In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show th…
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48 results for “big mapping class groups”
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 thesis introduces big mapping class groups and their structure.
problem Understanding mapping class groups of infinite-type surfaces.
method Systematic introduction and analysis of structure and topological generation.
result Key differences from finite-type mapping class groups.
On the large scale geometry of big mapping class groups of surfaces with a unique maximal endmath.GT
Characterizes and analyzes the large scale geometry of big mapping class groups of surfaces.
problem Analyzing the large scale geometry of big mapping class groups of surfaces with a unique maximal end.
method Building on previous work, the paper characterizes and analyzes the large scale geometry of big mapping class groups of surfaces with a unique maximal end.
result Proves that any locally CB big mapping class group is CB generated and gives an explicit criterion for determining which big mapping class groups are CB generated.
Study big mapping class groups and their co-Hopfian property, finding new examples and proving injective homomorphisms results.
problem Characterizing co-Hopfian property in big mapping class groups of infinite-type surfaces.
method Constructing examples, proving properties, exploring injective homomorphisms.
result First examples of injective endomorphisms of mapping class groups of infinite-type surfaces that fail to be surjective.
Classifies big mapping classes on infinite type surfaces.
problem Investigating geometric properties of hyperbolic structures on infinite type surfaces.
method Analyzing the space of convex hyperbolic structures with Fenchel-Nielsen topology and studying the dynamics of mapping classes.
result Classifies big mapping classes into three types based on their dynamics on Teichmüller subspaces.
The study examines when mapping class groups are quasi-isometric to graphs of curves.
problem When is the mapping class group of an infinite-type surface quasi-isometric to a graph of curves?
method Using the work of Rosendal, Mann, and Rafi, the study defines a necessary and sufficient condition called translatability for a mapping class group to be quasi-isometric to a graph of curves.
result The mapping class group of the plane minus a Cantor set is quasi-isometric to the loop graph defined by Bavard.
Paper shows mapping class groups are not extremely amenable except for specific cases.
problem Determining when mapping class groups are extremely amenable.
method Utilized Kechris-Pestov-Todorčević machinery.
result Big mapping class groups are not extremely amenable unless the surface is a sphere or a once-punctured sphere.
We study the action of (big) mapping class groups on the first homology of the corresponding surface. We give a precise characterization of the image of the induced homology representation.
We study mapping class groups of infinite type surfaces with isolated punctures and their actions on the loop graphs introduced by Bavard-Walker. We classify all of the mapping classes in these actions which are loxodromic with a WWPD action on the corresponding loop graph. The WWPD property is a weakening of Bestvina-…
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.
We show that if is a closed three manifold with a Heegaard splitting with sufficiently big "handlebody distance" then the subgroup of the mapping class group of the Heegaard surface, which extend to both handlebodies is finite. As a corollary, this implies that under the same hypothesis, the mapping class group of …
Study on self-similar surfaces and their mapping class groups generated by involutions.
problem When do big mapping class groups of self-similar surfaces generated by involutions?
method Investigation of self-similar surfaces with self-similar ends, focusing on infinite and one maximal ends.
result For self-similar surfaces with infinite maximal ends, their mapping class groups are generated by involutions and are uniformly perfect.
Proves left-orderability of mapping class groups of infinite-type surfaces.
problem Left-orderability of mapping class groups of infinite-type surfaces.
method Inductive construction of a stable Alexander system and ideal arc systems.
result Proves left-orderability using carefully chosen exhaustion by finite-type subsurfaces.
We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these `big' mapping class groups and shows that each finite-index subgroup has finite ou…
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.
We give a criterion to prove that some groups are not acylindrically hyperbolic. As an application, we prove that the mapping class group of an infinite type surface is not acylindrically hyperbolic.
By considering appropriate finite covering spaces of closed non-orientable surfaces, we construct linear representations of their mapping class group which have finite index image in certain big arithmetic groups.
Involutions generate mapping class groups of infinite surfaces.
problem Generating involutions for mapping class groups of infinite surfaces.
method Analyzing infinite surfaces with n ends, showing involutions generate groups for n ≥ 6 and n ≥ 3.
result Involutions generate mapping class groups for n ≥ 6 and n ≥ 3.
Study shows mapping class groups are one-ended for surfaces with at least one end.
problem Analyzing the number of ends in mapping class groups of surfaces.
method Proving the associated translatable curve graph is one-ended, quasi-isometric to the mapping class group.
result Mapping class groups are one-ended for surfaces with at least one end of discrete type.
Study shows certain mapping class groups cannot be realized as subgroup of homeomorphisms.
problem Proving non-realizability of specific mapping class groups.
method Analyzing compactly supported and full mapping class groups of surfaces with genus 3 or order 6 symmetries.
result Proven non-realizability of mapping class groups for surfaces with genus 3 or order 6 symmetries.
Study of CB generating sets for infinite-type surfaces.
problem Understanding CB generating sets for infinite-type surfaces.
method Constructing CB generating sets for specific infinite-type surfaces.
result Examples of surfaces with and without CB generating sets.
Let denote the mapping class group of the plane minus a Cantor set. We show that every action of on the circle is either trivial or semi-conjugate to a unique minimal action on the so-called simple circle.
Study on mapping class groups of infinite type surfaces.
problem Property for mapping class groups of infinite type surfaces.
method Analyzes the existence of elements and satisfying specific group properties.
result Establishes the existence of for any finite collection of non-trivial elements .
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 the relationship between orbit braid group and equivariant mapping class group on surfaces.
problem Understanding the relationship between mapping class groups and braid groups with group actions.
method Using the fibration and exact sequence.
result The conclusion is closely connected with the braid group of the quotient space.
We prove that the first integral cohomology of pure mapping class groups of infinite type genus one surfaces is trivial. For genus zero surfaces we prove that not every homomorphism to factors through a sphere with finitely many punctures. In fact we get an uncountable family of such maps.
We study two actions of big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. The first two parts of the paper are devoted to the definition of objects and tools needed to int…
We study when the mapping class group of an infinite-type surface admits an action with unbounded orbits on a connected graph whose vertices are simple closed curves on . We introduce a topological invariant for infinite-type surfaces that determines in many cases whether there is such an action. This allows us …
Proves homology of mapping class groups for infinite-type surfaces.
problem Homology of mapping class groups for infinite-type surfaces.
method Modification of Mather's argument and homological stability result.
result Homology of mapping class groups determined for binary tree surfaces.
We study the large-scale geometry of mapping class groups of surfaces of infinite type, using the framework of Rosendal for coarse geometry of non locally compact groups. We give a complete classification of those surfaces whose mapping class groups have local coarse boundedness (the analog of local compactness). When …
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.
Study on compact and finite-type support in mapping class group homology.
problem Understanding non-trivial classes supported on compact or finite-type subsurfaces.
method Use of shiftable subsurfaces and homological stability for finite-type surfaces.
result Almost-complete answer for surfaces with positive genus, partial answer for zero genus.
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.
Surface Houghton groupsmath.GT
Surface Houghton groups are studied for their mapping class properties.
problem Understanding the mapping class properties of surface Houghton groups.
method Analyzing the asymptotic rigidity and monodromy homeomorphisms of fibered components.
result Surface Houghton groups are of type but not of type .
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 of conjugacy classes in infinite-type surfaces' mapping class groups.
problem Characterizing conjugacy classes in infinite-type surfaces' mapping class groups.
method Model-theoretic methods developed by Kechris, Rosendal, and Truss.
result Detailed classification of conjugacy classes in mapping class groups of infinite-type surfaces.
The paper classifies dense conjugacy classes in mapping class groups of locally finite graphs.
problem Identifying which mapping class groups have dense conjugacy classes.
method Developed flux homomorphisms and combinatorial criteria for stability.
result A complete classification for self-similar locally finite graphs and a criterion for stability.
Study of infinite type surfaces' mapping class groups via hyperbolic structures.
problem Understanding mapping class groups of infinite type surfaces.
method Definition of a topology on a moduli space of marked hyperbolic structures.
result Continuous action of mapping class groups on the marked moduli space.
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.
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.
Study shows surfaces without certain curves have infinite orbit graph.
problem Characterizing surfaces with specific curve properties.
method Utilized tools from mapping class group geometry.
result Infinite-invariance index 1 surfaces lack good curve graphs.
This paper finds minimal sets of generators for mapping class groups of specific surfaces.
problem Finding minimal sets of generators for mapping class groups of infinite-type surfaces.
method Analyzing specific surfaces to determine minimal sets of generators.
result Minimal sets of generators for are identified for (3 elements), (4 elements), and (2 elements).
The study classifies normal subgroups of mapping class groups of surfaces with Cantor subsets.
problem Understanding the structure of normal subgroups in mapping class groups of surfaces with specific subsets.
method Proves two structure theorems: purity and inertia, characterizing normal subgroups.
result Characterizes finite-type normal subgroups of mapping class groups of surfaces with Cantor subsets.
Groups of homotopy equivalences of graphs help realize compact subgroups.
problem Realizing compact subgroups of homotopy equivalences of graphs.
method Introduced a Polish group topology on the group of proper homotopy equivalences and proved the Nielsen Realization theorem.
result Compact subgroups of homotopy equivalences can be realized by simplicial isomorphisms of graphs.
The paper shows uncountable integral homology for specific mapping class groups.
problem Integral homology of mapping class groups for infinite-type surfaces.
method Analyzing compactly-supported mapping class groups and Torelli groups.
result Integral homology is uncountable in all positive degrees for specific infinite-type surfaces.
Simplified Milnor-Schwarz lemma for geometric group theory.
problem Conditions for orbit maps to be quasi-isometries.
method Succinct treatment and applications to non-Archimedean groups.
result Sharpened results on mapping class groups and quasi-isometry classification.
Study on minimal torsion topological generators for mapping class groups of infinite-type surfaces.
problem Minimal topological generating sets of mapping class groups consisting of torsion elements.
method Investigation of minimal topological generating sets for consisting entirely of torsion elements, with special attention to involutions.
result Minimal topological generating sets for consisting of torsion elements are found for various .