VC dimensions of group CNNs are infinite for certain kernels and groups.
problem Estimating the generalization capacity of group convolutional neural networks.
method Identifying precise VC dimension estimates for simple sets of group CNNs.
result Two-parameter families of convolutional neural networks have an infinite VC dimension for infinite groups and certain kernels.
Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.
problem Properties of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
method Construction and study of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
result Hypoelliptic logarithmic Sobolev inequalities on the space.
Veech groups uniformize Teichmüller geodesic curves in Riemann moduli space. Recently, examples of infinitely generated Veech groups have been given. We show that these can even have infinitely many cusps and infinitely many infinite ends. We further show that examples exist for which each direction of an infinite end …
The paper defines infinite Schottky groups and their applications to infinite type surfaces.
problem Understanding group actions on infinite type surfaces.
method Definition and analysis of infinite Schottky groups and their properties.
result Every infinite type Riemann surface can be obtained as a quotient of a region of discontinuity of an infinite Schottky group.
Study infinite superelliptic curves and their Veech groups, providing geometric and algebraic insights.
problem Characterize Veech groups of infinite superelliptic curves.
method Analyzing geometric properties, differential equations, and group theory.
result Veech groups of infinite superelliptic curves are all matrices permuting branched points.
Study infinite genus surfaces and Schottky groups for uniformization.
problem Investigate infinite genus surfaces and Schottky groups for uniformization.
method Definitions and proofs for infinite genus surfaces and Schottky groups, showing uniformization by Schottky groups.
result Infinite genus surfaces and handlebodies can be topologically and quasiconformally uniformized by Schottky groups.
Most Finsler metrics have infinite-dimensional holonomy groups.
problem Understanding the holonomy groups of Finsler metrics.
method Analyzing the set of Finsler metrics on a manifold.
result An open dense subset of Finsler metrics have infinite-dimensional holonomy groups.
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.
Overview of infinite surface mapping class groups.
problem Understanding mapping class groups of infinite surfaces.
method Survey of recent research findings.
result Recent developments in mapping class groups of infinite surfaces.
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.
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.
The paper creates exotic 4-manifold structures with a specific group.
problem Producing exotic structures on 4-manifolds with infinite dihedral fundamental group.
method Using specific conditions on b2+ and b2−, the paper constructs these structures. result The existence of infinite exotic structures on 4-manifolds with infinite dihedral fundamental group.
Simplified presentations for non-orientable surface mapping class groups.
problem Presenting the mapping class group of non-orientable surfaces.
method Provided simpler infinite presentations.
result Simplified presentations for the group.
Study of mapping class groups on infinite graphs, focusing on their large-scale geometry.
problem Understanding the large-scale geometry of mapping class groups on infinite graphs.
method Using coarse geometry techniques, classify coarsely bounded groups and compute asymptotic dimension.
result Identify conditions for global and local coarsely bounded pure mapping class groups of infinite rank graphs.
Cyclohedra are a well-known infinite familiy of finite-dimensional polytopes that can be constructed from centrally symmetric triangulations of even-sided polygons. In this article we introduce an infinite-dimensional analogue and prove that the group of symmetries of our construction is a semidirect product of a degre…
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.
Study shows infinite manifold types for every group.
problem Finding manifold types for every finite group.
method Proved existence of infinitely many compact complex hyperbolic 2-manifolds.
result For every finite group, there are infinitely many isomorphism classes of compact complex hyperbolic 2-manifolds with automorphism group isomorphic to the group.
Big mapping class groups of infinite type surfaces have infinite asymptotic dimension.
problem Understanding asymptotic dimension of big mapping class groups of infinite type surfaces.
method Analyzing big mapping class groups with coarsely bounded generating sets and essential shifts.
result Big mapping class groups of infinite type surfaces have infinite asymptotic dimension.
This paper computes the quadratic Witt groups (the Wall L-groups) of the polynomial ring Z[t] and the integral group ring of the infinite dihedral group, with various involutions. We show that some of these groups are infinite direct sums of cyclic groups of order 2 and 4. The techniques used are quadratic linking form…
Study of flip graphs and their automorphism groups for infinite-type surfaces.
problem Understanding automorphism groups of flip graphs for infinite-type surfaces.
method Examined the relationship between mapping class groups and flip graphs for infinite-type surfaces.
result Extended mapping class groups are isomorphic to proper subgroups of automorphism groups of flip graphs.
Study shows quotient group is infinitely generated.
problem Understanding the structure of homology cobordism groups.
method Proved using Seifert fibered spaces.
result Quotient group is infinitely generated.
The study finds infinite geodesics on manifolds with specific homotopy group properties.
problem Existence of non-contractible closed geodesics on manifolds with certain homotopy group properties.
method Analyzes the homotopy groups and their action on the fundamental group to prove the existence of infinitely many closed geodesics.
result Infinitely many geometrically distinct closed geodesics exist under specific conditions on homotopy groups.
Study shows compact mapping class groups of infinite type surfaces are never perfect.
problem Characterizing the perfection of mapping class groups of infinite type surfaces.
method Analyzing the closure of compactly supported mapping class groups and Torelli groups, examining their abelianizations.
result The abelianization of the closure of compactly supported mapping class groups contains uncountable direct sums of rationals.
The paper classifies topological holonomy groups in SO(3).
problem Classifying holonomy groups in SO(3). method Analyzing and categorizing groups based on their properties.
result Different types of holonomy groups in SO(3) are identified and classified. The paper provides infinite presentations for surface groups.
problem Understanding fundamental groups of surfaces.
method Infinite presentations of fundamental groups using simple loops.
result Infinite presentations for fundamental groups of surfaces and non-orientable surfaces.
Convex-cocompact groups in infinite hyperbolic space are deformable.
problem Understanding deformability of convex-cocompact groups in infinite hyperbolic spaces.
method Proving convex-cocompact representations form an open set and using bending to deform them.
result Deformable convex-cocompact representations of surface groups not conjugate to exotic PSL(2,R) representations.
Book on infinite-dimensional Lie groups, covering basics and various classes.
problem Understanding Lie groups in infinite-dimensional spaces.
method Develops smooth manifolds and Lie groups in locally convex spaces, discussing various classes.
result Detailed exploration of infinite-dimensional Lie groups and their properties.
Study on mapping class groups of infinite type surfaces.
problem Property Pextnaive for mapping class groups of infinite type surfaces. method Analyzes the existence of elements g and hi satisfying specific group properties. result Establishes the existence of g for any finite collection of non-trivial elements hi. The paper extends Cartan development to infinite dimensional Lie groups.
problem Generalizing Cartan development to infinite dimensional Lie groups.
method Generalization of Cartan development to infinite dimensional manifolds and Lie groups.
result The tangent mapping of a Cartan development is another Cartan development.
New smooth structures found on certain 4D spaces.
problem Finding distinct smooth structures on specific 4D spaces.
method Constructing specific 4-manifolds with infinite fundamental group.
result Infinitely many non-diffeomorphic structures found.
We use controlled topology applied to the action of the infinite dihedral group on a partially compactified plane and deduce two consequences for algebraic K-theory. The first is that the family in the K-theoretic Farrell-Jones conjecture can be reduced to only those virtually cyclic groups which admit a surjection wit…
Study on infinite-type surfaces shows stable commutator length is continuous and defines open subgroups.
problem Understanding stable commutator length on infinite-type surfaces.
method Analyzing mapping class groups of infinite-type surfaces, showing continuity and openness of commutator subgroups.
result Stable commutator length defines a continuous function on commutator subgroups of infinite-type mapping class groups.
Study shows certain infinite-ended groups are not quasi-isometrically rigid.
problem Understanding when infinite-ended groups are quasi-isometrically rigid.
method Combining results on subgroups and hyperbolic groups, adapting Whyte's argument.
result Proves certain infinite-ended groups are not quasi-isometrically rigid.
The study of infinite groups through their finite quotients in geometry.
problem Understanding properties of infinite groups from their finite images.
method Analyzing infinite groups through their finite quotients and using low-dimensional topology.
result Recent results show how finite images can determine the group completely in some cases.
We show that the verbal width is infinite for acylindrically hyperbolic groups, which include hyperbolic groups, mapping class groups and Out(Fn).
We prove that rational homology of the Torelli group of genus g is infinite dimensional, provided g>6. This means that rational homology of the Torelli space of genus g>6 is infinite dimensional. The Torelli groups with marked points are also considered. In addition, we prove that rational homology of the subgroup of t…
Study mapping class groups of infinite type surfaces with noncompact boundaries.
problem Classify pure mapping class groups of infinite type surfaces.
method Developed a method to cut surfaces into simpler ones and combined recent results.
result Complete classification of perfect and uniformly perfect pure mapping class groups.
The Birman-Hilden theory is extended to infinite type surfaces and branched covers.
problem Extending Birman-Hilden theory to surfaces of infinite type and branched covers of infinite degree.
method Proving the Birman-Hilden property for fully ramified branched covering maps.
result The mapping class group of a non-orientable surface of infinite type can be realized as a subgroup of the mapping class group of its orientable double cover.
This paper provides an infinite presentation for a subgroup of mapping class groups of non-orientable surfaces.
problem Finite presentations for mapping class groups of non-orientable surfaces.
method Using Stukow's finite presentation and Birman exact sequences.
result An infinite presentation for the twist subgroup of the mapping class group of a compact non-orientable surface.
Study of pure mapping class groups on infinite graphs.
problem Classifying graphs with specific mapping class groups.
method Completely classified graphs with pure mapping class groups.
result Established semidirect product decomposition and computed first integral cohomology.
We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional g…
Study surfaces in 4-manifolds with cyclic fundamental group.
problem Characterize surfaces with infinite cyclic knot group.
method Algebraic topological criteria and homeomorphism proofs.
result Ambiently isotopic surfaces with the same genus have a homeomorphism.
The study finds conditions for groups acting on CAT(0) cube complexes to have infinite girth.
problem Conditions for groups acting on CAT(0) cube complexes to have infinite girth.
method Analyzes lattices in automorphism groups of finite dimensional CAT(0) cube complexes.
result Groups either have infinite girth or are {locally finite}-by-{virtually abelian}.
We study automorphisms of a relatively hyperbolic group G. When G is one-ended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgroups of GL_n(Z) fixing…
Multitwists cannot generate all compactly supported mapping class groups on infinite-type surfaces.
problem Generating compactly supported mapping class groups on infinite-type surfaces.
method Analyzing closure of compactly supported mapping class groups and their relation to multitwists.
result Closure of compactly supported mapping class groups is not generated by multitwists.
Infinite rank groups found in 3-manifolds with infinite fundamental groups.
problem Understanding the structure of diffeomorphism and homeomorphism groups of 3-manifolds with infinite fundamental groups.
method Analyzing actions of barbell diffeomorphisms on spaces of embedded arcs and configuration spaces.
result Groups of diffeomorphisms and homeomorphisms have infinite rank.
Study of mapping class groups of infinite graphs, focusing on their finiteness and commensurability.
problem Understanding the finiteness properties and commensurability of mapping class groups of infinite graphs.
method Investigation of asymptotically rigid mapping class groups, construction of explicit presentations, and analysis of algebraic and geometric properties.
result Graph Houghton groups are not commensurable with other known Houghton-type groups, defining a new class of groups.
We show that all non-trivial continuous endomorphisms of the circle group are topologically mixing. We also show that there exists a large infinite class of continuous endomorphisms of any n-dimensional torus group which are topologically mixing. Lastly, we prove that any continuous endomorphism on an abelian polish se…