arXiv research

A nontrivial element in a group is a generalized torsion element if some nonempty finite product of its conjugates is the identity. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental…

New findings on generating mapping class groups using pseudo-Anosov elements.

In this paper, we establish that, for statistically convex-cocompact actions, contracting elements are exponentially generic in counting measure. Among others, the following exponential genericity results are obtained as corollaries for the set of hyperbolic elements in relatively hyperbolic groups, the set of rank-1 e…

This note shows that if two elements of equal trace (e.g., conjugate elements) generate an arithmetic two-bridge knot or link group, then the elements are parabolic. This includes the figure-eight knot and Whitehead link groups. Similarly, if two conjugate elements generate the trefoil knot group, then the elements are…

Classifies reciprocal elements in Hecke groups, generalizing Sarnak's work.

New examples of hyperbolic links with generalized torsion elements found.

Proves mapping class group generated by two torsion elements for certain surfaces.

Study shows Morse elements are common in acylindrically hyperbolic groups.

Let $S_g$ be the closed oriented surface of genus g and let $\text{Mod}(S_g)$ be the mapping class group. When the genus is at least 3, $\text{Mod}(S_g)$ can be generated by torsion elements. We prove the follow results. For $g \geq 4$, $\text{Mod}(S_g)$ can be generated by 4 torsion elements. Three generators are invo…

We know that any element A of the group SO(3) can be represented as A = A1 A2 A1', where A1, A1' are elements of SO1(2)={A is an element of SO(3) | Ae1=e1}, and SO2(2)={A is an element of SO(3) | Ae2=e2} . This fact is known as Euler's angle. When this situation, a matrix A is called the generator. In the present paper…

We show that the mapping class group of a closed oriented surface of genus at least three is generated by 3 elements of order 3 and by 4 elements of order 4. Note that the mapping class group cannot be generated by finitely many torsion elements of same order if genus is equal to one or two.

The paper classifies 3-manifold groups with specific torsion elements.

The paper classifies reversible elements in Seifert-fibered spaces and braid groups.

New finite element method for complex forms in any dimension.

Wajnryb proved that the mapping class group of an orientable surface is generated by two elements. We prove that one of these generators can be taken as a Dehn twist. We also prove that the extended mapping class group is generated by two elements, again one of which is a Dehn twist. Another result we prove is that the…

Three elements generate balanced superelliptic mapping class groups.

Two elements generate all mappings of a nonorientable surface.

Two elements generate extended mapping class groups of certain surfaces.

The paper improves convergence rates of curvature approximations using Regge elements.

The study finds generalized torsion elements in 3-manifolds from specific knot surgeries.

We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper CAT(0) spaces, elements acting hyperbolically on the Bass-Serr…

The study finds infinitely many hyperbolic 3-manifolds with large rank and generalized torsion elements.

Paper maps Hamiltonians and line elements in manifolds.

New generalized torsion found in 3-manifolds.

The paper constructs hyperbolic elements in multiple spaces.

In Garside groups, axes of Morse elements are strongly contracting.

It is well known that any knot group is torsion-free, but it may admit a generalized torsion element. We show that the knot group of any negative twist knot admits a generalized torsion element. This is a generalization of the same claim for the knot $5_2$, which is the $(-2)$-twist knot, by Naylor and Rolfsen.

New findings on generating mapping class groups with specific torsion elements.

Researchers found that the twist subgroup can be generated by two elements for certain surface genera.

This paper studies parabolic quasi-Coxeter elements in complex reflection groups and their combinatorial properties.

Loxodromic elements are pseudo-Anosov on specific graphs.

Let $S_g$ be the closed oriented surface of genus g and let $\text{Mod}^{\pm}(S_g)$ be the extended mapping class group of $S_g$. When the genus is at least 5, we prove that $\text{Mod}^{\pm}(S_g)$ can be generated by two torsion elements. One of these generators is an order 2 element, and the other one is an order 4g+…

New findings on generating mapping class groups of nonorientable surfaces.

We consider interactive algorithms in the pool-based setting, and in the stream-based setting. Interactive algorithms observe suggested elements (representing actions or queries), and interactively select some of them and receive responses. Pool-based algorithms can select elements at any order, while stream-based algo…

Paper introduces Deep Sets for Symmetric Elements (DSS) layers for learning sets of symmetric elements.

We show that for any $k$ at least $6$ and $g$ sufficiently large, the mapping class group of a surface of genus $g$ can be generated by three elements of order $k$. We also show that this can be done with four elements of order $5$. We additionally prove similar results for some permutation groups, linear groups, and a…

Study on minimal torsion topological generators for mapping class groups of infinite-type surfaces.

It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds, and verify the conjecture for non-hyperbolic, geometric 3-manifolds. We also confirm the conjecture for some infinite fami…

We compute the growth series and the growth functions of reducible and pseudo-Anosov elements of the pure mapping class group of the sphere with four holes with respect to a certain generating set. We prove that the ratio of the number of pseudo-Anosov elements to that of all elements in a ball with center at the ident…

Random walks on metric spaces embed quasi-isometrically into the space.

Characterizes stably elliptic elements in Lie groups and their properties.

This paper studies the generic behavior of $k$-tuple elements for $k\ge 2$ in a proper group action with contracting elements, with applications towards relatively hyperbolic groups, CAT(0) groups and mapping class groups. For a class of statistically convex-cocompact action, we show that an exponential generic set of …

Let Mod_{g,b} denote the mapping class group of a surface of genus g with b punctures. Feng Luo asked in a recent preprint if there is a universal upper bound, independent of genus, for the number of torsion elements needed to generate Mod_{g,b}. We answer Luo's question by proving that 3 torsion elements suffice to ge…

The study shows pseudo-Anosovs are common in mapping class groups.

For random elements in free groups, we find a rank and set of subgroups.

This paper finds minimal sets of generators for mapping class groups of specific surfaces.

Let $Σ_{g,p}$ be a closed oriented surface of genus $g\geq 1$ with $p$ punctures. Let $\rm Mod(Σ_{\textit{g,p}})$ be the mapping class group of $Σ_{g,p}$. Wajnryb proved in [Wa] that for $p=0, 1$ $\rm Mod({Σ_{\textit{g,p}}})$ is generated by two elements. Korkmaz proved in [Ko] that one of these generators can be taken…

Given a finite set of $r$ points in a closed surface of genus $g$, we consider the torsion elements in the mapping class group of the surface leaving the finite set invariant. We show that the torsion elements generate the mapping class group if and only if $(g, r) \neq (2, 5k+4)$ for some integer $k$.