Lower bounds for faithful linear representations of subgroups of the mapping class group

Abstract This paper completely determines the non‐amenability of the mapping class groups of infinite‐type surfaces, the mapping class groups of locally finite infinite graphs of higher ranks, gives an example of non‐amenable stabiliser of a point at infinity of a coarsely bounded generated hyperbolic Polish group, and exhibits a class of mapping class groups of trees or rank‐one graphs that are amenable.

The combinatorial hierarchical hyperbolicity criterion is a very useful way of constructing new hierarchically hyperbolic spaces (HHSs). We show that, conversely, HHSs satisfying natural assumptions (satisfied, for example, by mapping class groups) admit a combinatorial HHS structure. This can be useful in constructions of new HHSs, and also our construction clarifies how to apply the combinatorial HHS criterion to suspected examples. We also uncover connections between HHS notions and lattice theory notions.

Abstract We describe the lower algebraic K -theory of the integral group ring of both the pure and full braid groups of the real projective plane $$\mathbb {R}P^2$$ R P 2 with 3 strings, as well as that of the integral group ring of the mapping class group of $$\mathbb {R}P^2$$ R P 2 with 3 marked points. In addition, we give a general formula for the algebraic K -theory groups of the group ring of the mapping class group of non-orientable surfaces with k marked points, where $$k\ge 3$$ k ≥ 3 .

Let $g$ be a non-negative integer, $\Sigma_{g}$ a closed orientable surface of genus $g$, and $\mathcal{M}_{g}$ its mapping class group. We classify all the group homomorphisms $\pi_{1}(\Sigma_{g}) \to G$ up to the action of $\mathcal{M}_{g}$ on $\pi_{1}(\Sigma_{g})$ in the following cases; (1) $G = PSL(2;\mathbb{Z})$, (2) $G = SL(2;\mathbb{Z})$. As an application of the case (2), we completely classify orientable $T^{2}$-bundles over closed orientable surfaces up to bundle isomorphism. In particular, we show that any orientable $T^{2}$-bundle over $\Sigma_{g}$ with $g \geq 1$ is isomorphic to the fiber connected sum of $g$ pieces of $T^{2}$-bundles over $T^{2}$. Moreover, the classification result in the case (1) can be generalized into the case where $G$ is the free product of a finite number of finite cyclic groups. We also apply it to an extension problem of maps from a closed surface to a connected sum of lens spaces.

Random walks on spaces with hyperbolic properties tend to sublinearly track geodesic rays which point in certain hyperbolic-like directions. Qing-Rafi-Tiozzo recently introduced the sublinearly Morse boundary and proved that this boundary is a quasi-isometry invariant which captures this notion of generic direction in a broad context. In this article, we develop the geometric foundations of sublinear Morseness in the mapping class group and Teichmüller space. We completely characterize sublinear Morseness in terms of the hierarchical structures of these spaces, and use this to prove that their sublinearly Morse boundaries admit continuous equivariant injections into the boundary of the curve graph. It was already known that the Gromov boundary of the curve graph is a Poisson model for sufficiently nice random walks of the mapping class group on itself and on Teichmüller space. As corollary, we prove that the corresponding hitting measure is fully supported on the image of the sublinearly Morse boundary, which was previously unknown. Our techniques include developing tools for modeling the hulls of median rays in hierarchically hyperbolic spaces via CAT(0) cube complexes. Part of this analysis involves establishing direct connections between the geometry of the curve graph and the combinatorics of hyperplanes in the approximating cube complexes.

On Generating Mapping Class Groups by Pseudo-Anosov Elements

The Johnson kernel is the subgroup of the mapping class group of a closed oriented surface that is generated by Dehn twists along separating simple closed curves. The rational abelianization of the Johnson kernel has been computed by Dimca, Hain and Papadima, and a more explicit form was subsequently provided by Morita, Sakasai and Suzuki. Based on these results, Nozaki, Sato and Suzuki used the theory of finite-type invariants of 3 -manifolds to prove that the torsion subgroup of the abelianized Johnson kernel is non-trivial. In this paper, we give a purely 2 -dimensional proof of the non-triviality of this torsion subgroup and provide a lower bound for its cardinality. Our main tool is the action of the mapping class group on the Malcev Lie algebra of the fundamental group of the surface. Using the same infinitesimal techniques, we also provide an alternative diagrammatic description of the rational abelianized Johnson kernel, and we include in the results the case of an oriented surface with one boundary component.

In this paper, we study the mapping class group action on the relative \operatorname{PSL}(2,\mathbb{R}) -character varieties of punctured surfaces. It is well known that Minsky’s primitive-stable representations form a domain of discontinuity for the \operatorname{Out}(F_{n}) -action on the \operatorname{PSL}(2,\mathbb{C}) -character variety. We define simple stability of representations of fundamental group of a surface into \operatorname{PSL}(2,\mathbb{R}) which is an analogue of the definition of primitive stability and prove that these representations form a domain of discontinuity for the \operatorname{MCG} -action. Our first main result shows that holonomies of hyperbolic cone surfaces are simple-stable. We also prove that holonomies of hyperbolic cone surfaces with exactly one cone-point of cone-angle less than \pi are primitive-stable, thus giving examples of an infinite family of indiscrete primitive-stable representations.

Abstract Homological mirror symmetry predicts that there is a relation between autoequivalence groups of derived categories of coherent sheaves on Calabi–Yau varieties and the symplectic mapping class groups of symplectic manifolds. In this paper, as an analogue of Dehn twists for closed oriented real surfaces, we study spherical twists for dg‐enhanced triangulated categories. We introduce the intersection number and relate it to group‐theoretic properties of spherical twists. Using an inequality analogous to a fundamental one in the theory of mapping class groups about the behavior of the intersection number via iterations of Dehn twists, we classify the subgroups generated by two spherical twists using the intersection number. As an application, we compute the center of autoequivalence groups of derived categories of K3 surfaces.

Stein-fillability and positivity in the mapping class group

Abstract We compare the marked length spectra of isometric actions of groups with non‐positively curved features. Inspired by the recent works of Butt, we study approximate versions of marked length spectrum rigidity. We show that for pairs of metrics, the supremum of the quotient of their marked length spectra is approximately determined by their marked length spectra restricted to an appropriate finite set of conjugacy classes. Applying this to fundamental groups of closed negatively curved Riemannian manifolds allows us to refine Butt's result. Our results, however, apply in greater generality and do not require the acting group to be hyperbolic. For example, we are able to compare the marked length spectra associated to mapping class groups acting on their Cayley graphs or on the curve graph.

Haruki Murakami’s Hard-Boiled Wonderland and the End of the World portrays a world where the “shadow”, the seat of memory, desire, and volition, is surgically removed, leaving behind a perfectly fluent but phenomenologically empty self. We argue that this literary structure mirrors a precise mathematical distinction in topological quantum matter. In a semisimple theory such as the semions of SU(2)1, there is a reducible component V(x) of the SL(2,C) character variety: a flat, abelian manifold devoid of parabolic singularities. By contrast, the non-semisimple completion introduces a neutral indecomposable excitation, the neglecton, whose presence forces the mapping class group from the standard braid group B2 to the affine braid group Aff2 and lifts the character variety to the Cayley cubic V(C), with its four parabolic loci. We propose that contemporary AI systems, including large language models, inhabit the shadowless regime of V(x): they exhibit coherence and fluency but lack any bulk degree of freedom capable of supporting persistent identity, non-contractible memory, or choice. To endow artificial systems with depth, one must introduce a structural asymmetry, a fixed, neutral defect analogous to the neglecton, that embeds computation in the non-semisimple geometry of the cubic. We outline an experimentally plausible architecture for such an “artificial ombre,” based on annular topological media with a pinned parabolic defect, realisable in fractional quantum Hall heterostructures, p+ip superconductors, or cold-atom simulators. Our framework suggests that consciousness, biological or artificial, may depend on or benefit from a bulk–boundary tension mediated by a logarithmic degree of freedom: a mathematical shadow that cannot be computed away. Engineering such a defect offers a new pathway toward AGI with genuine phenomenological depth.

Subgroup stability is a strong notion of quasiconvexity that generalizes convex cocompactness in a variety of settings.In this paper, we characterize stability of a subgroup by properties of its limit set on the Morse boundary.Given H < G, both finitely generated, H is stable exactly when all the limit points of H are conical, or equivalently when all the limit points of H are horospherical, as long as the limit set of H is a compact subset of the Morse boundary for G.We also demonstrate an application of these results in the settings of the mapping class group for a finite-type surface, Mod.S/.

We classify homomorphisms from the braid group on n strands to the pure mapping class group of a nonorientable surface of genus g . For n\ge 14 and g\le 2\lfloor{{\frac{n}{2}}}\rfloor+1 , every such homomorphism is either cyclic, or it maps standard generators of the braid group to either distinct Dehn twists or distinct crosscap transpositions, possibly multiplied by the same element of the centralizer of the image.

We extend the previous work [Ann. of Math. (2) 201 , 225–289 (2025)] by building a smooth complete manifold (M^{6},g,p) with \operatorname{Ric}\geq 0 and whose fundamental group \pi_1(M^6)=\mathbb{Q}/\mathbb{Z} is infinitely generated. The example is built with a variety of interesting geometric properties. To begin, the universal cover \widetilde M^{6} is diffeomorphic to S^{3}\times \mathbb{R}^{3} , which turns out to be rather subtle as this diffeomorphism is increasingly twisting at infinity. The curvature of M^{6} is uniformly bounded and in fact decaying polynomially. The example is locally noncollapsed, in that \operatorname{Vol}(B_{1}(x))&gt;v&gt;0 for all x\in M . Finally, the space is built so that it is almost globally noncollapsed. Precisely, for every \eta&gt;0 there exist radii r_{j}\to \infty such that \operatorname{Vol}(B_{r_j}(p))\geq r_j^{6-\eta} . The broad outline for the construction of the example will closely follow the scheme introduced in [Ann. of Math. (2) 201 , 225–289 (2025)]. The six-dimensional case requires a couple of new points, in particular the corresponding Ricci curvature control on the equivariant mapping class group is harder and cannot be done in the same manner.

We generalize the notion of asymptotic mapping class groups and allow them to surject to the Higman--Thompson groups, answering a question of Aramayona and Vlamis in the case of the Higman--Thompson groups. When the underlying surface is a disk, these new asymptotic mapping class groups can be identified with the ribbon and oriented ribbon Higman--Thompson groups. We use this model to prove that the ribbon Higman--Thompson groups satisfy homological stability, providing the first homological stability result for dense subgroups of big mapping class groups. Our result can also be treated as an extension of Szymik--Wahl's work on homological stability for the Higman--Thompson groups to the surface setting.

The surface Houghton groups $\mathcal{H}_{n}$ are a family of groups generalizing Houghton groups $H_n$, which are constructed as asymptotically rigid mapping class groups. We give a complete computation of the BNSR-invariants $Σ^{m}(P\mathcal{H}_{n})$ of their intersection with the pure mapping class group. To do so, we prove that the associated Stein--Farley cube complex is CAT(0), and we adapt Zaremsky's method for computing the BNSR-invariants of the Houghton groups. As a consequence, we give a criterion for when subgroups of $H_n$ and $P\mathcal{H}_{n}$ having the same finiteness length as their parent group are finite index. We also discuss the failure of some of these groups to be co-Hopfian.

Abstract Building on work of Harer, we construct a spine for the decorated Teichmüller space of a non‐orientable surface with at least one puncture and negative Euler characteristic. We compute its dimension, and show that the deformation retraction onto this spine is equivariant with respect to the pure mapping class group of the non‐orientable surface. As a consequence, we obtain a model for the classifying space for proper actions of the pure mapping class group of a punctured non‐orientable surface, which is of minimal dimension in the case there is a single puncture.

Symplectic groups, mapping class groups and the stability of bounded cohomology