Geometric Topology

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Showing new listings for Wednesday, 28 May 2025

New submissions (showing 5 of 5 entries)

[1] arXiv:2505.20620 [pdf, html, other]

Title: Links on incompressible surfaces and volumes

Corbin Reid

Comments: 19 pages, 7 figures

Subjects: Geometric Topology (math.GT)

We consider volumes of two families of links that have been the focus of recent results on geometry, namely weakly generalised alternating (WGA) links and fully augmented links (FAL). Both have known lower bounds on hyperbolic volume in terms of their diagram combinatorics, but less is known about upper bounds. In fact, Kalfagianni and Purcell recently found a family of WGA knots on a compressible surface for which there can be no upper bounds on volume in terms of twist number. They asked if upper volume bounds always exist on incompressible surfaces. We show the answer is no: we find infinite families of WGA and FALs on incompressible surfaces with no upper bound on volume in terms of twist number.

[2] arXiv:2505.20940 [pdf, html, other]

Title: On the isotopies of tangles in periodic 3-manifolds using finite covers

Yuka Kotorii, Sonia Mahmoudi, Elisabetta Matsumoto, Ken'ichi Yoshida

Comments: 33 pages, 8 figures

Subjects: Geometric Topology (math.GT)

A periodic tangle is a one-dimensional submanifold in $\mathbb{R}^3$ that has translational symmetry in one, two or three transverse directions. A periodic tangle can be seen as the universal cover of a link in the solid torus, the thickened torus, or the three-torus, respectively. Our goal is to study equivalence relations of such periodic tangles. Since all finite covers of a link lift to the same periodic tangle, it is necessary to prove that isotopies between different finite covers are preserved. In this paper, we show that if two links have isotopic lifts in a common finite cover, then they are isotopic. To do so, we employ techniques from 3-manifold topology to study the complements of such links.

[3] arXiv:2505.21113 [pdf, html, other]

Title: Pseudo-Anosov flows on hyperbolic L-spaces

John A. Baldwin, Steven Sivek, Jonathan Zung

Comments: 18 pages, 5 figures

Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS); Symplectic Geometry (math.SG)

We prove that for each $n\in\mathbb{N}$ there is a hyperbolic L-space with $n$ pseudo-Anosov flows, no two of which are orbit equivalent. These flows have no perfect fits and are thus quasigeodesic. In addition, our flows admit positive Birkhoff sections, which we argue implies that they give rise to $n$ universally tight contact structures whose lifts to any finite cover are non-contactomorphic. This argument involves cylindrical contact homology together with the work of Barthelmé, Frankel, and Mann on the reconstruction of pseudo-Anosov flows from their closed orbits. These results answer more general versions of questions posed by Calegari and Min--Nonino.

[4] arXiv:2505.21365 [pdf, html, other]

Title: Counting Reciprocal Hyperbolic Elements in Hecke Groups

Ara Basmajian, Blanca Marmolejo, Robert Suzzi Valli

Comments: 22 pages, 1figure

Subjects: Geometric Topology (math.GT); Combinatorics (math.CO); Group Theory (math.GR)

A reciprocal geodesic on a (2,k, $\infty$) Hecke surface is a geodesic loop based at an even order cone point p traversing its path an even number of times. Associated to each reciprocal geodesic is the conjugacy class of a hyperbolic element in the (2,k,$\infty$) Hecke group whose axis passes through a cone point that projects to p. Such an element is called a reciprocal hyperbolic element based at p.
In this paper, we determine the asymptotic growth rate and limiting constant (in terms of word length) of the number of primitive conjugacy classes of reciprocal hyperbolic elements in a Hecke group.

[5] arXiv:2505.21373 [pdf, html, other]

Title: Restricted (2+1)-TQFTs supported by thickened and solid tori

Dušan Đorđević, Danica Kosanović, Jovana Nikolić, Zoran Petrić

Comments: 26 pages

Subjects: Geometric Topology (math.GT); Category Theory (math.CT)

A faithful $(1+1)$ TQFT has recently been constructed, but the existence of a faithful $(2+1)$ TQFT remains an open question, that subsumes the hard problem of linearity of mapping class groups of surfaces. To circumvent the latter problem we construct a subcategory of the category of 3-cobordisms, containing disjoint unions of tori and simplest cobordisms between them. On this we define TQFTs that are able to distinguish pairs of torus bundles and lens spaces, previously shown not to be distinguishable by quantum invariants.

Cross submissions (showing 2 of 2 entries)

[6] arXiv:2505.20778 (cross-list from math.DG) [pdf, html, other]

Title: On Losik classes of diffeomorphism pseudogroups

Yaroslav V. Bazaikin, Yury D. Efremenko, Anton S. Galaev

Comments: 12 pages

Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

Let $P$ be a pseudogroup of local diffeomorphisms of an $n$-dimensional smooth manifold $M$. Following Losik we consider characteristic classes of the quotient $M/P$ as elements of the de~Rham cohomology of the second order frame bundles over $M/P$ coming from the generators of the Gelfand-Fuchs cohomology. We provide explicit expressions for the classes that we call Godbillon-Vey-Losik class and the first Chern-Losik class. Reducing the frame bundles we construct bundles over $M/P$ such that the Godbillon-Vey-Losik class is represented by a volume form on a space of dimension $2n+1$, and the first Chern-Losik class is represented by a symplectic form on a space of dimension $2n$. Examples in dimension 2 are considered.

[7] arXiv:2505.20960 (cross-list from math.GR) [pdf, html, other]

Title: Virtual homological torsion in graphs of free groups with cyclic edge groups

Dario Ascari, Jonathan Fruchter

Comments: 37 pages, 6 figures, comments are welcome

Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

Let $G$ be a hyperbolic group that splits as a graph of free groups with cyclic edge groups. We prove that, unless $G$ is isomorphic to a free product of free and surface groups, every finite abelian group $M$ appears as a direct summand in the abelianization of some finite-index subgroup $G'\le G$. As an application, we deduce that free products of free and surface groups are profinitely rigid among hyperbolic graphs of free groups with cyclic edge groups. We also conclude that partial surface words in a free group are determined by the word measures they induce on finite groups.

Replacement submissions (showing 4 of 4 entries)

[8] arXiv:2107.07544 (replaced) [pdf, other]

Title: On Boundaries of $\varepsilon$-neighbourhoods of Planar Sets, Part II: Global Structure and Curvature

Jeroen S. W. Lamb, Martin Rasmussen, Kalle Timperi

Comments: This article has been superseded by arXiv:2012.13515

Subjects: Metric Geometry (math.MG); Geometric Topology (math.GT)

We study the global topological structure and smoothness of the boundaries of $\varepsilon$-neighbourhoods $E_\varepsilon = \{x \in \mathbb{R}^2 \, : \, \textrm{dist}(x, E) \leq \varepsilon \}$ of planar sets $E \subset \mathbb{R}^2$. We show that for a compact set $E$ and $\varepsilon > 0$ the boundary $\partial E_\varepsilon$ can be expressed as a disjoint union of an at most countably infinite union of Jordan curves and a possibly uncountable, totally disconnected set of singularities. We also show that curvature is defined almost everywhere on the Jordan curve subsets of the boundary.

[9] arXiv:2108.10914 (replaced) [pdf, html, other]

Title: Lagrangian cobordism functor in microlocal sheaf theory I

Wenyuan Li

Comments: 43 pages, 13 figures. v2: Minor revisions on Intro, Sec 2.2 and other places (theorem numbers changed in Sec 3). v3: Fixed the proof of Thm 2.10 and Prop 3.3 (Lem 3.4 in the previous version) based on referees' comments, revisions on Sec 2.4, 4.1 and other places (section and theorem numbers changed in Sec 3; title changed). To appear in Journal of Topology. v4: Minor modifications

Subjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)

Given a Lagrangian cobordism $L$ of Legendrian submanifolds from $\Lambda_-$ to $\Lambda_+$, we construct a functor $\Phi_L^*: Sh^c_{\Lambda_+}(M) \rightarrow Sh^c_{\Lambda_-}(M) \otimes_{C_{-*}(\Omega_*\Lambda_-)} C_{-*}(\Omega_*L)$ between sheaf categories of compact objects with singular support on $\Lambda_\pm$ and its right adjoint on sheaf categories of proper objects, using Nadler-Shende's work. This gives a sheaf theory description analogous to the Lagrangian cobordism map on Legendrian contact homologies and the right adjoint on their unital augmentation categories. We also deduce some long exact sequences and new obstructions to Lagrangian cobordisms between high dimensional Legendrian submanifolds.

[10] arXiv:2503.19532 (replaced) [pdf, html, other]

Title: Non-factorizable ribbon Hopf Algebras

Quentin Faes, Maksymilian Manko

Comments: 39 pages, minor modifications of Definition 2.11 and Definition 2.20, some typos corrected

Subjects: Quantum Algebra (math.QA); Geometric Topology (math.GT)

Building on the work of Nenciu we provide examples of non-factorizable ribbon Hopf algebras, and introduce a stronger notion of non-factorizability. These algebras are designed to provide invariants of $4$-dimensional $2$-handlebodies up to 2-deformations. We prove that some of the invariants derived from these examples are invariants dependent only on the boundary or on the presentation of the fundamental group of the 2-handlebody.

[11] arXiv:2504.06054 (replaced) [pdf, html, other]

Title: Thermodynamic formalism for Quasi-Morphisms: Bounded Cohomology and Statistics

Pablo D. Carrasco, Federico Rodriguez-Hertz

Comments: Some typos and inaccuracies fixed

Subjects: Dynamical Systems (math.DS); Geometric Topology (math.GT)

For a compact negatively curved space, we develop a notion of thermodynamic formalism and apply it to study the space of quasi-morphisms of its fundamental group modulo boundedness. We prove that this space is Banach isomorphic to the space of Bowen functions corresponding to the associated Gromov geodesic flow, modulo a weak notion of Livsic cohomology.
The results include that each such unbounded quasi-morphism is associated with a unique invariant measure for the flow, and this measure uniquely characterizes the cohomology class. As a consequence, we establish the Central Limit Theorem for any unbounded quasi-morphism with respect to Markov measures, the invariance principle, and the Bernoulli property of the associated equilibrium state.