Francesco Matucci, Altair Santos de Oliveira-Tosti
Monod introduced in [14] a family of Thompson-like groups which provides natural counterexamples to the von Neumann-Day conjecture. We construct a characterization of conjugacy and invariant and use them to compute centralizers in one group of this family.
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Let $G$ be an algebraic group of type $G_2$ over a field $k$ of characteristic $\neq 2,3$. In this paper we calculate centralizers of semisimple elements in anisotropic $G_2$. Using these, we show explicitly that there are six conjugacy classes of centralizers in the compact real form of $G_2$.
The set of all centralizers of elements in a finite group $G$ is denoted by $Cent(G)$ and $G$ is called $n-$centralizer if $|Cent(G)| = n$. In this paper, the structure of centralizers in a non-abelian finite group $G$ with this property that $\frac{G}{Z(G)} \cong Z_{p^2} \rtimes Z_{p^2}$ is obtained. As a consequence, it is proved that such a group has exactly $[(p+1)^2+1]$ element centralizers and the structure of the commuting conjugacy class graph of $G$ is completely determined.
Let G be a group. Two elements x and y in G are said to be in the same z-class if their centralizers in G are conjugate within G. In this paper, we prove that the number of z-classes in the group of upper triangular matrices is infinite provided that the field is infinite and size of the matrices is at least 6, and finite otherwise.
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 $\mathscr{C}$ be a classical group defined over a finite field. We present comprehensive theoretical solutions to the following closely related problems: 1) List a representative for each conjugacy class of $\mathscr{C}$. 2) Given $x \in \mathscr{C}$, describe the centralizer $C_{\mathscr{C}}(x)$ of $x$ in $\mathscr{C}$, by giving its group structure and a generating set. 3) Given $x,y \in \mathscr{C}$, establish whether $x$ and $y$ are conjugate in $\mathscr{C}$ and, if they are, find explicit $z \in \mathscr{C}$ such that $z^{-1}xz = y$. We also formulate practical algorithms to solve th…
We explore when generator-conjugate homomorphisms are conjugate and when element-conjugate homomorphisms are conjugate from abelian or dihedral groups to the symmetric group. We completely determine when such homomorphisms are conjugate in the case where the source group has two generators by studying centralizer actions in the target group.
Let $G$ be a group. Two elements $x,y \in G$ are said to be in the same $z$-class if their centralizers in $G$ are conjugate within $G$. Consider $\mathbb F$ a perfect field of characteristic $\neq 2$, which has a non-trivial Galois automorphism of order $2$. Further, suppose that the fixed field $\mathbb F_0$ has the property that it has only finitely many field extensions of any finite degree. In this paper, we prove that the number of $z$-classes in the unitary group over such fields is finite. Further, we count the number of $z$-classes in the finite unitary group $U_n(q)$, and prove that …
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.
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.
Verification of the conjugacy classes and ordinary character table of the Monster
本文在Monster群唯一性假设下，利用196883维忠实表示重新计算其共轭类与特征表，并验证了p<11的中心化子特征表及类融合。
A criterion for Tits alternative on the centralizer of a matrix
本文提出一个矩阵中心化子在GL(n,Z)中为多项式循环群的充要条件，即不包含非阿贝尔自由子群，并给出了使矩阵可交换的简单条件，从而在GL(n,Z)中建立有效的蒂茨替代准则，并应用于保持非退化有理双线性形式的算术群的共轭问题。
To classify the finite dimensional pointed Hopf algebras with $G= {\rm HS}$ or ${\rm Co3}$ we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software {\rm GAP}.
This manuscript represents the author's PhD dissertation thesis.The first part studies decision problems in Thompson's groups F,T,V and some generalizations. The simultaneous conjugacy problem is determined to be solvable for Thompson's group F and suitable larger groups of piecewise-linear homeomorphisms of the unit interval. We determine algorithms to compute roots and centralizers in these groups and to detect periodic points and their behavior by looking at a particular diagram associated to an element. In the second part, we describe the structure of subgroups of the group of all homeomor…
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。
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.
Let $G$ be a group endowed with a solution to the conjugacy problem and with an algorithm which computes the centralizer in $G$ of any element of $G$. Let $H$ be a subgroup of $G$. We give some conditions on $H$, under which we provide a solution to the conjugacy problem in $H$. We apply our results to some Artin groups and braid type groups. In particular, we give explicit solutions to the conjugacy problem in the Artin groups of type $\tilde{A}_n$ and $\tilde{C}_n$.
We describe a self-homeomorphism $R$ of the Cantor set $X$ and then show that its conjugacy class in the Polish group $H(X)$ of all homeomorphisms of $X$ forms a dense $G_δ$ subset of $H(X)$. We also provide an example of a locally compact, second countable topological group which has a dense conjugacy class.
We characterize the double centralizer of all parabolic subgroups of the braid groups. We apply this result to provide a new and potentially more efficient solution to the subgroup conjugacy problem for parabolic subgroups. In the course of the proof we also characterize the centralizer for all parabolic subgroups.
The moduli space for a flat G-bundle over the two-torus is completely determined by its holonomy representation. When G is compact, connected, and simply connected, we show that the moduli space is homeomorphic to a product of two tori mod the action of the Weyl group, or equivalently to the conjugacy classes of commuting pairs of elements in G. Since the component group for a non-simply connected group is given by some finite dimensional subgroup in the centralizer of an n-tuple, we use diagram automorphisms of the extended Dynkin diagram to prove properties of centralizers of pairs of elemen…
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.
We prove the Arad-Herzog conjecture for various families of finite simple groups- if A and B are nontrivial conjugacy classes, then AB is not a conjugacy class. We also prove that if G is a finite simple group of Lie type and A and B are nontrivial conjugacy classes, either both semisimple or both unipotent, then AB is not a conjugacy class. We also prove a strong version of the Arad-Herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes. As a conseque…
For any odd prime $p$ we consider representations of a group of order $p$ in the symplectic group $Sp(p-1,Z[1/n])$ of $(p-1)\times(p-1)$-matrices over the ring $Z[1/n]$, $0\neq n\in N$. We construct a relation between the conjugacy classes of subgroups $P$ of order $p$ in the symplectic group and the ideal class group in the ring $Z[1/n]$. This is used for the study of these classes. In particular we determine the centralizer $C(P)$ and $N(P)/C(P)$ where $N(P)$ denotes the normalizer.
Let G be a connected, reductive group over an algebraically closed field of good characteristic. For u in G unipotent, we describe the conjugacy classes in the component group A(u) of the centralizer of u. Our results extend work of the second author done for simple, adjoint G over the complex numbers. When G is simple and adjoint, the previous work of the second author makes our description combinatorial and explicit; moreover, it turns out that knowledge of the conjugacy classes suffices to determine the group structure of A(u). Thus we obtain the result, previously known through case-checki…
Let G be a simple algebraic group of adjoint type over an algebraically closed field of bad characteristic. We show that its sheets of conjugacy classes are parametrized by G-conjugacy classes of pairs (M,O) where M is the identity component of the centralizer of a semisimple element in G and O is a rigid unipotent conjugacy class in M, in analogy with the good characteristic case.
We give a new method to compute the centralizer of an element in Artin braid groups and, more generally, in Garside groups. This method, together with the solution of the conugacy problem given by the authors in a previous paper, are two main steps for solving conjugacy systems, thus breaking recently discovered cryptosystems based in braid groups. We also present the result of our computations, where we notice that our algorithm yields surprisingly small generating sets for the centralizers.
Set theory, logic, and homeomorphism groups of manifolds
研究公理集合论与流形同胚群一阶理论的关系，通过构建性公理（V = L）证明同胚群的一阶刚性等，在正则假设下得出不同结论，还表明无穷公式可确定同胚类和流形类型。
We show that several classes of groups G of PL-homeomorphisms of the real line admit non-trivial homomorphisms from G to the additive group of reals that are fixed by every automorphism of G. The classes of groups enjoying the stated property include the generalisations of Thompson's group F studied by K. S. Brown, M. Stein, S. Cleary, and Bieri-Strebel but also the class of groups investigated by Bieri-Neumann-Strebel in [BNS87, Theorem 8.1]. It follows that every automorphism of a group in one of these classes has infinitely many associated twisted conjugacy classes.
Double coset problem for parabolic subgroups of braid groups
针对辫群中所有具有连通Coxeter图的抛物子群，首次给出双陪集问题的解，并推广至所有抛物子群。
In this paper, we compute the number of z-classes (conjugacy classes of centralizers of elements) in the symmetric group S_n, when n is greater or equal to 3 and alternating group A_n, when n is greater or equal to 4. It turns out that the difference between the number of conjugacy classes and the number of z-classes for S_n is determined by those restricted partitions of n-2 in which 1 and 2 do not appear as its part. And, in the case of alternating groups, it is determined by those restricted partitions of n-3 which has all its parts distinct, odd and in which 1 (and 2) does not appear as it…
A group is said to have the $R_\infty$ property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether $G$ has the $R_\infty$ property when $G$ is a finitely generated torsion-free nilpotent group. As a consequence, we show that for every positive integer $n\ge 5$, there is a compact nilmanifold of dimension $n$ on which every homeomorphism is isotopic to a fixed point free homeomorphism. As a by-product, we give a purely group theoretic proof that the free group on two generators has the $R_\infty$ property. The $R_{\infty}$ property for virtu…
Conjugacy in finite classical groups
本文针对有限域上典型群，系统解决了共轭类代表元列举、中心化子结构描述与共轭判定及显式共轭元构造三个基本问题，并给出理论方案及 Magma 完整实现，为计算群论中的特征标表计算和一般共轭问题求解提供关键支撑。
Topological Full Groups of Irreducible Edge Shifts have Solvable Conjugacy Problem
本研究利用链图和交换环半群等方法，解决了不可约边移位的拓扑全群的共轭问题，相关技术可推广至包含Houghton群、类Thompson群$QV$的更大边移位分段典范同胚群类。
We show that conjugacy of reversible cellular automata is undecidable, whether the conjugacy is to be performed by another reversible cellular automaton or by a general homeomorphism. This gives rise to a new family of finitely-generated groups with undecidable conjugacy problems, whose descriptions arguably do not involve any type of computation. For many automorphism groups of subshifts, as well as the group of asynchronous transducers and the homeomorphism group of the Cantor set, our result implies the existence of two elements such that every finitely-generated subgroup containing both ha…
In this paper we study free mappings of the plane, that is orientation preserving fixed point free homeomorphisms of $\mathbb{R}^2$. We provide a necessary and sufficient condition under which two free mappings of the plane that are embedded in flows are conjugate to one another using Haefliger-Reeb theory of plane foliations.
Springer categories for regular centralizers in well-generated complex braid groups
本文研究了复杂辫群中正则中心化子的弹簧类别，通过Garside结构证明了关于中心的定理，并提供了Hurwitz-like的呈现方法。
This paper concerns the enumeration of simultaneous conjugacy classes of tuples of commuting unitary matrices and of commuting symplectic matrices over a finite field $\mathbf{F}_q$ of odd size. For any given conjugacy class, the orbits for the action of its centralizer group on itself by conjugation are called branches. We determine the branching rules for the unitary groups $U_2(\mathbf{F}_q), U_3(\mathbf{F}_q)$, and for the symplectic groups $Sp_2(\mathbf{F}_q), Sp_4(\mathbf{F}_q)$.
A canonical section method for conjugacy classes of GL_n(O_2)
本文提出GLₙ(O₂)共轭类的典范截面方法，证明其类方程仅依赖剩余域基数，该论证早于并独立于后续相关分类方法，因具独立意义而见刊。
Let $N \geq 2$ and let $\mathrm{Out}(F_N)$ be the outer automorphism group of a nonabelian free group of rank $N$. Let $\mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$ be the finite index subgroup of $\mathrm{Out}(F_N)$ which is the kernel of the natural action of $\mathrm{Out}(F_N)$ on $H_1(F_N,\mathbb{Z}/3\mathbb{Z})$. We show that $\mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$ is an $R$-group, that is, for every $ϕ,ψ\in \mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$, if there exists $k \in \mathbb{N}^*$ such that $ϕ^k=ψ^k$, then $ϕ=ψ$. This answers a question of Handel and Mosher. We then use the fact that $\mathr…
We examine the Polish group $H_{+}^{AC}$ of order-preserving self-homeomorphisms $f$ of the interval $\left[ 0,1 \right]$ for which both $f$ and $f^{-1}$ are absolutely continuous; in particular, we establish two results. First, we prove that $H_{+}^{AC}$ is topologically $2$-generated; in fact, it is generically $2$-generated, i.e., there is a dense $G_δ$ set of pairs $\left( f,g \right) \in H_{+}^{AC} \times H_{+}^{AC}$ for which $\left\langle f,g \right\rangle$ is dense. Secondly, we prove that $H_{+}^{AC}$ admits a dense $G_δ$ conjugacy class, and we explicitly characterize the elements th…
We show that the group of homeomorphisms of the Cantor set $H(K)$ has ample generics, that is, for every $m$ the diagonal conjugacy action $g\cdot(h_1,h_2,..., h_m)=(gh_1g^{-1},gh_2g^{-1},..., gh_mg^{-1})$ of $H(K)$ on $H(K)^m$ has a comeager orbit. This answers a question of Kechris and Rosendal. We show that the generic tuple in $H(K)^m$ can be taken to be the limit of a certain projective Fraisse family. We also present a proof of the existence of the generic homeomorphism of the Cantor set in the context of the projective Fraisse theory.
Rearrangement Groups of Fractals: Structure and Conjugacy
本文系统研究分形同胚构成的重排群，给出有限生成与单性结果、共轭问题充分条件、不变生成否定结论、有理粘合关系及稳定子群刻画，并证明其均可嵌入 Thompson 群 $V$。
Let $G$ be a group. Two elements $x, y$ are said to be {\it $z$-equivalent} if their centralizers are conjugate in $G$. The class equation of $G$ is the partition of $G$ into conjugacy classes. Further decomposition of conjugacy classes into $z$-classes provides an important information about the internal structure of the group. Let $I(\H^n)$ denote the group of isometries of the hyperbolic $n$-space. We show that the number of $z$-classes in $I(\H^n)$ is finite. We actually compute their number, cf. theorem 1.3. We interpret the finiteness of $z$-classes as accounting for the finiteness of "d…
Brauer and Fowler noted restrictions on the structure of a finite group G in terms of the order of the centralizer of an involution t in G. We consider variants of these themes. We first note that for an arbitrary finite group G of even order, we have |G| is less than the number of conjugacy classes of the Fitting subgroup times the order of the centralizer to the fourth power of any involution in G. This result does require the classification of the finite simple groups. The groups SL(2,q) with q even shows that the exponent 4 cannot be replaced by any exponent less than 3. We do not know at …
Conjugacy problems in higher-dimensional Thompson groups
该研究证明了n≥2时Brin-Thompson群$nV$中元素的拓扑共轭性不可判定，还得到了Cantor空间同胚群有限生成超群中Whitehead问题及群论共轭的不可判定性。
Letting tau denote the inverse transpose automorphism of GL(n,q), a formula is obtained for the number of g in GL(n,q) so that gg^{tau} is equal to a given element h. This generalizes a result of Gow and Macdonald for the special case that h is the identity. We conclude that for g random, gg^{tau} behaves like a hybrid of symplectic and orthogonal groups. It is shown that our formula works well with both cycle index generating functions and asymptotics, and is related to the theory of random partitions. The derivation makes use of models of representation theory of GL(n,q) and of symmetric fun…
We construct a "higher dimensional" version 2V of Thompson's group V. Like V it is an infinite, finitely presented, simple subgroup of the homeomorphism group of the Cantor set, but we show that it is not isomorphic to V by showing that the actions on the Cantor set are not topologically conjugate: 2V has an element with "chaotic" action, while V cannot have such an element. A theorem of Rubin is then applied which shows that for these two groups, isomorphism would imply topological conjugacy.
Strata of Weyl groups
本文研究了韦尔群的层结构，通过修改后的中心化群给出显式描述，并提出关于超层的猜想。
We describe the centralizer of irreducible representations from a finitely generated group $Γ$ to $PSL(p,\mathbb{C})$ where $p$ is a prime number. This leads to a description of the singular locus (the set of conjugacy classes of representations whose centralizer strictly contains the center of the ambient group) of the irreducible part of the character variety $χ^i(Γ,PSL(p,\mathbb{C}))$. When $Γ$ is a free group of rank $l\geq 2$ or the fundamental group of a closed Riemann surface of genus $g\geq 2$, we give a complete description of this locus and prove that this locus is exactly the set of…
There are well-known monomorphisms between the Artin groups of finite type $\arA_n$, $\arB_n=\arC_n$ and affine type $\tilde \arA_{n-1}$, $\tilde\arC_{n-1}$. The Artin group $A(\arA_n)$ is isomorphic to the $(n+1)$-strand braid group $B_{n+1}$, and the other three Artin groups are isomorphic to some subgroups of $B_{n+1}$. The inclusions between these subgroups yield monomorphisms $A(\arB_n)\to A(\arA_n)$, $A(\tilde \arA_{n-1})\to A(\arB_n)$ and $A(\tilde \arC_{n-1})\to A(\arB_n)$. There are another type of monomorphisms $A(\arB_d)\to A(\arA_{md-1})$, $A(\arB_d)\to A(\arB_{md})$ and $A(\arB_d)…
We show that a natural quotient of the projective Fraisse limit of a family that consists of finite rooted trees is the Lelek fan. Using this construction, we study properties of the Lelek fan and of its homeomorphism group. We show that the Lelek fan is projectively universal and projectively ultrahomogeneous in the class of smooth fans. We further show that the homeomorphism group of the Lelek fan is totally disconnected, generated by every neighbourhood of the identity, has a dense conjugacy class, and is simple.
Let G be a Poincare duality group of dimension n. For a given element g in G, let C_g denote its centralizer subgroup. Let L_G be the graded abelian group defined by (L_G)_p = oplus_{[g]}H_{p+n}(C_g) where the sum is taken over conjugacy classes of elements in G. In this paper we construct a multiplication on L_G directly in terms of intersection products on the centralizers. This multiplication makes L_G a graded, associative, commutative algebra. When G is the fundamental group of an aspherical, closed oriented n manifold M, then (L_G)_* = H_{*+n}(LM), where LM is the free loop space of M. W…
Finite subgroups of the profinite completion of good groups
本文证明剩余有限好群的有限 $p$-子群共轭类可自然双射对应到其有限完备化，并刻画中心化子与正规化子的闭包性质，且推广至有限可解子群并给出几何群论应用。
Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre's concept of G-complete reducibility for subgroups of G. The separability hypothesis appears in many general theorems concerning G-complete reducibility. We demonstrate that many of these results fail without this hypothesis. On the other hand, we prove that if G is a connected reductive group and p is very go…
Solvable Groups and Affine Actions on the Line
本文研究有限生成可解群在实区间上的保向作用，证明C^1微分同胚作用必半共轭于仿射作用，并给出同胚情形下的局部半共轭结果及仿射作用族的鲁棒性条件。
Dense and comeager conjugacy classes in zero-dimensional dynamics
研究了可数群在Cantor空间上的最小和拓扑transitive作用的波兰空间，探讨了共轭类的存在性，发现自由群的最小作用和概率测度保持作用存在稠密且来ager共轭类，而可数非有限生成阿贝尔群则不存在。
Let $\tilde f\colon(S^2,\tilde A)\to(S^2,\tilde A)$ be a Thurston map and let $M(\tilde f)$ be its mapping class biset: isotopy classes rel $\tilde A$ of maps obtained by pre- and post-composing $\tilde f$ by the mapping class group of $(S^2,\tilde A)$. Let $A\subseteq\tilde A$ be an $\tilde f$-invariant subset, and let $f\colon(S^2,A)\to(S^2,A)$ be the induced map. We give an analogue of the Birman short exact sequence: just as the mapping class group $\mathrm{Mod}(S^2,\tilde A)$ is an iterated extension of $\mathrm{Mod}(S^2,A)$ by fundamental groups of punctured spheres, $M(\tilde f)$ is an …
We introduce and study the notion of a chain group of homeomorphisms of a one-manifold, which is a certain generalization of Thompson's group $F$. The resulting class of groups exhibits a combination of uniformity and diversity. On the one hand, a chain group either has a simple commutator subgroup or the action of the group has a wandering interval. In the latter case, the chain group admits a canonical quotient which is also a chain group, and which has a simple commutator subgroup. On the other hand, every finitely generated subgroup of $\operatorname{Homeo}^+(I)$ can be realized as a subgr…
We study symmetric random walks on finitely generated groups of orientation-preserving homeomorphisms of the real line. We establish an oscillation property for the induced Markov chain on the line that implies a weak form of recurrence. Except for a few special cases, which can be treated separately, we prove a property of "global stability at a finite distance": roughly speaking, there exists a compact interval such that any two trajectories get closer and closer whenever one of them returns to the compact interval. The probabilistic techniques employed here lead to interesting results for t…
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