Groups of importance in group theory have flexible stability properties.
problem Stability and flexibility of groups in geometric and combinatorial group theory.
method Establishing Kirchberg's Local Lifting Property and Lubotzky--Shalom's Property FD for specific groups.
result Groups like 3-manifold groups, limit groups, and certain one-relator groups are very flexibly stable. This paper generalizes property (QT) to a broader class of groups.
problem Proving property (QT) for a wider range of groups.
method Using projection complex machinery and hierarchical hyperbolic groups.
result Established sufficient conditions for groups to have property (QT).
Groups with Property (T) have fiber products with Property (T).
problem When does the fiber product of groups with Property (T) have Property (T)?
method Analyzing fiber products of groups with Property (T).
result Fiber products of groups with Property (T) also have Property (T).
The paper explores the twisted Rokhlin property in mapping class groups of surfaces.
problem Classifying surfaces whose mapping class groups have the twisted Rokhlin property.
method Generalizing the Rokhlin property to the twisted version, the authors classify surfaces based on their mapping class groups' properties.
result The mapping class groups of connected orientable infinite-type surfaces without boundaries have the twisted Rokhlin property, while those of other surfaces do not.
Survey on foliations and diffeomorphism groups.
problem Relationship between algebraic and homotopical properties.
method Survey and analysis of existing literature.
result Explains the connection between diffeomorphism groups and foliations.
Study classifies 2-manifolds with special homeomorphism groups.
problem Classifying 2-manifolds with specific homeomorphism groups.
method Introduced virtual Rokhlin property; classified homeomorphism groups of 2-manifolds.
result 2-manifolds with the virtual Rokhlin property have specific homeomorphism groups.
Proves congruence subgroup property for mapping class groups of hyperbolic surfaces.
problem Residual finiteness of hyperbolic groups and congruence subgroup property for mapping class groups.
method Assumption of residual finiteness of hyperbolic groups leads to proof of congruence subgroup property.
result Congruence subgroup property for mapping class groups of hyperbolic surfaces.
We say that a group has property R∞ if any group automorphism has an infinite number of twisted conjugacy classes. Fel'shtyn and Goncalves prove that the solvable Baumslag-Solitar groups BS(1,m) have property R∞. We define a solvable generalization Γ(S) of these groups which we show to have proper…
The paper studies groups with proper actions on finite products of hyperbolic spaces.
problem Characterizing groups with proper actions on finite products of hyperbolic spaces.
method Lineal actions and central extensions of groups.
result Groups with property (PH) or (PH') have specific properties related to their actions on hyperbolic spaces.
Groups with hyperbolic properties don't have strong Property (T).
problem Proving groups with hyperbolic properties don't have strong Property (T).
method Constructing an unbounded affine representation with polynomial growth.
result Groups with hyperbolic properties do not have strong Property (T).
Study on finiteness properties of handlebody mapping class groups.
problem Understanding finiteness properties of asymptotically rigid handlebody groups.
method Introduced asymptotically rigid mapping class groups and determined their finiteness properties based on the space of ends of handlebodies.
result Homology of these groups coincides with stable homology of handlebody groups in some cases.
Survey on algebraic fibers of group extensions and their finiteness properties.
problem Existence and finiteness of algebraic fibers in group extensions.
method Analysis of abstract and pro-p groups. result Finiteness properties of algebraic fibers in group extensions.
Survey on handlebody groups and their properties.
problem Understanding handlebody groups and their properties.
method Survey and recent trends in geometric group theory.
result Application of the cheap α-rebuilding property to handlebody group homology growth. Study compares and unifies finiteness properties of locally compact groups.
problem Understanding finiteness properties of locally compact groups.
method Comparing and unifying three families of finiteness properties: type Cn, coarse (n−1)-connectedness, and type Fn. result All three families lead to the same notion for locally compact groups.
Proves graph 3-manifold groups have two specific properties.
problem Understanding fundamental groups of graph 3-manifolds.
method Constructing sequences of covers to prove properties.
result Graph 3-manifold groups are virtually poly-free and in Lex family.
Bi-orderable groups from left-orderable ones, showing non-profinite properties.
problem Non-profinite properties in bi-orderability and generalized torsion elements.
method Using fiber products to construct bi-orderable groups, showing non-profinite properties.
result Bi-orderability and generalized torsion elements are not profinite properties.
Non-isomorphic groups with similar profinite completions found.
problem Finding non-isomorphic groups with similar profinite completions.
method Exhibited infinitely many pairs of non-isomorphic groups with specific properties.
result Groups with Property FA and non-trivial actions on trees have isomorphic profinite completions.
3-manifold groups have a property that allows them to act on quasi-trees.
problem Proving property (QT) for 3-manifold groups.
method Analyzing the geometry of 3-manifolds and using quasi-trees.
result Compact, orientable 3-manifold groups have property (QT) if they don't support Sol or Nil geometry.
Proves 3D Poincaré duality groups without property (T)
problem Residually finite 3D Poincaré duality groups and property (T)
method Using coboundary expansion and recent results on 3-manifold groups
result 3D Poincaré duality groups without property (T)
New hyperbolic groups exhibit unusual finiteness properties.
problem Finding groups with specific finiteness properties.
method Fibre product construction and homomorphisms to Z and Z2. result Examples of hyperbolic groups with kernels of type Fk but not Fk+1. This is a survey of the recent work in algorithmic and asymptotic properties of groups. I discuss Dehn functions of groups, complexity of the word problem, Higman embeddings, and constructions of finitely presented groups with extreme properties (monsters).
The paper studies groups with specific actions on hyperbolic spaces and finds that subgroups are either amenable or contain a free group.
problem Characterizing subgroups of groups with specific actions on hyperbolic spaces.
method Analyzing groups with property (PPH) and (PPT) and their subgroups.
result Any finitely generated subgroup of a finitely generated group with property (PPH) either is amenable or contains \(F_2\).
Proves Congruence Subgroup Property for two types of groups.
problem Proving Congruence Subgroup Property for specific groups.
method Elementary proof of Johnson filtration and geometric subsurface inclusions.
result Proves Congruence Subgroup Property for nilpotent quotients and subsurface subgroups.
Groups with certain properties have invariant subalgebra rigidity.
problem Invariant subalgebra rigidity in groups with specific properties.
method Analyzing normal subgroups and invariant subalgebras in groups.
result Torsion-free acylindrically hyperbolic groups and hyperbolic groups have the relative ISR property.
Random groups with high density have Property (T).
problem Determining the conditions for Property (T) in random groups.
method Analyzing k-gonal random groups and their limits. result Property (T) holds with high probability for high densities.
Paper studies R∞-property for braid groups on surfaces, proving it for most cases.
problem Investigating the R∞-property for braid groups on orientable surfaces.
method Analyzing surface pure and full braid groups to determine the R∞-property.
result Most surface braid groups have the R∞-property, with exceptions noted.
Groups with special properties always have fixed points.
problem Groups acting on finite CW-complexes without fixed points.
method Exhibited specific groups with strong fixed-point properties.
result Groups with finite generation and torsion-freeness have global fixed points.
Uniform proof of property R_infinity for specific Artin-Tits groups.
problem Establishing property R_infinity for various Artin-Tits groups.
method Uniform short proof for multiple groups using similar techniques.
result Property R_infinity confirmed for specified Artin-Tits groups.
New group found not satisfying quasi-isometric triviality property.
problem Identifying groups that do not satisfy a specific quasi-isometric property.
method Analyzing central extensions and cohomology properties of groups.
result Found a finitely generated group that does not satisfy Property QITB.
Accessible groups with infinitely many ends have infinitely many twisted conjugacy classes.
problem Characterizing groups with infinitely many ends and their conjugacy classes.
method Analyzing accessible groups and relatively hyperbolic groups to deduce properties.
result Groups with infinitely many ends have infinitely many twisted conjugacy classes.
We examine residual properties of word-hyperbolic groups, adapting a method introduced by Darren Long to study the residual properties of Kleinian groups.
A generalized Baumslag-Solitar group is the fundamental group of a graph of groups all of whose vertex and edge groups are infinite cyclic. Levitt proves that any generalized Baumslag-Solitar group has property R-infinity, that is, any automorphism has an infinite number of twisted conjugacy classes. We show that any g…
The paper bounds abnormal and Goh-abnormal sets for metabelian Lie groups with polarizations.
problem Bounding abnormal and Goh-abnormal sets for metabelian Lie groups.
method Analyzing rank 2 polarizations and sub-Riemannian structures on metabelian Lie groups.
result Metabelian Lie groups with polarizations satisfy the minimizing Sard property.
New group theory insights on knot surgery results.
problem Understanding non-simply connected 3-manifolds from Dehn surgery.
method Group theoretic analysis of Property P conjecture variations.
result New group theoretic perspectives on Dehn filling.
The paper explores properties of continuous actions on manifolds, proving bounds on subgroup size and fixed points.
problem Properties of continuous finite group actions on topological manifolds.
method Analyzes properties including Jordan property and almost fixed point property, proving bounds on subgroup size.
result Existence of a constant C such that for any continuous action of a finite group G on a manifold X, there is a subgroup H with [G:H] ≤ C and a fixed point.
Paper proves naturally reductive property is inaudible for certain manifolds.
problem Proving naturally reductive property is inaudible for specific manifolds.
method Using a pair of non-compact 11-dimensional generalized Heisenberg groups, the paper proves the naturally reductive property is inaudible.
result The naturally reductive property is inaudible for certain manifolds.
New findings on group actions and stabilizers of infinite sets.
problem Understanding finiteness properties of stabilizers in group actions.
method Analyzing oligomorphic actions and permutational wreath products.
result Existence of finite subsets with non-FP∞ stabilizers. Pseudo-automorphisms are birational transformations acting as regular automorphisms in codimension 1. We import ideas from geometric group theory to prove that a group of birational transformations that satisfies a fixed point property on CAT(0) cubical complexes, for example a discrete countable group with Kazhdan Pro…
Characterizes groups with specific boundary properties.
problem Groups with Schottky set boundaries.
method Study relatively hyperbolic group pairs with Schottky boundaries.
result Groups with boundaries where Schottky sets have 1 or 2 component incidence graphs.
The study examines groups with a specific automorphism property using BNS-invariant.
problem When does a group have the finitely generated fixed subgroup property?
method Using BNS-invariant to analyze groups of the form GimesZm. result Provides partial answers and examples for groups with the property.
The paper explores higher property T in lattices and its connections to geometric phenomena.
problem Understanding higher property T in lattices and related geometric phenomena.
method Operator-algebraic characterizations of higher property T and connections to lattice geometry.
result Unified framework for understanding higher property T and related geometric phenomena.
New hyperbolic 3-pseudomanifolds with unique properties.
problem Understanding cubulable groups in hyperbolic 3-manifolds.
method Constructing compact hyperbolic 3-manifolds with specific boundary conditions.
result Found groups that are word hyperbolic but not cubulable.
Study presentations of groups that can be generalised over continuous open group monomorphisms.
problem Investigate presentations of groups that can be generalised over continuous open group monomorphisms.
method Systematic study of presentations with generalisation properties, focusing on right-angled Artin groups (RAAGs).
result Establish high connectivity properties for universal Salvetti-type complexes and novel examples of LC groups with prescribed compactness properties.
The study finds optimal curvature pinching in Heintze groups.
problem Exploring curvature properties in Heintze groups.
method Examining metric properties of rank-one symmetric spaces, proving existence of metrics on Heintze groups of Carnot-type.
result Optimal curvature pinching is demonstrated in a special case.
Let Γ be a discrete group with property (T) of Kazhdan. We prove that any Riemannian isometric action of Γ on a compact manifold X is locally rigid. We also prove a more general foliated version of this result. The foliated result is used in our proof of local rigidity for standard actions of higher rank semisi…
Study spectral properties of sub-Laplacians in Carnot groups.
problem Spectral properties of sub-Laplacians in Carnot groups.
method Proved pure point spectrum and spectral gap; applied to small ball problem and heat content.
result Proved existence of spectral gap and pure point spectrum.
New groups constructed from tree automorphisms, proving finiteness.
problem Finiteness properties of self-similar and Röver groups.
method Braiding of self-similar groups, use of d-ary cloning systems, analysis of embedded disks. result Construction of a new group (braided Röver group) of type F∞. We explain and generalise a construction due to Gromov to realise geometric small cancellation groups over graphs of groups as fundamental groups of non-positively curved 2-dimensional complexes of groups. We then give conditions so that the hyperbolicity and some finiteness properties of the small cancellation quotien…