The paper solves conditions for metacyclic actions on surfaces, including upper bounds and subgroup classifications.
problem Conditions for metacyclic actions on surfaces and their properties.
method Derives necessary and sufficient conditions for conjugates of torsion elements to generate finite metacyclic subgroups.
result Complete solution to liftability of periodic mapping classes under finite cyclic covers, including upper bounds and subgroup classifications.
The paper explores infinite metacyclic subgroups in mapping class groups of surfaces.
problem Existence and properties of infinite metacyclic subgroups in mapping class groups.
method Provided necessary and sufficient conditions for the existence of infinite metacyclic subgroups.
result Established the existence of specific infinite metacyclic subgroups and derived bounds on their periodic generators.
The paper characterizes finite metacyclic subgroups in mapping class groups of surfaces.
problem Characterizing finite metacyclic subgroups in mapping class groups of surfaces.
method Derives necessary and sufficient conditions for conjugates of torsion elements to generate finite split non-abelian metacyclic subgroups.
result Complete classification of certain finite metacyclic subgroups in Mod(Sg) for g≥3. Algebraic methods prove knot primality using Floer homology.
problem Proving the primality of knots.
method Knot Floer homology, metacyclic representations, and twisted homology.
result Primality tests have proven primality for over 99.6% of knots.
The study counts critical points in knot cobordisms using abelian and metacyclic invariants.
problem Counting critical points in knot cobordisms.
method Using homological invariants from cyclic and metacyclic branched covering spaces.
result For each pair of integers g and n, there exists a ribbon knot K with at least n critical points of each index in any genus g cobordism from K to its reverse.
The study examines Lee metrics on groups and their properties.
problem Characterizing groups that admit Lee metrics.
method Analyzing conditions for groups to have or not have Lee metrics, studying specific families of groups, and providing tables for groups of order ≤ 31.
result Conditions for groups to have Lee metrics, including specific families and non-cyclic groups.
Let p be an odd prime and D_p a dihedral group of order 2p. Let ρ: G(K) --> D_p --> GL(p,Z) be a non-abelian representation of the knot group G(K) of a knot K in 3-sphere. Let Δ_{ρ,K} (t) be the twisted Alexander polynomial of K associated to ρ. Let H(p) is the set of 2-bridge knots K, such that G(K) is mapped onto a n…
Let p be an odd prime. We construct a non-abelian extension Γ of S1 by Z/p×Z/p, and prove that any finite subgroup of Γ acts freely and smoothly on S2p−1×S2p−1. In particular, for each odd prime p we obtain free smooth actions of infinitely many non-metacyclic rank two p-groups on $…
A {\em solvable} cover of a graph is a regular cover whose covering transformation group is solvable. In this paper, we show that a solvable cover of a graph can be decomposed into layers of abelian covers, and also, a lift of a given automorphism of the base graph of a solvable cover can be decomposed into layers of l…
A {\em balanced} spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to Kinoshita \cite{ki}), and examine their behavior under some common operations on the graph. We…
Conditions for Baumslag-Solitar subgroups in mapping class groups.
problem Characterizing Baumslag-Solitar subgroups in mapping class groups.
method Analyzing necessary and sufficient conditions for subgroup embeddings.
result Conditions for embedding BS(p,q) in Mod(Sg), including ∣p∣=∣q∣ and reducibility. New link groups are derived from torus necklaces, connecting braid groups to reflection groups.
problem Understanding the relationship between braid groups and reflection groups.
method Constructing torus necklaces and linking them to braid groups of J-reflection groups. result Link groups of torus necklaces are precisely braid groups of J-reflection groups, with meridians as braid reflections. The study proves super-rigidity of Gromov's random monster group for various types of groups.
problem Super-rigidity of Gromov's random monster group in various group types.
method Proof of morphisms having finite image and introduction of hereditary super-rigidity.
result Gromov's random monster group has super-rigidity and hereditary super-rigidity with respect to certain groups.
We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi--direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi--direct product of free groups. From these results we obtain a normal form of words…
Virtual twin groups map to symmetric groups, revealing automorphism structure.
problem Understanding homomorphisms between virtual twin groups and symmetric groups.
method Using irreducible right-angled Coxeter groups and right-angled Artin groups.
result A complete description of homomorphisms between virtual twin groups and symmetric groups, including the structure of the automorphism group of VTn. Characterizes group connections on group bundles.
problem Understanding connections on group bundles.
method Characterizes connections as affine spaces and uses the Ambrose-Singer theorem.
result Group connections form an affine space over cocycles.
Study on totally symmetric sets with group applications.
problem Understanding totally symmetric sets and their group applications.
method Survey of existing theory and applications to various groups.
result Exploration of totally symmetric sets in multiple group contexts.
Affine cactus groups are CAT(0) and hyperbolic.
problem Characterizing geometric properties of affine cactus groups.
method Analyzing CAT(0) and hyperbolic properties through group theory.
result Affine cactus groups of degree three are hyperbolic.
The study restricts groups in graph of groups structures.
problem Realizing groups as fundamental groups of graph of groups with restricted vertex groups.
method Analyzes restrictions on groups that can be realized and applies to manifold construction.
result Places constraints on groups that can be realized in graph of groups structures.
New Garside structures found for torus knot groups and related braid groups.
problem Finding Garside structures for torus knot groups and related braid groups.
method Introducing a new Garside monoid M(n,m) for (n,m)-torus knot groups and other braid groups. result New Garside structures for (n,m)-torus knot groups and related braid groups are constructed. Logarithmic separation profile in hyperbolic groups shows hierarchical structure.
problem Understanding hierarchical structure in hyperbolic groups with logarithmic separation.
method Proving groups with logarithmic separation split over cyclic groups and providing counterexamples.
result Not all groups with hierarchical structure have logarithmic separation profile.
New Garside structures derived from groups, leading to new group properties.
problem Creating Garside structures from groups and Artin groups.
method Method for turning direct product of a group G by Z into a Garside group.
result Proved new cases of K(π,1)-conjecture for some hyperbolic type Artin groups.
We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pur…
The group of 2-by-2 matrices with integer entries and determinant ±>1 can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, name…
The study restricts normal subgroups of Kähler groups, proving specific cases and general restrictions.
problem Characterizing normal subgroups of Kähler groups.
method Analyzing embeddings and conjugation actions of surface groups and one-ended hyperbolic groups.
result Restrictions on normal subgroups of Kähler groups, including virtual direct products and surface group properties.
New reflection groups derived from torus knots with finite meridians.
problem Understanding reflection groups derived from torus knot groups with finite meridians.
method Using the theory of J-groups and Coxeter groups, study quotients of torus knot groups.
result Classification of toric reflection groups and their properties.
Graphically discrete groups have strong rigidity properties.
problem Understanding the rigidity of group actions on graphs.
method Introducing graphical discreteness and proving rigidity properties.
result Free products of graphically discrete groups are action rigid.
Paper proves vanishing homology groups for certain hyperbolic groups.
problem Understanding homology groups of specific hyperbolic groups.
method Using twisted Wirtinger presentations to prove homology group vanishing.
result Second homology groups vanish for certain Gromov hyperbolic groups.
Study fundamental groups of geometric transformation groups using loop spaces.
problem Understanding fundamental groups of geometric transformation groups.
method Use differential forms on loop spaces to prove infinite fundamental groups.
result Proves infinite fundamental groups for specific geometric transformation groups.
Simple construction of Lie 2-groups from loop group extensions.
problem Constructing Lie 2-groups from loop group extensions.
method Using conjugation action of loop group on its central extension.
result Simple construction of string 2-group as a strict Fréchet Lie 2-group.
Study knot invariants using automorphism groups of free nilpotent groups.
problem Developing knot invariants using automorphism groups.
method Nilpotently p-localization of knot groups and automorphism groups of free nilpotent groups. result Maps from outer automorphism groups yield knot invariants.
We discuss dense embeddings of surface groups and fully residually free groups in topological groups. We show that a compact topological group contains a nonabelian dense free group of finite rank if and only if it contains a dense surface group. Also, we obtain a characterization of those Lie groups which admit a dens…
We exhibit a family of infinite, finitely-presented, nilpotent-by-abelian groups. Each member of this family is a solvable S-arithmetic group that is related to Baumslag-Solitar groups, and everyone of these groups has a quasi-isometry group that is virtually a product of a solvable real Lie group and a solvable p-adic…
The study shows that certain groups can be uniquely identified by their finite abelian summands.
problem Identifying groups based on their finite abelian summands.
method Analyzing hyperbolic groups as graphs of free groups with cyclic edge groups.
result Free products of free and surface groups are profinitely rigid.
Survey on Coxeter groups for Lie group examples.
problem Understanding Coxeter groups and their applications.
method Constructing discrete subgroups of Lie groups using Coxeter groups.
result Coxeter groups provide new examples in discrete subgroups of Lie groups.
In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various subgroups of them such as the Torelli group, the IA outer automorphism group of …
The paper studies actions on Bass-Serre trees and identifies new C∗-simple groups.
problem Investigating actions of fundamental groups on Bass-Serre trees and their C∗-algebraic properties. method Analyzing boundary actions of fundamental groups of graphs of groups on their Bass-Serre trees.
result Identification of new families of C∗-simple groups, including tubular groups and certain graphs of groups. In this article we define the twisted product of groups as the generalization of the semidirect product of groups. We will find the necessary and sufficient condition in order that the twisted product of groups to be a group. In particular, for two copies of the same group, the twisted product of group by itself throug…
We find finite presentations for the automorphism group of the Artin pure braid group and the automorphism group of the pure braid group associated to the full monomial group.
Study the relationship between orbit braid group and equivariant mapping class group on surfaces.
problem Understanding the relationship between mapping class groups and braid groups with group actions.
method Using the fibration F0GMightarrowF(M/G,n) and exact sequence. result The conclusion is closely connected with the braid group of the quotient space.
The paper characterizes crystallographic groups derived from virtual braid and twin groups.
problem Characterizing crystallographic groups from virtual braid and twin groups.
method Analyzing quotients of virtual braid and twin groups by their commutator subgroups.
result The quotients of virtual braid and twin groups by their commutator subgroups are crystallographic groups.
Origamis with specific groups have Veech groups that surject onto SL(2, Z/nZ).
problem Characterizing Veech groups of origamis as totally non-congruence groups.
method Using results on SL(2, Z/nZ) and properties of origamis' deck transformation groups.
result Origamis with certain triangle group quotients have Veech groups that surject onto SL(2, Z/nZ).
New groups are hyperbolic and rigid in mapping class groups.
problem Understanding the structure of Veech groups in mapping class groups.
method Showed that Veech groups are hierarchically hyperbolic and quasi-isometrically rigid.
result Veech groups are hierarchically hyperbolic and quasi-isometrically rigid.
Outer automorphism group of hyperbolic groups is HHG under certain conditions.
problem Characterizing the outer automorphism group of hyperbolic groups.
method Proving finite-index subgroups are central extensions of orbifold mapping class groups with bounded Euler class.
result Outer automorphism group of a one-ended hyperbolic group is virtually a hierarchically hyperbolic group.
Outer automorphism groups of Coxeter groups are trivial except for small ranks.
problem Triviality of outer automorphism groups of Coxeter groups.
method Using Guirardel-Levitt outer space for free products, proving triviality and cyclic order for specific ranks.
result Outer automorphism groups are trivial except for small ranks.
Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asy…
Flawed groups are shown to include all finitely generated groups isomorphic to free products of nilpotent groups.
problem Characterizing flawed groups and understanding their topological properties.
method Analyzing finitely presented groups and their deformation retracts onto subspaces of character varieties.
result All finitely generated groups isomorphic to free products of nilpotent groups are flawed.
Computes the component group of arbitrary real algebraic groups.
problem Computing the component group of arbitrary real algebraic groups.
method Structure results on algebraic groups and Galois cohomology methods.
result The group of connected components π0G(R) is an elementary Abelian 2-group.