This paper explores groups acting on the circle with invariant laminations, called laminar groups.
problem Understanding groups acting on the circle with invariant laminations.
method Review of Thurston's theory of universal circles and follow-up work on invariant laminations.
result Groups acting on the circle with invariant laminations are called laminar groups.
New actions of mapping class groups on circles are classified.
problem Understanding actions of mapping class groups on circles.
method Minimal actions on the simple circle.
result Every action of the mapping class group on the circle is semi-conjugate to a unique minimal action on the simple circle.
This paper characterizes Fuchsian groups acting on the circle with invariant laminations.
problem Characterize Fuchsian groups acting on the circle with invariant laminations.
method Proves a structure theorem for hyperbolic 2-orbifolds and characterizes Fuchsian groups.
result Proves a complete generalization of the previous result for Fuchsian groups.
Surprising circles found in Coxeter group boundaries.
problem Embedded circles in Morse boundaries of Coxeter groups.
method Analysis of Morse boundaries and defining graphs.
result Circles not arising from visible Fuchsian subgroups.
We prove a true bootstrapping result for convergence groups acting on a Peano continuum. We give an example of a Kleinian group H which is the amalgamation of two closed hyperbolic surface groups along a simple closed curve. The limit set Lambda H is the closure of a `tree of circles' (adjacent circles meeting in pairs…
Summary of pure cactus groups and circle points.
problem Understanding pure cactus groups and configuration spaces.
method Summarizes previous work on the topic.
result Summary of results from previous papers and thesis.
Circle graph automorphisms match circle's and are strongly universal.
problem Identifying the automorphism group of the circle.
method Proving the circle graph's automorphism group coincides with the circle's and showing the circle graph's rational chords form a strongly universal element.
result The circle graph's automorphism group is strongly universal.
Investigates local indicability of groups with circle homology presentations.
problem Conditions for local indicability in groups with circle homology presentations.
method Generalizes results for two-relator presentations to circle homology presentations.
result Extends results on local indicability to LOT groups and non-cycle-free Adian presentations.
Study on fundamental groups of framed circle embeddings in 4-manifolds.
problem Understanding the fundamental groups of spaces of framed embeddings of a circle in 4-manifolds.
method Investigates the fundamental groups of spaces of embeddings of S1imesD3 in 4-manifolds, focusing on framed immersed circles. result Investigates the fundamental groups of spaces of framed embeddings of a circle in 4-manifolds, providing insights into the structure of these groups.
Explains the pure cactus group of degree three and its relation to four points on a circle.
problem Understanding the relationship between cactus groups and configuration spaces.
method Provides an explicit description of the pure cactus group of degree three and its connection to the configuration space of four points on a circle.
result Explicitly describes the relationship between the pure cactus group of degree three and the configuration space of four points on the circle.
New actions found on exotic spheres using group theory.
problem Understanding smooth transformations on exotic spheres.
method Recent progress in stable homotopy groups of spheres and group theory.
result Smooth circle and cyclic group actions on exotic spheres produced.
A new homomorphism connects group actions on circles to Euler classes.
problem Understanding group actions on circles and their implications.
method Using crossed homomorphisms and Poincaré translation numbers.
result Relates the Euler class of actions to a specific homomorphism.
Adapts pivoting technique to circle homeomorphisms for proofs.
problem Probabilistic Tits alternative and exponential synchronization.
method Adapts Gou{ë}zel's pivoting technique.
result Different proofs of probabilistic Tits alternative and exponential synchronization.
The paper describes orbits of circle-valued functions on a 2-torus.
problem Understanding the fundamental groups of orbits of circle-valued functions.
method Algebraic description of fundamental groups of orbits of circle-valued smooth functions.
result An algebraic description of fundamental groups of orbits of circle-valued smooth functions.
Study of circle configurations in the plane, proving aspherical space and computing fundamental groups.
problem Understanding the space of configurations of circles in the plane.
method Proved the space is aspherical and computed fundamental groups of its components.
result Fundamental groups are iterated semidirect products of braid groups, with structure dictated by a finite rooted tree.
A 5-manifold fibers over a circle with nonpositive curvature.
problem Finding nonpositively curved manifolds that fiber over a circle.
method Exhibited a specific 5-manifold with nonpositive curvature.
result The fiber is a 4-manifold with nonhyperbolic fundamental group.
Finite index subgroups of certain groups cannot act faithfully on the circle.
problem Finite index subgroups of specific groups cannot act faithfully on the circle.
method Analyzing C1 actions on the circle for finite index subgroups of mapping class groups, automorphism groups, and outer automorphism groups. result No orientation preserving C1 action of finite index subgroups of these groups on the circle can be faithful. For a circle packing P on the sphere invariant under a geometrically finite Kleinian group, we compute the asymptotic of the number of circles in P of spherical curvature at most T which are contained in any given region.
New findings on mapping class group actions on the circle, improving critical regularity.
problem Improving understanding of mapping class group actions on the circle.
method Analyzing actions of non-solvable groups and finite index subgroups of mapping class groups.
result Critical regularity of mapping class groups is at most one for surfaces of complexity at least three.
Paper distinguishes 2-knots with circle actions using fundamental groups.
problem Distinguishing 2-knots with circle actions.
method Using fundamental groups of 3-orbifolds of cyclic type.
result Proves that most pairs of 1-knots yield distinct branched twist spins.
Let P be a locally finite circle packing in the plane invariant under a non-elementary Kleinian group Gamma and with finitely many Gamma-orbits. When Gamma is geometrically finite, we construct an explicit Borel measure on the plane which describes the asymptotic distribution of small circles in P, assuming that either…
New example of hyperbolic 6-manifold with circle-valued Morse function.
problem Finding Morse functions on hyperbolic manifolds.
method Constructing a circle-valued Morse function on a hyperbolic 6-manifold.
result First example of a hyperbolic 6-manifold with a perfect circle-valued Morse function.
Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C^1 action of the mapping class group of S on the circle is trivial. The techniques used in the proof of this result permit us to show that products of Kazhdan groups and certain latti…
We prove that rigid representations of the fundamental group of a surface into the group of oreintation-preserving homeomorphisms of the circle are geometric, thereby establishing a converse statement of a theorem by the first author.
New geometric interpretation of a group class using circle action and rotation numbers.
problem Understanding a specific class in the mapping class group.
method Action of the punctured mapping class group on the circle and rotation numbers.
result Construction matches Furuta and Trapp's using winding numbers.
The boundary of a Möbius manifold carries a canonical Möbius structure. This enables one to define the cobordism group of n-dimensional (closed) Möbius manifolds. The purpose of this note is to show that the cobordism group of Möbius circles is zero, i.e., every Möbius circle bounds a Möbius surface. We also complete…
New topological Riemann-Roch theorem for circle fibrations.
problem Topological Riemann-Roch theorem for complex line bundles on circle fibrations.
method Construction of central extensions and application to algebraic K-theory.
result Equality of specific cohomology elements in the third cohomology group.
Minimal action on universal circle for foliations on 3-manifolds.
problem Action of fundamental group on universal circle of foliations.
method Analyzes uniform foliations on 3-manifolds, proving minimality and transitivity.
result Action is minimal and transitive on pairs of different points.
Spaces of circle embeddings in curved surfaces indexed by trees.
problem Classifying spaces of braided automorphism groups of trees.
method Indexed connected components with finite rooted trees, constructed strong deformation retract.
result Connected components are classifying spaces of braided automorphism groups.
Schenkel proved that the automorphism group of a flat Minkowski plane is a Lie group of dimension at most 6 and described planes whose automorphism group has dimension at least 4 or one of whose kernels has dimension 3. We extend these results to the case of toroidal circle planes.
The paper studies mapping class groups of 3-manifolds fibered over surfaces.
problem Understanding the structure of mapping class groups of 3-manifolds.
method Examines an exact sequence involving the mapping class groups and relates it to the Birman exact sequence.
result Determines conditions under which the exact sequence splits.
Study of symplectomorphisms on ruled surfaces under circle actions.
problem Homotopy type of equivariant symplectomorphisms on rational ruled surfaces.
method Analysis of action on compatible and invariant almost complex structures, use of Delzant's and Karshon's classifications.
result Equivariant symplectomorphisms are homotopy equivalent to tori or their pushout.
We define a "circle Euler characteristic" of a circle action on a compact manifold or finite complex X. It lies in the first Hochschild homology group of ZG where G is the fundamental group of X. It is analogous in many ways to the ordinary Euler characteristic. One application is an intuitively satisfying formula for …
We propose a program to study groups acting faithfully on S^1 in terms of number of pairwise transverse dense invariant laminations. We give some examples of groups which admit a small number of invariant laminations as an introduction to such groups. Main focus of the present paper is to characterize Fuchsian groups i…
We prove that a circle bundle over a closed oriented aspherical manifold with hyperbolic fundamental group admits a self-map of absolute degree greater than one if and only if it is virtually trivial. This generalizes in every dimension the case of circle bundles over hyperbolic surfaces, for which the result was known…
Study on regularity of exceptional actions and moduli of continuity for circle diffeomorphisms.
problem Regularity of exceptional actions of groups on the circle.
method Analysis of C1,α diffeomorphisms with free orbits and bounded derivatives. result Existence of exceptional C1,α diffeomorphisms under certain conditions on α. We present recent results on counting and distribution of circles in a given circle packing invariant under a geometrically finite Kleinian group and discuss how the dynamics of flows on geometrically finite hyperbolic 3 manifolds are related. Our results apply to Apollonian circle packings, Sierpinski curves, Schott…
Computes homotopy groups of embedding spaces of arcs or circles in 4-manifolds.
problem Computing homotopy groups of embedding spaces of arcs or circles in 4-manifolds.
method Computes homotopy groups using examples and answers questions posed by Arone and Szymik.
result Fundamental group of embedding spaces is isomorphic to the second homology group of the manifold.
We give examples of symplectic actions of a cyclic group, inducing a trivial action on homology, on four-manifolds that admit Hamiltonian circle actions, and show that they do not extend to Hamiltonian circle actions. Our work applies holomorphic methods to extend combinatorial tools developed for circle actions to stu…
We contribute to the classification of toroidal circle planes and flat Minkowski planes possessing three-dimensional connected groups of automorphisms. When such a group is an almost simple Lie group, we show that it is isomorphic to PSL(2,R). Using this result, we describe a framework for the full cl…
Geometrically constructs Virasoro-Bott group from circle diffeomorphisms.
problem Constructing the Virasoro-Bott group from circle diffeomorphisms.
method Analogous to loop group construction, using disc diffeomorphisms with special boundary conditions.
result Identifies Virasoro-Bott group as a quotient of disc diffeomorphisms with identified Lie algebra.
We study groups of circle diffeomorphisms whose action on the cylinder C=S1×S1∖Δ preserves a volume form. We first show that such a group is topologically conjugate to a subgroup of PSL(2,R), then discuss the existence of a differentiable conjugacy.
We study surface groups Γ in SO(4,1), which is the group of Mobius tranformations of S3, and also the group of isometries of H4. We consider such Γ so that its limit set ΛΓ is a quasi-circle in S3, and so that the quotient (S3−ΛΓ)/Γ is a circle bundle over a surface. This circle bundl…
We continue the study of linear families of contact forms on 3-manifolds begun in our paper `Contact geometry and complex surfaces'. The present paper introduces Teichmuller and moduli spaces for so-called taut contact circles. By constructing a developing map for taut contact circles, we show that these geometrically …
We prove a homological stability theorem for unlinked circles in 3-manifolds and give an application to certain groups of diffeomorphisms of 3-manifolds.
Study complex deformations of the circle using group cohomology and Virasoro algebra.
problem Complexification of circle diffeomorphism group and its geometric properties.
method Real-analytic maps, group cohomology, Witt algebra, Frölicher structures.
result Virasoro uniformization theorem for moduli spaces of Riemann surfaces.
The paper examines the boundedness of bundle diffeomorphism groups over a circle.
problem Investigating the boundedness of bundle diffeomorphism groups over a circle.
method Distinguishing an integer k and constructing a function to analyze the bundle diffeomorphism group.
result The bundle diffeomorphism group is uniformly perfect when k ≥ 1 and unbounded when k = 0.
We determine the rank of the fundamental group of those hyperbolic 3-manifolds fibering over the circle whose monodromy is a sufficiently high power of a pseudo-Anosov map. Moreover, we show that any two generating sets with minimal cardinality are Nielsen equivalent.