Mapping class groups have a unique Polish group structure
We prove that mapping class groups of surfaces and of locally finite connected graphs support a unique Polish group structure.
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arXiv subjects
Nicholas G. Vlamis
Publications and source records attributed to Nicholas G. Vlamis.
Strongly bounded generation in transformation groups
Word metrics on finitely generated groups have canonical quasi-isometry classes, making quasi-isometry invariants genuine group invariants. Rosendal generalized this phenomenon to topological groups through CB-generation, but in the general topological setting the resulting quasi-isometry invariants are not invariants of the underlying abstract group. Specializing to the discrete case yields what we call SB-generated groups, where the invariants are genuinely algebraic. We show that SB-generation arises naturally in transformation groups by identifying several broad families of examples: the identity component of homeomorphism groups of closed manifolds, certain big mapping class groups, and homeomorphism groups of compact well-ordered spaces with successor limit capacity. These results demonstrate that SB-generation provides a robust extension of finite generation.
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Non-planar ends are continuously unforgettable
We show that continuous epimorphisms between a class of subgroups of mapping class groups of orientable infinite-genus 2-manifolds with no planar ends are always induced by homeomorphisms. This class of subgroups includes the pure mapping class group, the closure of the compactly supported mapping classes, and the full mapping class group in the case that the underlying manifold has a finite number of ends or is perfectly self-similar. As a corollary, these groups are Hopfian topological groups.
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Algebraic and geometric properties of homeomorphism groups of ordinals
We study the homeomorphism groups of ordinals equipped with their order topology, focusing on successor ordinals whose limit capacity is also a successor. This is a rich family of groups that has connections to both permutation groups and homeomorphism groups of manifolds. For ordinals of Cantor--Bendixson degree one, we prove that the homeomorphism group is strongly distorted and uniformly perfect, and we classify its normal generators. As a corollary, we recover and provide a new proof of the classical result that the subgroup of finite permutations in the symmetric group on a countably infinite set is the maximal proper normal subgroup. For ordinals of higher Cantor--Bendixson degree, we establish a semi-direct product decomposition of the (pure) homeomorphism group. When the limit capacity is one, we further compute the abelianizations and determine normal generating sets of minimal cardinality for these groups.
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Covers of surfaces
We study the homeomorphism types of certain covers of (always orientable) surfaces, usually of infinite-type. We show that every surface with non-abelian fundamental group is covered by every noncompact surface, we identify the universal abelian covers and the $\mathbb{Z}/n\mathbb{Z}$-homology covers of surfaces, and we show that non-locally finite characteristic covers of surfaces have four possible homeomorphism types.
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Orientation-preserving homeomorphisms of Euclidean space are commutators
We prove that every orientation-preserving homeomorphism of Euclidean space can be expressed as a commutator of two orientation-preserving homeomorphisms. We give an analogous result for annuli. In the annulus case, we also extend the result to the smooth category in the dimensions for which the associated sphere has a unique smooth structure. As a corollary, we establish that every orientation-preserving diffeomorphism of the real line is the commutator of two orientation-preserving diffeomorphisms.
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Homeomorphism groups of telescoping 2-manifolds are strongly distorted
Building on the work of Mann and Rafi, we introduce an expanded definition of a telescoping 2-manifold and proceed to study the homeomorphism group of a telescoping 2-manifold. Our main result shows that it is strongly distorted. We then give a simple description of its commutator subgroup, which is index one, two, or four depending on the topology of the manifold. Moreover, we show its commutator subgroup is uniformly perfect with commutator width at most two, and we give a family of uniform normal generators for its commutator subgroup. As a consequence of the latter result, we show that for an (orientable) telescoping 2-manifold, every (orientation-preserving) homeomorphism can be expressed as a product of at most 17 conjugate involutions. Finally, we provide analogous statements for mapping class groups.
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Homeomorphism groups of 2-manifolds with the virtual Rokhlin property
We introduce and motivate the definition of the virtual Rokhlin property for topological groups. We then classify the 2-manifolds whose homeomorphism groups have the virtual Rokhlin property. We also establish the analogous result for mapping class groups of 2-manifolds.
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Homeomorphism groups of self-similar 2-manifolds
The class of self-similar 2-manifolds consists of manifolds exhibiting a type of homogeneity akin to the 2-sphere and the Cantor set, and includes both the 2-sphere and the 2-sphere with a Cantor set removed. This chapter aims to provide a narrative thread between recent results on the structure of homeomorphism groups/mapping class groups of self-similar 2-manifolds, and also connections to classical structural results on the homeomorphism group of the 2-sphere and the Cantor set. In order to do this, we provide a survey of recent results, an exposition on classical results about homeomorphism groups, provide a treatment of the structure of stable sets, and prove extensions/strengthenings of the recent results surveyed. Of particular note, we establish the following theorems: (1) A characterization of homeomorphisms of (orientable) perfectly self-similar 2-manifolds that normally generate the group of (orientation-preserving) homeomorphisms -- a strengthening of a result of Malestein-Tao. (2) The homeomorphism group of a perfectly self-similar 2-manifold is strongly distorted -- an extension of a result of Calegari-Freedman for spheres. (3) The homeomorphism group of a perfectly tame 2-manifold is Steinhaus, and hence has the automatic continuity property -- an extension of a result of Mann in dimension two -- providing the first examples of homeomorphism groups of infinite-genus 2-manifolds with the Steinhaus property.
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Mapping class groups with the Rokhlin property
We classify the connected orientable 2-manifolds whose mapping class groups have a dense conjugacy class. We also show that the mapping class group of a connected orientable 2-manifold has a comeager conjugacy class if and only if the mapping class group is trivial.
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Ends of non-metrizable manifolds: a generalized bagpipe theorem
We initiate the study of ends of non-metrizable manifolds and introduce the notion of short and long ends. Using the theory developed, we provide a characterization of (non-metrizable) surfaces that can be written as the topological sum of a metrizable manifold plus a countable number of "long pipes" in terms of their spaces of ends; this is a direct generalization of Nyikos's bagpipe theorem.
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Bounded geometry with no bounded pants decomposition
We construct a quasiconformally homogeneous hyperbolic Riemann surface-other than the hyperbolic plane-that does not admit a bounded pants decomposition. Also, given a connected orientable topological surface of infinite type with compact boundary components, we construct a complete hyperbolic metric on the surface that has bounded geometry but does not admit a bounded pants decomposition.
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Isometry groups of infinite-genus hyperbolic surfaces
Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannan metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-genus 2-manifolds with no planar ends. Surprisingly, we show there is an uncountable class of such 2-manifolds where every countable group can be realized as an isometry group (namely, those with self-similar end spaces). We apply this result to obtain obstructions to standard group theoretic properties for the groups of homeomorphisms, diffeomorphisms, and the mapping class groups of such 2-manifolds. For example, none of these groups satisfy the Tits Alternative; are coherent; are linear; are cyclically or linearly orderable; or are residually finite. As a second application, we give an algebraic rigidity result for mapping class groups.
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Three perfect mapping class groups
We prove that the mapping class group of a surface obtained from removing a Cantor set from either the 2-sphere, the plane, or the interior of the closed 2-disk has no proper countable-index subgroups. The proof is an application of the automatic continuity of these groups, which was established by Mann. As corollaries, we see that these groups do not contain any proper finite-index subgroups and that each of these groups have trivial abelianization.
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There are no exotic ladder surfaces
It is an open problem to provide a characterization of quasiconformally homogeneous Riemann surfaces. We show that given the current literature, this problem can be broken into four open cases with respect to the topology of the underlying surface. The main result is a characterization in one of the these open cases; in particular, we prove that every quasiconformally homogeneous ladder surface is quasiconformally equivalent to a regular cover of a closed surface (or, in other words, there are no exotic ladder surfaces).
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The first integral cohomology of pure mapping class groups
It is a classical result of Powell that pure mapping class groups of connected, orientable surfaces of finite type and genus at least three are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology group associated to the pure mapping class group of any connected orientable surface of genus at least 2 in terms of the surface's simplicial homology. In order to do this, we show that pure mapping class groups of infinite-genus surfaces split as a semi-direct product.
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Big mapping class groups: an overview
We survey recent developments on mapping class groups of surfaces of infinite topological type.
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Big mapping class groups acting on homology
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.
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