ such that all $d$ in $D$ have $2$-dimensional commutator space $[V,d]$.
Unipotent linear suspensions of free groups
受自由群外自同构共轭问题驱动，本文发展了由幂单线性增长自同构产生的 free-by-cyclic 群的算法理论，计算了其规范分裂与自同构群，证明了有效凝聚性，解决了混合 Whitehead 问题，并最终应用于解决该类自由群外自同构的共轭问题。
We show that for any $n$ and $q$, the number of real conjugacy classes in $\mathrm{PGL}(n, \mathbb{F}_q)$ is equal to the number of real conjugacy classes of $\mathrm{GL}(n, \mathbb{F}_q)$ which are contained in $\mathrm{SL}(n, \mathbb{F}_q)$, refining a result of Lehrer, and extending the result of Gill and Singh that this holds when $n$ is odd or $q$ is even. Further, we show that this quantity is equal to the number of real conjugacy classes in $\mathrm{PGU}(n, \mathbb{F}_q)$, and equal to the number of real conjugacy classes of $\mathrm{U}(n, \mathbb{F}_q)$ which are contained in $\mathrm{…
On the traces of the product of 2 linear similarity classes
研究了特殊线性群SL(n,K)中两个非标量共轭类乘积的迹性质，证明当n≥4时，乘积包含任意迹的矩阵，或当n=3且K为有限域时也成立。
This article is a contribution to the following problem: does there exist a Polish non-archimedean group (equivalently: automorphism group of a Fraisse limit) that is extremely amenable, and has ample generics. As Fraisse limits whose automorphism groups are extremely amenable must be ordered, i.e., equipped with a linear ordering, we focus on ordered Fraisse limits. We prove that automorphism groups of the universal ordered boron tree, and the universal ordered poset have a comeager conjugacy class but no comeager $2$-dimensional diagonal conjugacy class. We also provide general conditions im…
We prove that the conjugacy problem in right-angled Artin groups (RAAGs), as well as in a large and natural class of subgroups of RAAGs, can be solved in linear-time. This class of subgroups contains, for instance, all graph braid groups (i.e. fundamental groups of configuration spaces of points in graphs), many hyperbolic groups, and it coincides with the class of fundamental groups of ``special cube complexes'' studied independently by Haglund and Wise.
The existence of closed orbits of real algebraic groups on real algebraic varieties is established. As an application, it is shown that if G is a real reductive linear group with Iwasawa decomposition G= KAN, then every unipotent subgroup of G is conjugate to a subgroup of N.
Let $q$ be a power of a prime $p$. Let $P$ be a parabolic subgroup of the general linear group $\GL_n(q)$ that is the stabilizer of a flag in $\FF_q^n$ of length at most 5, and let $U = O_p(P)$. In this note we prove that, as a function of $q$, the number $k(U)$ of conjugacy classes of $U$ is a polynomial in $q$ with integer coefficients.
Irreducible representations of conjugacy quandles
研究共轭quandle的不可约表示，通过群的特征标与线性表示的乘积来描述，证明其 enveloping group 的嵌入性质及完美群时的同构性。
Conjugacy problem in T-RAAGs
本文提出了解决扭曲右角Artin群（T-RAAGs）共轭问题的可行算法，并展示了T-RAAGs是双自动的，提供了另一种解决方案。
We prove several results on products of conjugacy classes in finite simple groups. The first result is that there always exists a uniform generating triple. This result and other ideas are used to solve a 1966 conjecture of Peter Neumann about the existence of elements in an irreducible linear group with small fixed space. We also show that there always exist two conjugacy classes in a finite non-abelian simple group whose product contains every nontrivial element of the group. We use this to show that every element in a non-abelian finite simple group can be written as a product of two rth po…
We describe the relation between two characterizations of conjugacy in groups of piecewise-linear homeomorphisms, discovered by Brin and Squier in [2] and Kassabov and Matucci in [5]. Thanks to the interplay between the techniques, we produce a simplified point of view of conjugacy that allows us to easily recover centralizers and lends itself to generalization.
Limits of Baumslag-Solitar groups and dimension estimates in the space of marked groups
该研究证明了Baumslag-Solitar群的极限的多项代数性质，给出其表现、同构分类、自同构描述及相关判定问题讨论，并证明其在双生成元带标群空间中具有非零Hausdorff维数。
Let G be a finite group. Define a relation ~ on the conjugacy classes of G by setting C ~ D if there are representatives c \in C and d \in D such that cd = dc. In the case where G has a normal subgroup H such that G/H is cyclic, two theorems are proved concerning the distribution, between cosets of H, of pairs of conjugacy classes of G related by ~. One of the proofs involves an interesting application of the famous Marriage Theorem of Philip Hall. The paper concludes by discussing some aspects of these theorems and of the relation ~ in the particular cases of symmetric and general linear grou…
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