We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as subgroups of word hyperbolic groups. As an …
Infinite genus surfaces have diverse Veech groups.
problem Characterizing Veech groups of infinite genus surfaces.
method Proving every countable subgroup without contracting elements is a Veech group of a surface with infinitely many topological types.
result There are no restrictions on the topological type of surfaces to realize all possible uncountable Veech groups.
The paper proves no exotic actions of diffeomorphism groups on 1-manifolds.
problem The study tackles the exotic actions of diffeomorphism groups on 1-dimensional manifolds.
method The approach involves showing any nontrivial homomorphism has a standard form, with countably many embeddings and conjugate actions.
result The groups Diff^r_c(M) have no countable index subgroups, solving a conjecture of Matsumoto.
We prove that two countable locally finite-by-abelian groups G,H endowed with proper left-invariant metrics are coarsely equivalent if and only if their asymptotic dimensions coincide and the groups are either both finitely-generated or both are infinitely generated. On the other hand, we show that each countable group…
We present an alternative approach to the result of Guentner, Higson, and Weinberger concerning the Baum-Connes conjecture for finitely generated subgroups of SL(2,C). Using finite-dimensional methods, we show that the Baum-Connes assembly map for such groups is an isomorphism.
We classify Veech groups of tame non-compact flat surfaces. In particular we prove that all countable subgroups of GL+(2,R) avoiding the set of mappings of norm less than 1 appear as Veech groups of tame non-compact flat surfaces which are Loch Ness monsters. Conversely, a Veech group of any tame flat surf…
Generalizes manifold results for Lie groups, proving equivariant homotopy type.
problem Hilbert-Smith conjecture for topological G-manifolds. method Generalization of previous results, verification of n-classifying spaces.
result Any Palais-proper topological G-manifold has the equivariant homotopy type of a countable proper G-CW complex. Perfect mapping class groups of specific surfaces have no proper subgroups.
problem Characterizing the structure of mapping class groups of surfaces with removed Cantor sets.
method Automatic continuity of the groups, proven by Mann.
result These groups have no proper finite-index subgroups and trivial abelianization.
The paper explores centers of subgroups in mapping class groups and their relation to free groups and Tits alternatives.
problem Investigating centers of subgroups in mapping class groups and their properties.
method Similar techniques to show the presence of nonabelian free groups and failure of Tits alternatives.
result No big mapping class group satisfies the strong Tits alternative, and many have trivial centers.
We study Farrell Nil-groups associated to a finite order automorphism of a ring R. We show that any such Farrell Nil-group is either trivial, or infinitely generated (as an abelian group). Building on this first result, we then show that any finite group that occurs in such a Farrell Nil-group occurs with infinite mu…
We prove that every right-angled Artin group embeds into the C∞ diffeomorphism group of the real line. As a corollary, we show every limit group, and more generally every countable residually RAAG group, embeds into the C∞ diffeomorphism group of the real line.
We study the finitely generated Hausdorff spectrum of spinal automorphism groups acting on rooted trees. Given any α∈[0,1], we construct a branch group Gα such that Gα has a finitely generated subgroup H where H has Hausdorff dimension α in G. Using results by Barnea, Shalev and Klopsch we further de…
Groups can be embedded in larger, simpler groups with special properties.
problem Embedding all groups into simpler, more manageable groups.
method Constructing a finitely generated, hopfian, and complete group G∗ from any countable group G. result The constructed group G∗ has trivial center and all finite subgroups are conjugate to those in G. New framework shows C∗-simplicity for groups without certain subalgebras.
problem Characterizing C∗-simplicity of groups. method Introducing confined subalgebras and Uniformly Recurrent States.
result A countable discrete group is C∗-simple if it has no non-trivial amenable confined subalgebras. We present a simple approach to questions of topological orbit equivalence for actions of countable groups on topological and smooth manifolds. For example, for any action of a countable group Γ on a topological manifold where the fixed sets for any element are contained in codimension two submanifolds, every orbit e…
The paper extends end concepts to arbitrary groups and spaces.
problem Extending end concepts to arbitrary groups and spaces.
method Description of maximal coarse compactification and geometric proof of end properties.
result Generalization of Stallings' theorem and definition of ends for coarse spaces.
Analytic torsion defined for non-compact Lie groups and discrete subgroups.
problem Defining and calculating analytic torsion for non-compact Lie groups and their discrete subgroups.
method Localised analytic torsion and relative analytic torsion defined for Lie groups of type I, using representations and discrete subgroups.
result Relative analytic torsion of (G,Γ) coincides with Lott L2 analytic torsion of a covering space. This note proves equivariant de Rham cohomology for quotient spaces.
problem Computing de Rham cohomology of quotient spaces under group actions.
method Equivariant identification of de Rham complexes using foliation theory.
result Canonical isomorphism of de Rham complexes for quotient spaces.
We introduce a geometric invariant, called finite decomposition complexity (FDC), to study topological rigidity of manifolds. We prove for instance that if the fundamental group of a compact aspherical manifold M has FDC, and if N is homotopy equivalent to M, then M x R^n is homeomorphic to N x R^n, for n large enough.…
New subgroups of mapping class groups constructed for infinite-type surfaces.
problem Constructing new subgroups of mapping class groups for infinite-type surfaces.
method Utilization of special homeomorphisms called shift maps and multipush maps.
result Countably (and uncountably in certain cases) many non-conjugate embeddings of subgroups into mapping class groups.
The paper constructs surfaces with infinite-genus Veech groups from finite subgroups of GL(2, R).
problem Constructing surfaces with specific Veech groups from given subgroups.
method PSV construction and modifications to produce tame translation surfaces.
result Ends of constructed surfaces can be decomposed into a group's ends and a dense subset.
Proper proximality proved for various groups on non-positive curvature spaces.
problem Proper proximality of groups acting on non-positive curvature spaces.
method Established proper proximality for groups acting on CAT(0) spaces and hierarchically hyperbolic groups. result Proper proximality of many groups including mapping class groups and subgroups of curve graphs.
Classifies manifolds and discrete subgroups of Lie groups using descriptive set theory.
problem Classifying manifolds and discrete subgroups of Lie groups.
method Descriptive set theory and Borel complexity computations.
result Complexity of homeomorphism problems for manifolds and conjugacy relations for subgroups.
Proves properties of arithmetic lattices and hyperbolic manifolds.
problem Properties of arithmetic lattices and hyperbolic manifolds.
method Study of normalizers of lattices and subgroup growth theory.
result Every arithmetic lattice has the property of being the normalizer of many sublattices.
A graph product kernel means the kernel of the natural surjection from a graph product to the corresponding direct product. We prove that a graph product kernel of countable groups is special, and a graph product of finite or cyclic groups is virtually cocompact special in the sense of Haglund and Wise. The proof of th…
We prove that any countable discrete and torsion free subgroup of a general linear group over an arbitrary field or a similar subgroup of an almost connected Lie group satisfies the integral algebraic K-theoretic (split) Novikov conjecture over \cpt and §, where \cpt denotes the C^*-algebra of compact operators and §de…
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.
New group structure from interval homeomorphisms.
problem Understanding homeomorphisms of the interval.
method Introducing chain groups and studying their properties.
result Uncountably many isomorphism types of chain groups and subgroups.
Study shows complex K3 surfaces have infinite free abelian subgroup in their diffeomorphism group.
problem Understanding the structure of diffeomorphism groups of complex K3 surfaces.
method Used families of Seiberg-Witten invariants and moduli spaces of Einstein metrics.
result Proved the existence of a free abelian subgroup of countably infinite rank in the identity component of the diffeomorphism group.
Countable modular groups found on surfaces with infinite type.
problem Finding modular groups of infinite type surfaces.
method Proving countable modular groups for orientable infinite type surfaces.
result Every orientable infinite type surface has a countable modular group.
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.
Researchers create earring spaces from metric spaces to study fundamental groups.
problem Understanding fundamental groups of constructed spaces from metric spaces.
method Attach copies of the circle to a dense subset of a separable metric space to form an earring space, then analyze its fundamental group.
result The fundamental group of the earring space is isomorphic to a subgroup of the Hawaiian earring group under specific conditions.
Given a constant magnetic field on Euclidean space Rp determined by a skew-symmetric (p×p) matrix Θ, and a Zp-invariant probability measure μ on the disorder set Σ which is by hypothesis a Cantor set, where the action is assumed to be minimal, the corresponding Integrated Density…
Study chaotic behavior in homeomorphism groups of countable products of spaces.
problem Investigate chaotic behavior in homeomorphism groups of countable products of various metrizable topological spaces.
method Construct numerous examples of chaotic groups of homeomorphisms of countable products of spaces.
result New chaotic groups of homeomorphisms of countable products of various metrizable topological spaces are discovered.
New bounds on group asymptotic dimension derived from finitely amenable actions.
problem Bounding the asymptotic dimension of groups using finitely amenable actions.
method Introduced finitely F-amenable actions and used them to derive upper bounds on asymptotic dimension. result Upper bounds on asymptotic dimension of groups based on finitely amenable actions.
Study homeomorphism groups of ordinals, proving strong distortion and normal generators.
problem Understanding algebraic and geometric properties of homeomorphism groups of ordinals.
method Analyzing successor ordinals with connections to permutation groups and manifolds.
result Proves strong distortion and normal generators for homeomorphism groups of ordinals.
Finite groups can be represented as origami automorphisms, extended to countable groups.
problem Representing countable groups as automorphisms of origamis.
method Considering origamis on the Loch Ness monster.
result Every countable group can be represented as origami automorphisms.
Let G be a simply connected, solvable Lie group and Γ a lattice in G. The deformation space D(Γ,G) is the orbit space associated to the action of $\Aut(G)$ on the space X(Γ,G) of all lattice embeddings of Γ into G. Our main result generalises the classical rigidity theorems of Mal'tsev…
The study classifies subgroups of outer automorphisms of free products.
problem Classifying subgroups of outer automorphisms of free products.
method Geometric tool: boundaries of relative factor graphs and equivalence classes of arational trees.
result Every finitely generated subgroup either contains a relatively fully irreducible automorphism or virtually preserves a conjugacy class.
Free group automorphisms group rigidity proven.
problem Proving rigidity of Out(F_N).
method Measure equivalence rigidity, new canonical splittings.
result Superrigidity of Out(F_N).
Study on diffeomorphism groups with Hölder continuity properties.
problem Characterizing subgroups of diffeomorphism groups with specific Hölder continuity.
method Analyzes groups of C^k diffeomorphisms on circles or intervals with Hölder continuous derivatives.
result Existence of finitely generated subgroups with no injective homomorphisms into higher Hölder groups.
New insights into ends of quotient spaces and graphs.
problem Understanding the ends of quotient spaces and graphs.
method Analyzing infinite volume ends of quotient spaces and graphs.
result Quotient spaces and graphs have exactly one infinite volume end under certain conditions.
We classify up to coarse equivalence all countable abelian groups of finite torsion free rank. The Q-cohomological dimension and the torsion free rank are the two invariants that give us such classification. We also prove that any countable abelian group of finite torsion free rank is coarsely equivalent to Z^n + H whe…
Mixing endomorphisms found on toroidal groups and their products.
problem Finding topologically mixing endomorphisms on toroidal groups and their products.
method Analyzing continuous endomorphisms on toroidal groups and their countable products.
result Proves existence of infinitely many topologically mixing endomorphisms on countable infinite toroidal groups.
Extends inf-convolution to countable risk measures for risk sharing.
problem Limited inf-convolution theory to finite sets of risk measures.
method Extends inf-convolution to countable sets, investigates properties and results.
result Generalizes known properties and results to countable case.
Unified approach to studying hyperbolic groups using stable subspaces and Morse boundaries.
problem Understanding the geometric and algebraic properties of hyperbolic groups.
method Unified approach to viewing geodesic metric spaces as unions of stable subspaces, using quasi-convex subsets and direct limits of Gromov boundaries.
result Unified understanding of stable subgroups and Morse boundaries, leading to new quasi-isometry invariant dimensions.
Every countable compact subset of sphere is tame.
problem Characterizing compact subsets of spheres.
method Proving homeomorphic complements imply homeomorphic subsets.
result Wild subspaces like Antoine contain Cantor sets.
Let W be an infinite Coxeter group. We initiate the study of the set E of limit points of "normalized" positive roots (representing the directions of the roots) of W. We show that E is contained in the isotropic cone of the bilinear form B associated to a geometric representation, and illustrate this property with nume…