The paper classifies reversible elements in Seifert-fibered spaces and braid groups.
problem Classifying reversible elements in Seifert-fibered spaces and braid groups.
method Classification of reversible elements in Fuchsian groups, application to Seifert-fibered groups, and analysis of 3-torsion elements.
result Classification and analysis of reversible and 3-torsion elements in Seifert-fibered spaces and braid groups.
Classifies reciprocal elements in Hecke groups, generalizing Sarnak's work.
problem Classifying reciprocal elements in Hecke groups.
method Classifying and parametrizing reciprocal classes in Hecke groups Γp for p≥3. result Generalizes Sarnak's result on reciprocal elements in the modular group.
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…
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 paper classifies 3-manifold groups with specific torsion elements.
problem Classifying 3-manifold groups with generalized torsion elements of order two.
method Analyzing the fundamental groups of 3-manifolds and their conjugates.
result 3-manifold groups with generalized torsion elements of order two have been classified.
New findings on generating mapping class groups using pseudo-Anosov elements.
problem Generating mapping class groups using specific types of elements.
method Proving the generation of mapping class groups by pseudo-Anosov elements and conjugate reducible but not periodic elements.
result The mapping class group can be generated by two conjugate pseudo-Anosov elements with arbitrarily large dilatations for surfaces of genus greater than or equal to nine.
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…
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…
Study shows Morse elements are common in acylindrically hyperbolic groups.
problem Understanding generic elements in acylindrically hyperbolic groups.
method Analyzing Morse elements and outer automorphisms.
result Morse elements are common in acylindrically hyperbolic groups.
Classifies reversible and strongly reversible elements in quaternionic groups.
problem Classifying reversible and strongly reversible elements in quaternionic groups.
method Proves elements are reversible if and only if they are products of skew-involutions (resp. involutions).
result Proves elements are reversible if and only if they are products of skew-involutions (resp. involutions).
New group with non-loxodromic Morse element found.
problem Finding non-loxodromic Morse elements in groups.
method Small-cancellation techniques to construct a Morse local-to-global group.
result Found an infinite-order Morse element that is not loxodromic.
Characterizes periodic elements in Artin-Tits groups via stability conditions.
problem Understanding periodic elements in Artin-Tits groups.
method Dynamical characterization via 2-Calabi-Yau category and stability conditions.
result An element is periodic if and only if it has a fixed point in the stability manifold.
The paper classifies reversible and strongly reversible elements in Hermitian isometry groups.
problem Classifying reversible and strongly reversible elements in Hermitian isometry groups.
method Classification through group theory and algebraic manipulation.
result New classification of strongly reversible elements in Sp(n).
Two elements generate extended mapping class groups of certain surfaces.
problem Generating extended mapping class groups with specific elements.
method Analyzing finite order elements and isotopy classes of homeomorphisms.
result Extended mapping class groups of certain surfaces are generated by two elements of finite order.
Three elements generate balanced superelliptic mapping class groups.
problem Generating balanced superelliptic mapping class groups.
method Proving groups are generated by three elements through normalizers and liftable mapping class groups.
result Balanced superelliptic mapping class groups are generated by three elements.
Persistent elements are ubiquitous in knot groups, especially for hyperbolic knots.
problem Identifying persistent elements in knot groups under Dehn fillings.
method Combining techniques from knot theory and hyperbolic geometry, including Dehn fillings and automorphisms.
result Persistent elements are structurally pervasive in knot groups, not just rare exceptions.
The study calculates the growth rate of reciprocal hyperbolic elements in Hecke groups.
problem Counting reciprocal hyperbolic elements in Hecke groups.
method Analyzes conjugacy classes of hyperbolic elements associated with reciprocal geodesics.
result Determines the asymptotic growth rate and limiting constant of primitive conjugacy classes of reciprocal hyperbolic elements.
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…
New examples of hyperbolic links with generalized torsion elements found.
problem Finding generalized torsion elements in the fundamental groups of hyperbolic links.
method Analyzing the Weeks manifold, figure-eight sister manifold, and Whitehead sister link to identify generalized torsion elements.
result First examples of hyperbolic links with link groups admitting generalized torsion elements.
Characterizes stably elliptic elements in Lie groups and their properties.
problem Understanding stably elliptic elements in Lie groups and their geometric and algebraic properties.
method Characterization through fixed point algebra and Weyl group action; relates to maximal invariant cones and compactness of order intervals.
result Connected components of stably elliptic elements can be described using Weyl group action on a compactly embedded Cartan subalgebra.
Proves mapping class group generated by two torsion elements for certain surfaces.
problem Generating mapping class group with two torsion elements.
method Analyzes surfaces of different genera and orders, proving generation by two elements of specific orders.
result Mapping class group generated by two torsion elements for g≥6 and other genera. The study shows pseudo-Anosovs are common in mapping class groups.
problem Counting pseudo-Anosovs in mapping class groups.
method Using weakly contracting isometries and Morse elements.
result Pseudo-Anosovs are generic in mapping class groups.
Two elements generate all mappings of a nonorientable surface.
problem Generating the mapping class group of a nonorientable surface.
method Proving two elements generate the mapping class group for g≥13. result The mapping class group of a nonorientable surface of genus g≥13 can be generated by exactly two elements. 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…
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.
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…
New findings on generating mapping class groups with specific torsion elements.
problem Understanding the structure of mapping class groups through torsion elements.
method Analyzing the orders of torsion elements required to generate mapping class groups of surfaces of varying genus.
result For specific genera, mapping class groups can be generated by two torsion elements of particular orders.
In Garside groups, axes of Morse elements are strongly contracting.
problem Understanding the dynamics of Morse elements in Garside groups.
method Analyzing the Cayley graph of Garside groups modulo their center, using Garside generators.
result Morse elements act loxodromically on the additional length graph of Garside groups.
The paper shows how contracting elements in groups lead to large quotients with specific growth rates.
problem Understanding the growth rates of group actions with contracting elements.
method Using extension lemma, rotating families theory, and quasi-tree construction.
result There exist sequences of quotient groups with growth rates approaching the original group's growth rate.
Extended logarithm for solvable elements in mapping class groups.
problem Logarithm of Johnson map extension to solvable elements.
method Extension to exponential solvable elements in mapping class groups using solvable Lie groups.
result Solvability of extended logarithm.
Positive Thompson links are proven for oriented subgroup elements.
problem Proving properties of Thompson links with positive elements.
method Analyzing elements of the oriented subgroup of the Thompson group.
result Positive oriented Thompson links are established.
The study finds infinitely many hyperbolic 3-manifolds with large rank and generalized torsion elements.
problem Finding hyperbolic 3-manifold groups with large rank and generalized torsion elements.
method Constructing specific hyperbolic 3-manifolds with given properties.
result Infinitely many hyperbolic 3-manifolds with generalized torsion elements of arbitrarily large order.
Loxodromic elements are pseudo-Anosov on specific graphs.
problem Characterizing loxodromic elements in specific groups.
method Analyzing subgroups acting on multiarc and curve graphs, and the handlebody group on disk graphs.
result Loxodromic elements are pseudo-Anosov on witness graphs.
Study growth rates of subgroups in groups with a constricting element.
problem Understanding growth rates of subgroups in groups with a constricting element.
method Examining the spectrum of relative and quotient exponential growth rates of quasi-convex subgroups.
result Determine when growth rates of subgroups are strictly smaller or coincide with the group's growth rate.
The study finds generalized torsion elements in 3-manifolds from specific knot surgeries.
problem Identifying generalized torsion elements in 3-manifolds from knot surgeries.
method Using the JSJ-decomposition of the 3-manifold and bi-orderability properties.
result Existence and properties of generalized torsion elements in specific 3-manifolds.
The paper studies cohomology of groups with contracting elements.
problem Understanding the cohomology of groups with specific elements.
method Proving infinite-dimensional relative bounded cohomology for groups with contracting elements.
result The cohomology is infinite-dimensional for groups with contracting elements.
This paper classifies reversible and strongly reversible elements in affine groups.
problem Classifying reversible and strongly reversible elements in affine groups.
method Identifying affine transformations and using conjugacy by involutions.
result Classification of reversible and strongly reversible elements in affine groups.
This paper studies parabolic quasi-Coxeter elements in complex reflection groups and their combinatorial properties.
problem Characterizing and studying parabolic quasi-Coxeter elements in complex reflection groups.
method Defining and characterizing parabolic quasi-Coxeter elements, studying collections of reduced reflection factorizations and relative generating sets.
result Computing cardinalities of collections of reduced reflection factorizations and relative generating sets for large families of parabolic quasi-Coxeter elements.
Let G be a Garside group with Garside element Δ, and let Δm be the minimal positive central power of Δ. An element g∈G is said to be 'periodic' if some power of it is a power of Δ. In this paper, we study periodic elements in Garside groups and their conjugacy classes. We show that the periodicity of an…
The paper proves conditions for the isomorphism between standard and dual Artin groups.
problem Conditions for the isomorphism between standard and dual Artin groups.
method Analyzes Coxeter systems and their actions on reduced words to prove isomorphisms.
result Proves conditions for the isomorphism between standard and dual Artin groups.
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)=(2,5k+4) for some integer k.
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 52, which is the (−2)-twist knot, by Naylor and Rolfsen.
The paper classifies and decomposes quaternionic projective transformations.
problem Classifying and decomposing elements of the projective linear group PSL(3,H). method Algebraic characterization of dynamical types using reversibility, decomposition of elements into simple elements.
result Offered a complete classification for elements of SL(3,R). We prove that a ``bouillabaisse'' surface (translation surface which has two transverse parabolic elements) has totally real trace field. As a corollary, non trivial Veech groups which have no parabolic elements do exist. The proof follows Veech's viewpoint on Thurston's construction of pseudo-Anosov diffeomorphisms.
Artin-Tits groups act on a certain delta-hyperbolic complex, called the "additional length complex". For an element of the group, acting loxodromically on this complex is a property analogous to the property of being pseudo-Anosov for elements of mapping class groups. By analogy with a well-known conjecture about mappi…
A positive integer m will be called a {\it finitistic order} for an element γ of a group Γ if there exist a finite group G and a homomorphism h:Γ→G such that h(γ) has order m in G. It is shown that up to conjugacy, all but finitely many elements of a given finitely generated, torsion-free Kleinian gr…
The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. The authors and Putman proved that this group is generated by Dehn twists about separating curves fixed by t…
New method counts link components from Thompson group elements.
problem Counting connected components of positive Thompson links.
method Defining Thompson permutations from Thompson group elements.
result Number of orbits equals number of link components.