arXiv research

Autoconj automates conjugacy exploitation in Python without a DSL.

Smoothly conjugate perturbations of certain toral automorphisms.

Margalit and Schleimer constructed nontrivial roots of the Dehn twist about a nonseparating curve. We prove that the conjugacy classes of roots of the Dehn twist about a nonseparating curve correspond to the conjugacy classes of periodic maps with certain conditions. Futhermore, we give data set which determine the con…

Proposes PG-DA for Bayesian MMNL estimation to handle non-conjugacy.

Study Agol cycles for pseudo-Anosov 3-braids.

Constructs equivariant analytic torsion for proper actions on manifolds.

We prove that fundamental groups of non-orientable 3-manifolds have a solvable conjugacy problem, and construct an algorithm. Together with our earlier work on the conjugacy problem in groups on orientable geometrizable 3-manifolds, all $π_1$ of (geometrizable) 3-manifolds have a solvable conjugacy problem. In corollar…

Study weak conjugacy in surface homeomorphisms.

Study of conjugacy classes in infinite-type surfaces' mapping class groups.

Classifies manifolds with dense conjugacy classes in their mapping class groups.

The paper generalizes knot signatures to tori using representations and invariants.

We use the theory of group actions on profinite trees to prove that the fundamental group of a finite, 1-acylindrical graph of free groups with finitely generated edge groups is conjugacy separable. This has several applications: we prove that positive, $C'(1/6)$ one-relator groups are conjugacy separable; we provide a…

Study conjugacy classes of parabolic diffeomorphisms fixing the origin.

In this article, we study connections between representation theory and efficient solutions to the conjugacy problem on finitely generated groups. The main focus is on the conjugacy problem in conjugacy separable groups, where we measure efficiency in terms of the size of the quotients required to distinguish a distinc…

We show that every non-decreasing function $f\colon \mathbb N\to \mathbb N$ bounded from above by $a^n$ for some $a\ge 1$ can be realized (up to a natural equivalence) as the conjugacy growth function of a finitely generated group. We also construct a finitely generated group $G$ and a subgroup $H\le G$ of index 2 such…

Criterion for stopping conjugacy class enumeration in triangle groups.

This paper explores links from Thompson's group conjugacy classes.

Paper derives asymptotic formula for conjugacy classes in groups with contracting elements.

We obtain new variations of the original McShane identity for those SL(2,C)-representations of the once punctured torus group which satisfy the Bowditch conditions, and also for those fixed up to conjugacy by an Anosov mapping class of the torus and satisfying the relative Bowditch conditions.

In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a conformal automorphism of prime order which reflects the action the automorphism in an…

New algorithm broadens BART models applicability.

We prove that the fundamental group of any Seifert 3-manifold is conjugacy separable. That is, conjugates may be distinguished in finite quotients or, equivalently, conjugacy classes are closed in the pro-finite topology.

The paper characterizes finite metacyclic subgroups in mapping class groups of surfaces.

We prove that under fairly general conditions an iterated exchange move gives infinitely many non-conjugate braids. As a consequence, every knot has infinitely many conjugacy classes of n-braid representations if and only if it has one admitting an exchange move.

Proves cosets of certain subgroups in hyperbolic 3-manifold groups are conjugacy distinguished.

Study uses braid conjugacy class invariants to find ribbon obstructions for knots.

The cycling operation endows the super summit set $S_x$ of any element $x$ of a Garside group $G$ with the structure of a directed graph $Γ_x$. We establish that the subset $U_x$ of $S_x$ consisting of the circuits of $Γ_x$ can be used instead of $S_x$ for deciding conjugacy to $x$ in $G$, yielding a faster and more pr…

The paper counts conjugacy classes in Out(F_N) using hyperbolic spaces.

The study examines conditions for symmetric and alternating subgroups in mapping class groups of surfaces.

Given a group 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 $φ$-conjugacy classes. One says that $Γ$ has the $R_\infty$-property if there are infinitely many $φ$-conjugacy classes for every automorphism $φ$ of $Γ$. I…

Proves any three or more knots can form a genus-zero link in a 3-manifold.

We show that there is a family of pseudo-Anosov braids independently parameterized by the braid index and the (canonical) length whose smallest conjugacy invariant sets grow exponentially in the braid index and linearly in the length and conclude that the conjugacy problem remains exponential in the braid index under t…

There are recent cryptographic protocols that are based on Multiple Simultaneous Conjugacy Problems in braid groups. We improve an algorithm, due to Sang Jin Lee and Eonkyung Lee, to solve these problems, by applying a method developed by the author and Nuno Franco, originally intended to solve the Conjugacy Search Pro…

Study conjugacy growth in free groups and products of finite groups.

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 ty…

The study examines aperiodicity properties of automorphism groups of free products of groups.

We give a complete description of conjugacy classes of finite subgroups of the mapping class group of the sphere with r marked points. As a corollary we obtain a description of conjugacy classes of maximal finite subgroups of the hyperelliptic mapping class group. In particular, we prove that for a fixed genus g there …

New growth rate for pseudo-Anosov conjugacy classes in Teichmüller space.

Let $\mathcal M (Σ, \mathcal P)$ be the mapping class group of a punctured oriented surface $(Σ, \mathcal P)$ (where $\mathcal P$ may be empty), and let $\mathcal T_p(Σ,\mathcal P)$ be the kernel of the action of $\mathcal M (Σ, \mathcal P)$ on $H_1 (Σ\setminus \mathcal P, \mathbb F_p)$. We prove that $\mathcal T_p(Σ, …

The paper classifies dense conjugacy classes in mapping class groups of locally finite graphs.

We give a quadratic lower bound on the dimension of the space of conjugacy classes of subgroups of SL(n,R) that are limits under conjugacy of the diagonal subgroup. We give the first explicit examples of abelian n-1 dimensional subgroups of SL(n,R) which are not such a limit, however all such abelian groups are limits …

Enhances HDP-HMM for state transitions between similar states.

There exist combable groups in which the conjugacy problem is unsolvable. The isomorphism problem is unsolvable for certain recursive sequences of finite presentations of combable groups.

Counting subgroups of a surface using convex core lengths.

New bounds reveal double exponential growth in conjugacy classes of fully irreducibles.

The paper finds formulas for word lengths and conjugacy classes in surface groups.

Solvability of the conjugacy problem for relatively hyperbolic groups was announced by Gromov [Hyperbolic groups, MSRI publications 8 (1987)]. Using the definition of Farb of a relatively hyperbolic group in the strong sense [B Farb, Relatively hyperbolic groups, Geom. Func. Anal. 8 (1998) 810-840], we prove this asser…

The paper solves the conjugacy problem in a specific braid group quotient and finds infinite virtually cyclic subgroups.