Let [Formula: see text] be a surface of finite type with [Formula: see text], and a representation [Formula: see text] from the fundamental group [Formula: see text] into [Formula: see text]. We define the trace systole of [Formula: see text], denoted [Formula: see text] as follows : [Formula: see text] When [Formula: see text] is endowed with a hyperbolic structure, the trace systole of the holonomy representation is naturally related to the usual systolic length of the hyperbolic surface, which is one of the motivations for this study. The function [Formula: see text] is bounded above on relative character varieties of [Formula: see text], and in this paper we compute explicitly the optimal bounds for the one-holed torus, the four-holed sphere and the non-orientable surface of genus [Formula: see text]. The proofs rely on the correspondence between representations of these surface groups and so-called Markoff maps which were introduced by Bowditch. From this, we infer various consequences on the optimal systolic inequalities of certain hyperbolic manifolds and also on non-Fuchsian representations for these surfaces.
Harer proved that the braid group Bn is a duality group with a dualizing module called the Steinberg module St(B-n). He also gave a finite Bn-module resolution of St(B-n) that yields a finite free B-n-module presentation of St(B-n), which we call the Harer presentation of St(B-n). This presentation has 1 /n ((2n-2) (n-1) )(the (n - 1)th Catalfan number) generators and ((2n-2) (n-1) ) relations. In this paper, we will simplify this presentation to arrive at a B-n-module presentation of St(B-n) with one generator and [n+1/ 2 ]relations.
In this paper, we provide an explicit construction of continuous paths of SL2(& Ropf;)-representations of the knot groups of (-2, 3, 2n + 1)-pretzel knots. As an application, we show that the fundamental group of the 3-manifold obtained from the 3-sphere by m l-surgery along the (-2, 3, 2n + 1)-pretzel knot, where n >= 3 is an integer and n not equal 4, is left-orderable if m\l < 2 & LeftFloor;2n+4\3 & RightFloor;.
We investigate a question posed by Matui concerning the existence of minimal ample groupoids that are neither almost finite nor purely infinite, along with several natural variations of this problem. We begin by noting that there exists minimal, effective, ample transformation groupoids with this property. Moreover, such examples can be chosen to be either principal or amenable. We then construct new examples of essentially principal ample groupoids that are likewise neither almost finite nor purely infinite. These examples rely on the recent twisted topological groupoid construction of Palmer and Wu. In particular, our examples do not arise from transformation groupoids.
In this paper, we extend a result of S. Ivanov to prove that if (M,g) is a genus G orientable surface with a single boundary component S-1, and if (D,g(0)) is a disc such that interior points are connected by unique minimizing geodesics and d((D,g0))(x,y) >= d((M,g))(x,y) for all x,y is an element of partial derivative M = partial derivative D, then (1 + 2G/ pi) Area(M,g) >= Area(D,g(0)).
We prove several Myers-type theorems by assuming some exponential decays of the m-Bakry-& Eacute;mery Ricci curvature when m is a positive constant, a negative constant, or infinity. Our results are new even when the m-Bakry-& Eacute;mery Ricci curvature is reduced to the Ricci curvature and greatly improve previous Myers-type theorems via m-Bakry-& Eacute;mery Ricci curvature.
In this paper, we prove that for manifolds with negative curvature bounded away from 0 of infinite volume and bounded geometry, the bounded fundamental class, defined via integration of the volume form over straight top-dimensional simplices, vanishes if and only if the Cheeger isoperimetric constant is positive. This gives a partial affirmative answer to a conjecture of Kim and Kim. Furthermore, we show that for all manifolds with negative curvature bounded away from 0 of infinite volume, the positivity of the Cheeger constant implies the vanishing of the bounded volume class, solving one direction of the conjecture in full generality.
Let (M,g) be a C-infinity-smooth, n-dimensional Riemannian manifold that is diffeomorphic to & Ropf;(n) and admits a properly discontinuous, cocompact, isometric, fixed-point-free action by a group. This work proves the existence of a C-infinity equivariant isometric embedding of M into some Euclidean space & Ropf;(q )with q =max{s(n )+ 2n,s(n) + n + 5} which is the same as the optimal dimension bound in Matthias Gunther's results.
The relative Novikov conjecture for manifolds with boundary states that the relative higher signatures of such manifolds are invariant under orientation-preserving homotopy equivalences of pairs. It was proved to be true by Deng et al. when the fundamental group of the whole manifold admits a coarse embedding into Hilbert space, and the fundamental group of the boundary is a-T-menable.In this paper, we study a case in which the fundamental group of the whole manifold does not necessarily admit a coarse embedding into a Hilbert space. More precisely, we prove that if (M,partial derivative M) and (N,partial derivative N) are orientation-preserving homotopy equivalent, pi(1)(partial derivative M) is a normal subgroup of pi(1)(M), and both pi(1)(partial derivative M) and pi(1)(M)/pi(1)(partial derivative M) admit coarse embeddings into Hilbert space, then the relative higher signatures of (M,partial derivative M) and (N,partial derivative N) are equal.
We introduce a local deformation called the virtualized n-gon move for virtual knots and links (n >= 2), inspired by virtualized Delta-move introduced by [T. Nakamura, Y. Nakanishi, S. Satoh and K. Wada, Virtualized Delta moves for virtual knots and links, preprint (2024), arXiv:2401.12506]. We show that the virtualized n-gon move is an unknotting operation for virtual knots for any n >= 3, and give a necessary and sufficient condition for two virtial links to be related by a finite sequence of virtualized n-gon moves.
In this paper, we generalize the Dirac-dual-Dirac method to Hecke pairs with equivariant coarse embeddings and establish the K-theoretic isomorphisms between the maximal and reduced equivariant Roe algebras. We also extend these results to prove the Baum-Connes conjecture in this context.
In this paper, we develop the theory of probabilistic variants of the one-category and diagonal topological complexity, which bound the classical LS-category and topological complexity from below. Unlike any other classical or probabilistic invariants, these invariants are rigid on spaces with finite fundamental group. On Eilenberg–Mac Lane spaces, we identify these new invariants with distributional category and complexity, respectively, and use them to illuminate aspects of the behavior of the latter invariants on aspherical spaces and products of spaces. We also study their properties on covering maps, [Formula: see text]-isomorphisms, [Formula: see text]-spaces, and closed essential manifolds, and consequently, obtain the first examples of closed manifolds beyond the real projective spaces on which the distributional theory disagrees with the classical one.
In previous works, we studied the intersection homotopy groups associated to a Goresky and MacPherson perversity p and a filtered space X. They are defined as the homotopy groups of simplicial sets introduced by Gajer. We particularized to locally conical spaces of Siebenmann (called CS sets) and established a topological invariance for them when the regular part remains unchanged. Here, we consider coarsenings, made of two structures of CS sets on the same topological space, the strata of one being a union of strata of the other. We endow them with a general perversity and its pushforward, where the adjective "general" means that the perversities are defined on the poset of the strata and not only according to their codimension. If the perversity verifies a growing property analogous to that of the original perversities of Goresky and MacPherson, we also find an invariance theorem for the intersection homotopy groups of a coarsening, under the above restriction on the regular parts. An invariance is shown too in some cases where singular strata become regular in the coarsening, for Thom-Mather spaces.
This paper presents a systematic quantitative study of contact rigidity phenomena based on the contact Hamiltonian Floer theory established in [W. J. Merry and I. Uljarevi & cacute;, Maximum principles in symplectic homology, Israel J. Math. 229 (2019) 39-65]. Our quantitative approach applies to arbitrary admissible contact Hamiltonian functions on the contact boundary M = partial derivative W of a weakly(+)-monotone symplectic manifold W. From a theoretical standpoint, we develop a comprehensive contact spectral invariant theory. As applications, the properties of these invariants enable us to establish several fundamental results: contact big fiber theorem, sufficient conditions for orderability, existence results of translated points. Furthermore, we uncover a non-traditional filtration structure on contact Hamiltonian Floer groups, which we formalize through the introduction of a novel type of persistence modules, called gapped modules, that are only parametrized by a partially ordered set. Among the various properties of contact spectral invariants, we highlight that the triangle inequality is derived through an innovative analysis of a pair-of-pants construction in the contact-geometric framework.
In this paper, we study periodic orbits in the spatial rotating Kepler problem from a symplectic-topological perspective. Our first main result provides a complete classification of these orbits via a natural parametrization of the space of Kepler orbits, using angular momentum and the Laplace–Runge–Lenz vector. We then compute the Conley–Zehnder indices of non-degenerate orbits and the Robbin–Salamon indices of degenerate families, establishing their contributions to symplectic homology via the Morse–Bott spectral sequence. To address coordinate degeneracies in the spatial setting, we introduce a new coordinate system based on the Laplace–Runge–Lenz vector. These results offer a full symplectic-topological profile of the three-dimensional rotating Kepler problem and connect it to generators of symplectic homology.
In this paper, we introduce regular closed subgraphs of Katsura's topological graphs and use them to generalize the notion of an adjunction space from topology. Our construction attaches a topological graph onto another via a regular factor map. We prove that under suitable assumptions the C-& lowast;-algebra of the adjunction graph is a pullback of the C-& lowast;-algebras of the topological graphs being glued. Our results generalize certain pushout-to-pullback theorems proved in the context of discrete directed graphs. Our theorem applied to homeomorphism C-& lowast;-algebras recovers a special case of the well-known result stating that pullbacks of & Zopf;-C-& lowast;-algebras induce pullbacks of the respective crossed product C-& lowast;-algebras. Furthermore, we show that the C-& lowast;-algebras of odd-dimensional quantum balls of Hong and Szyma & nacute;ski (which are known not to be graph C-& lowast;-algebras) are topological graph C-& lowast;-algebras and we recover the pullback structure of C-& lowast;-algebras of odd-dimensional quantum spheres by gluing the topological graphs associated to the C-& lowast;-algebras of the corresponding odd-dimensional quantum balls.
Previous work of the authors established the rigorous limiting behavior of minimizing capillary surfaces to minimizers of the Alt–Caffarelli functional as the capillary angle tends to zero. We prove here that in this limit, the capillary area-density converges to the Weiss energy density. We apply this to obtain angle-independent curvature estimates and regularity results for capillary minimizers.
We call a central extension bounded if its Euler class is represented by a bounded cocycle. We prove that a bounded central extension of a hierarchically hyperbolic group (HHG) is still a HHG; conversely if a central extension is a HHG, then the extension is bounded, and under a further mild assumption the quotient is commensurable to a HHG. Motivated by questions on hierarchical hyperbolicity of quotients of mapping class groups, we therefore consider the general problem of determining when a quotient of a bounded central extension is still bounded, which we prove to be equivalent to an extendability problem for quasihomomorphisms. Finally, we show that quotients of the 4-strands braid group by suitable powers of a pseudo-Anosov are HHG, and in fact bounded central extensions of some HHG. We also speculate on how to extend the previous result to all mapping class groups.
In this paper, we show that the spectral gap of the first cohomological Laplacian Delta(1) for Sp(2n)(& Zopf;) follows once a slightly stronger assumption holds for some Sp(2m)(& Zopf;), where n >= m. As an application of this result, we provide explicit lower bounds for the maximal positive lambda such that Delta(1) - lambda I is a sum of squares in the group ring of some quotients of Sp(2n)(& Zopf;) for any n >= 2.
