Big Map Research Articles

Big mapping class groups are not extremely amenable

This paper uses the renowned Kechris-Pestov-Todorcevic machinery to show that (big) mapping class groups are not extremely amenable unless the underlying surface is a sphere or a once-punctured sphere, or equivalently when the mapping class group is trivial. The same techniques also show that the pure mapping class groups, as well as compactly supported mapping class groups, of a surface with genus at least one can never be extremely amenable.

On the homology of big mapping class groups

AbstractWe prove that the mapping class group of the one‐holed Cantor tree surface is acyclic. This in turn determines the homology of the mapping class group of the once‐punctured Cantor tree surface (i.e. the plane minus a Cantor set), in particular answering a recent question of Calegari and Chen. We in fact prove these results for a general class of infinite‐type surfaces called binary tree surfaces. To prove our results we use two main ingredients: one is a modification of an argument of Mather related to the notion of dissipated groups; the other is a general homological stability result for mapping class groups of infinite‐type surfaces.

Non-realizability of some big mapping class groups

In this note, we prove that the compactly supported mapping class group of a surface containing a genus 3 3 subsurface has no realization as a subgroup of the homeomorphism group. We also prove that for certain surfaces with order 6 6 symmetries, their mapping class groups have no realization as a subgroup of the homeomorphism group. Examples of such surfaces include the plane minus a Cantor set and the sphere minus a Cantor set.

Self-Similar Surfaces: Involutions and Perfection

We investigate the problem of when big mapping class groups are generated by involutions. Restricting our attention to the class of self-similar surfaces, which are surfaces with self-similar ends spaces, as defined by Mann and Rafi, and with 0 or infinite genus, we show that when the set of maximal ends is infinite, then the mapping class groups of these surfaces are generated by involutions, normally generated by a single involution, and uniformly perfect. In fact, we derive this statement as a corollary of the corresponding statement for the homeomorphism groups of these surfaces. On the other hand, among self-similar surfaces with one maximal end, we produce infinitely many examples in which their big mapping class groups are neither perfect nor generated by torsion elements. These groups also do not have the automatic continuity property.

Involution Generators of the Big Mapping Class Group

Involution Generators of the Big Mapping Class Group

Big Mapping Class Groups and the Co-Hopfian Property

We study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples of surfaces without boundary whose (pure) mapping class groups are not co-Hopfian; these are first such examples of injective endomorphisms of mapping class groups that fail to be surjective. We then prove that, subject to some topological conditions on the domain surface, any continuous injective homomorphism between (arbitrary) big mapping class groups that sends Dehn twists to Dehn twists is induced by a subsurface embedding. Finally, we explore the extent to which, in stark contrast to the finite-type case, superinjective maps between curve graphs impose no topological restrictions on the underlying surfaces.

Graphs of curves and arcs quasi-isometric to big mapping class groups

Following the works of Rosendal and Mann and Rafi, we try to answer the following question: when is the mapping class group of an infinite-type surface quasi-isometric to a graph whose vertices are curves on that surface? With the assumption of tameness as defined by Mann and Rafi, we describe a necessary and sufficient condition, called translatability, for a geometrically non-trivial big mapping class group to admit such a quasi-isometry. In addition, we show that the mapping class group of the plane minus a Cantor set is quasi-isometric to the loop graph defined by Bavard, which we believe represents the first known example of a hyperbolic mapping class group that is not virtually free.

Big mapping class groups with uncountable integral homology

We prove that, for any infinite-type surface S , the integral homology of the closure of the compactly-supported mapping class group \smash{\overline{\mathrm{PMap}_c(S)}} and of the Torelli group \mathcal{T}(S) is uncountable in every positive degree. By our results in [arXiv:2211.07470] and other known computations, such a statement cannot be true for the full mapping class group \mathrm{Map}(S) for all infinite-type surfaces S . However, we are still able to prove that the integral homology of \mathrm{Map}(S) is uncountable in all positive degrees for a large class of infinite-type surfaces S . The key property of this class of surfaces is, roughly, that the space of ends of the surface S contains a limit point of topologically distinguished points. Our result includes in particular all finite-genus surfaces having countable end spaces with a unique point of maximal Cantor–Bendixson rank \alpha , where \alpha is a successor ordinal. We also observe an order- 10 element in the first homology of the pure mapping class group of any surface of genus 2 , answering a recent question of G. Domat.

Automatic Continuity for Homeomorphism Groups and Big Mapping Class Groups

We show that for any manifold M, compact or homeomorphic to the interior of a compact manifold, the homeomorphism group of M has automatic continuity. The same is true for the relative homeomorphism group Homeo(M,X) where X is homeomorphic to the union of a Cantor set and a (possibly, empty) finite set, and the big mapping class group Homeo(M,X)/Homeo0(M,X). In other words, the algebraic structure of these groups is extremely rigid and determines their topology in a very strong way.

Surface Houghton groups

AbstractFor every $$n\ge 2$$ n ≥ 2 , the surface Houghton group$${\mathcal {B}}_n$$ B n is defined as the asymptotically rigid mapping class group of a surface with exactly n ends, all of them non-planar. The groups $${\mathcal {B}}_n$$ B n are analogous to, and in fact contain, the braided Houghton groups. These groups also arise naturally in topology: every monodromy homeomorphism of a fibered component of a depth-1 foliation of closed 3-manifold is conjugate into some $${\mathcal {B}}_n$$ B n . As countable mapping class groups of infinite type surfaces, the groups $$\mathcal {B}_n$$ B n lie somewhere between classical mapping class groups and big mapping class groups. We initiate the study of surface Houghton groups proving, among other things, that $$\mathcal {B}_n$$ B n is of type $$\text {F}_{n-1}$$ F n - 1 , but not of type $$\text {FP}_{n}$$ FP n , analogous to the braided Houghton groups.

Improved value iteration network for path planning

In this study, a series of improved path planning algorithms are designed for path planning tasks in autonomous control based on deep reinforcement learning. The Value Iteration Network (VIN) is used to deal with the path planning problem. Origin VIN performs well on small size maps, but when it comes to a bigger size of map on test set, the success rate decreased. In order to solve the problem that origin VIN lacks long-distance multi-step planning ability on large maps and generalization ability is insufficient, a three-step improvement was made. First of all, in view of the inconvenient data flow and the disappearance of gradients caused by the network being too deep, the jump connection structure is used to obtain the deeper VIN, in which the accuracy of the experiment is improved. Secondly, with the purpose of solving the problem that the complexity of the model is greatly increased due to the deepening of the network, Batch normalization is used to obtain a new network with dueling architecture plus batch normalization layer, which further accelerates the convergence speed of the network. Third, to deal with the global path planning problem on the big map, the hierarchical network structure is adopted for hierarchical value iteration, and the Hierarchical Structure VIN is obtained. In Hierarchical Structure VIN, the long-term planning ability and generalization ability of the algorithm have been significantly improved, and the algorithm could figure out the large-scale and complex path planning problem.

Large-scale geometry of big mapping class groups

Large-scale geometry of big mapping class groups

Two simultaneous actions of big mapping class groups

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 introduce these two actions; in particular, we define and prove the existence of equators for infinite type surfaces, we define the hyperbolic graph and the circle needed for the actions, and we describe the Gromov-boundary of the graph using the embedding of its vertices in the circle. The third part focuses on some fruitful relations between the dynamics of the two actions. For example, we prove that loxodromic elements (for the first action) necessarily have rational rotation number (for the second action). In addition, we are able to construct non-trivial quasimorphisms on many subgroups of big mapping class groups, even if they are not acylindrically hyperbolic.

Loxodromic elements in big mapping class groups via the Hooper–Thurston–Veech construction

Let $S$ be an infinite-type surface and $p\in S$. We show that the Thurston-Veech construction for pseudo-Anosov elements, adapted for infinite-type surfaces, produces infinitely many loxodromic elements for the action of $Mod(S;p)$ on the loop graph $L(S;p)$ that do not leave any finite-type subsurface $S'\subset S$ invariant. Moreover, in the language of Bavard-Walker, Thurston-Veech's construction produces loxodromic elements of any weight. As a consequence of Bavard and Walker's work, any subgroup of $Mod(S;p)$ containing two "Thurston-Veech loxodromics" of different weight has an infinite-dimensional space of non-trivial quasimorphisms.

Twisted conjugacy in big mapping class groups

Let G be a group and φ be an automorphism of G. Two elements x,y of G are said to be φ-twisted conjugate if y=gxφ(g)−1 for some g∈G. A group G has the R∞-property if the number of φ-twisted conjugacy classes is infinite for every automorphism φ of G. In this paper, we prove that the big mapping class group MCG(S) possesses the R∞-property under some suitable conditions on the infinite-type surface S. As an application, we also prove that the big mapping class group possesses the R∞-property if and only if it satisfies the S∞-property.

Normal subgroups of big mapping class groups

Let S S be a surface and let Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) be the mapping class group of S S permuting a Cantor subset K ⊂ S K \subset S . We prove two structure theorems for normal subgroups of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) . (Purity:) if S S has finite type, every normal subgroup of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) either contains the kernel of the forgetful map to the mapping class group of S S , or it is ‘pure’ — i.e. it fixes the Cantor set pointwise. (Inertia:) for any n n element subset Q Q of the Cantor set, there is a forgetful map from the pure subgroup PMod ⁡ ( S , K ) \operatorname {PMod}(S,K) of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) to the mapping class group of ( S , Q ) (S,Q) fixing Q Q pointwise. If N N is a normal subgroup of Mod ⁡ ( S , K ) \operatorname {Mod}(S,K) contained in PMod ⁡ ( S , K ) \operatorname {PMod}(S,K) , its image N Q N_Q is likewise normal. We characterize exactly which finite-type normal subgroups N Q N_Q arise this way. Several applications and numerous examples are also given.

Irregular Shaped Small Nodule Detection Using a Robust Scan Statistic.

The spatial scan statistics based on the Poisson and binomial models are the most common methods to detect spatial clusters in disease surveillance. These models rely on Monte-Carlo simulation which are time consuming. Moreover, frequently, datasets present over-dispersion which cannot be handled by them. Thus, we have the following goals. First, we propose irregularly shaped spatial scan for the Bell, Poisson, and binomial. The Bell distribution has just one parameter but it is capable of handling over-dispersed datasets. Second, we apply these scan statistics to big maps. A fast version, without Monte-Carlo simulation, for the proposed Poisson and binomial scans is introduced. Intensive simulation studies are carried out to assess the quality of the proposals. In addition, we show the time improvement of the fast scan versions over their traditional ones. Finally, we end the paper with an application on the detection of irregular shape small nodules in a medical image.Supplementary InformationThe online version contains supplementary material available at 10.1007/s12561-022-09353-7.

“Big Map Idea”: Diasporic Currents in South African Music

ABSTRACT South African jazz historiography has tended to regard Cape Town and Johannesburg as the principal axes of musical practice within the country’s borders. This discourse tends to marginalize such practice in Durban as the most obviously “British colonial” of South Africa’s big cities. While Durban’s history has shaped its unique population, its unique demographic makeup has in turn influenced local musicking practices. This essay examines notions of diaspora under three broad headings: diasporic currents within the country, diasporic flows from “outside” (indentured labor from India for Natal’s sugar plantations), and phenomenological reflections on the interwovenness of place and culture.

Stable commutator length on big mapping class groups

We study stable commutator length on mapping class groups of certain infinite-type surfaces. In particular, we show that stable commutator length defines a continuous function on the commutator subgroups of such infinite-type mapping class groups. We furthermore show that the commutator subgroups are open and closed subgroups and that the abelianizations are finitely generated in many cases. Our results apply to many popular infinite-type surfaces with locally coarsely bounded mapping class groups.

Graphs of curves for surfaces with finite-invariance index \(1\)

In this note we make progress toward a conjecture of Durham–Fanoni–Vlamis, showing that every infinite-type surface with fi­ni­te-invariance index \(1\) and no nondisplaceable compact subsurfaces fails to have a good graph of curves, that is, a connected graph where vertices represent homotopy classes of essential simple closed curves and with a natural mapping class group action having infinite diameter orbits. Our arguments use tools developed by Mann–Rafi in their study of the coarse geometry of big mapping class groups.