Limits of conjugacy classes under iterates of Hyperbolic elements of $\mathsf{Out(\mathbb{F})}$
Pritam Ghosh
备注：
Several typos corrected; a misleading/ conflicting notation has been changed in the statement of the main theorem and the results which depend on it; introduction reorganized; references updated
For a free group $\mathbb{F}$ of finite rank such that $\text{rank}(\mathbb{F})\geq 3$, we prove that the set of weak limits of a conjugacy class in $\mathbb{F}$ under iterates of some hyperbolic $ϕ\in\mathsf{Out(\mathbb{F})}$ is equal to the collection of generic leaves and singular lines of $ϕ$. As an application we describe the ending lamination set for a hyperbolic extension of $\mathbb{F}$ by a hyperbolic subgroup of $\mathsf{Out(\mathbb{F})}$ in a new way and use it to prove results about Cannon-Thurston maps for such extensions. We also use it to derive conditions for quasiconvexity of finitely generated, infinite index subgroups of $\mathbb{F}$ in the extension group. These results generalize similar results obtained by Mahan Mj, Kapovich-Lustig and use different techniques.
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 construct new non-associative key establishment protocols for all left self-distributive (LD), multi-LD-, and mutual LD-systems. The hardness of these protocols relies on variations of the (simultaneous) iterated LD-problem and its generalizations. We discuss instantiations of these protocols using generalized shifted conjugacy in braid groups and their quotients, LD-conjugacy and $f$-symmetric conjugacy in groups. We suggest parameter choices for instantiations in braid groups, symmetric groups and several matrix groups.
S. Gersten announced an algorithm that takes as input two finite sequences $\vec K=(K_1,\dots, K_N)$ and $\vec K'=(K_1',\dots, K_N')$ of conjugacy classes of finitely generated subgroups of $F_n$ and outputs: (1) $\mathsf{YES}$ or $\mathsf{NO}$ depending on whether or not there is an element $θ\in \mathsf{Out}(F_n)$ such that $θ(\vec K)=\vec K'$ together with one such $θ$ if it exists and (2) a finite presentation for the subgroup of $\mathsf{Out}(F_n)$ fixing $\vec K$. S. Kalajdžievski published a verification of this algorithm. We present a different algorithm from the point of view of Culle…
$\mathcal{C}$-Hereditarily conjugacy separable groups and wreath products
School of Computer and Information Sciences, University of Newcastle（纽卡斯尔大学计算机与信息科学学院）；Department of Computer Science, Faculty of Science, University of South Bohemia in České Budějovice（南波希米亚大学理学院计算机科学系）；IMPAN（波兰科学院数学研究所）
In adjoint reductive groups $H$ of type $\mathsf{D}$ we show that for every semisimple element $s$, its centralizer splits over its connected component, i.e., $C_H(s) = C_H(s)^\circ \rtimes \check A$ for some complement $\check A$ with strong stability properties. We derive several consequences about the action of automorphisms on semisimple conjugacy classes. This helps to parametrize characters of the finite groups $\mathsf{D}_{l,\text{sc}}(q)$ and $^2\mathsf{D}_{l,\text{sc}}(q)$ and describe the action of automorphisms on them. It is also a contribution to the final proof of the McKay conje…
In his earlier work, the author introduced a group theory question that arises in the study of iterated Galois groups of post-critically finite quadratic polynomials. In this paper, we prove the first non-trivial results on this question.
In this paper a relation between iterated cyclings and iterated powers of elements in a Garside group is shown. This yields a characterization of elements in a Garside group having a rigid power, where 'rigid' means that the left normal form changes only in the obvious way under cycling and decycling. It is also shown that, given X in a Garside group, if some power X^m is conjugate to a rigid element, then m can be bounded above by ||Δ||^3. In the particular case of braid groups, this implies that a pseudo-Anosov braid has a small power whose ultra summit set consists of rigid elements. This s…
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 …
Let $G$ be a nonabelian, simple group with a nontrivial conjugacy class $C \subseteq G$. Let $K$ be a diagram of an oriented knot in $S^3$, thought of as computational input. We show that for each such $G$ and $C$, the problem of counting homomorphisms $π_1(S^3\setminus K) \to G$ that send meridians of $K$ to $C$ is almost parsimoniously $\mathsf{\#P}$-complete. This work is a sequel to a previous result by the authors that counting homomorphisms from fundamental groups of integer homology 3-spheres to $G$ is almost parsimoniously $\mathsf{\#P}$-complete. Where we previously used mapping class…
A new class of groups $\mathcal{C}$, containing all coherent RAAGs and all toral relatively hyperbolic groups, is defined. It is shown that, for a group $G$ in the class $\mathcal{C}$, the $\mathbb{Z}[t]$-exponential group $G^{\mathbb{Z}[t]}$ may be constructed as an iterated centraliser extension. Using this fact, it is proved that $G^{\mathbb{Z}[t]}$ is fully residually $G$ (i.e. it has the same universal theory as $G$) and so its finitely generated subgroups are limit groups over $G$. If $\mathbb{G}$ is a coherent RAAG, then the converse also holds - any limit group over $\mathbb{G}$ embeds…
We present a new operation to be performed on elements in a Garside group, called cyclic sliding, which is introduced to replace the well known cycling and decycling operations. Cyclic sliding appears to be a more natural choice, simplifying the algorithms concerning conjugacy in Garside groups and having nicer theoretical properties. We show, in particular, that if a super summit element has conjugates which are 'rigid' (that is, which have a certain particularly simple structure), then the optimal way of obtaining such a rigid conjugate through conjugation by positive elements is given by it…
The cycling operation is a special kind of conjugation that can be applied to elements in Artin's braid groups, in order to reduce their length. It is a key ingredient of the usual solutions to the conjugacy problem in braid groups. In their seminal paper on braid-cryptography, Ko, Lee et al. proposed the {\it cycling problem} as a hard problem in braid groups that could be interesting for cryptography. In this paper we give a polynomial solution to that problem, mainly by showing that cycling is surjective, and using a result by Maffre which shows that pre-images under cycling can be computed…
