The paper explores non-amenability in infinite-type surfaces and graphs.
problem Determining non-amenability in mapping class groups of infinite-type surfaces and graphs.
method Analyzes mapping class groups of infinite-type surfaces and graphs, provides examples and exhibits classes of groups.
result Completely determines non-amenability of mapping class groups of infinite-type surfaces and graphs.
New computations show various properties of bounded cohomology in finitely presented groups.
problem Understanding bounded cohomology properties in finitely presented groups.
method Computational and theoretical analysis of bounded cohomology.
result Existence of finitely presented non-amenable boundedly acyclic groups and groups with uncountable bounded cohomology.
Study irregular behavior of ball averages for non-amenable group actions on foliations.
problem Exploring irregular behavior of ball averages for non-amenable group actions.
method Introducing a new mechanism based on group structure to analyze irregular behavior.
result First examples of codimension one foliations with non-existent length averages.
The Dirichlet random walk on manifolds has a positive escape rate if the cover is non-amenable.
problem Analyzing the stochastic behavior of Dirichlet random walks on manifolds.
method Defining a recursive process on Galoisian covers and proving a theorem about the escape rate.
result The escape rate is positive if and only if the cover is non-amenable.
We prove the existence of Veech groups having a critical exponent strictly greater than any elementary Fuchsian group (i.e. >21) but strictly smaller than any lattice (i.e. <1). More precisely, every affine covering of a primitive L-shaped Veech surface X ramified over the singularity and a non-periodic …
We show that surface groups are flexibly stable in permutations. This is the first non-trivial example of a non-amenable flexibly stable group. Our method is purely geometric and relies on an analysis of branched covers of hyperbolic surfaces. Along the way we establish a quantitative variant of the LERF property for s…
We study hyperbolic cohomology classes in the general context of simplicial complexes and prove homological invariance statements for them. We relate the existence of hyperbolic cohomology classes to the non-amenability of the fundamental group. In degree two we clarify the relation between hyperbolic and atoroidal cla…
Let X and Y be infinite graphs, such that the automorphism group of X is nonamenable, and the automorphism group of Y has an infinite orbit. We prove that there is no automorphism-invariant measure on the set of spanning trees in the direct product X times Y. This implies that the minimal spanning forest corresponding …
We prove that every countable family of countable acylindrically hyperbolic groups has a common finitely generated acylindrically hyperbolic quotient. As an application, we obtain an acylindrically hyperbolic group Q with strong fixed point properties: Q has property FLp for all p∈[1,+∞), and every ac…
Groups with specific properties have vanishing ℓ2-Betti numbers.
problem Understanding ℓ2-Betti numbers for certain groups. method Introduced cheap 1-rebuilding property and used structure theorem of Tucker-Drob.
result First ℓ2-Betti numbers vanish for specified groups. A symplectic form is called hyperbolic if its pull-back to the universal cover is a differential of a bounded one-form. The present paper is concerned with the properties and constructions of manifolds admitting hyperbolic symplectic forms. The main results are: * If a symplectic form represents a bounded cohomology cl…
Recent research in coarse geometry revealed similarities between certain concepts of analysis, large scale geometry, and topology. Property A of G.Yu is the coarse analog of amenability for groups and its generalization (exact spaces) was later strengthened to be the large scale analog of paracompact spaces using parti…
Study non-vanishing ℓ2-Betti numbers for specific groups.
problem Calculating non-vanishing ℓ2-Betti numbers for certain groups. method Using Euler characteristics, higher Kazhdan projections, and Baum-Connes assembly map.
result Non-vanishing calculations for delocalised ℓ2-Betti numbers. Researchers calculate spectral dimension of complex networks using renormalization group theory.
problem Understanding diffusion properties in complex systems.
method Renormalization group theory applied to graph Laplacians of simplicial complexes.
result Spectral dimension decreases with randomness in topological structure.
Measure-scaling quasi-isometries on graphs have specific scaling groups.
problem Understanding the scaling groups of graphs under quasi-isometries.
method Analyzing measure-scaling quasi-isometries on graphs and their properties.
result The scaling group of a graph is invariant under measure-scaling quasi-isometries.
Study of quasimorphisms and bounded cohomology in braided Thompson groups.
problem Investigate quasimorphisms and bounded cohomology in braided versions of Thompson groups.
method Analyze quasimorphisms and bounded cohomology of various braided Thompson groups.
result Found infinite-dimensional spaces of quasimorphisms in some braided Thompson groups and trivial second bounded cohomology in others.
Classifies quasi-isometry types of lamplighter groups over one-ended groups.
problem Classifying the quasi-isometry types of lamplighter groups over one-ended groups.
method New geometric interpretation of lamplighter groups using quasi-median spaces.
result Complete classification of quasi-isometry types of wreath products F≀H. The firefighter game problem on locally finite connected graphs was introduced by Bert Hartnell. The game on a graph G can be described as follows: let fn be a sequence of positive integers; an initial fire starts at a finite set of vertices; at each (integer) time n≥1, fn vertices which are not on fire b…
Boundary rigidity defined for hyperbolic spaces, tied to geometric properties.
problem Understanding boundary rigidity in Gromov hyperbolic spaces.
method Analyzing properties of Gromov hyperbolic spaces and their boundaries.
result Boundary rigidity is equivalent to positive Cheeger isoperimetric constant and non-amenability.
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.
The paper describes geometrically how certain groups act on surfaces.
problem Understanding the geometric structure of virtual Schottky groups.
method Geometric structural decomposition of virtual Schottky groups.
result Provides a geometrical structural decomposition for specific virtual Schottky 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.