math.GT daily digest: 8 new submissions for 2 July 2026

1. The topology of Schottky spaces in higher dimensions

• Authors: Donggyun Seo
• arXiv: 2607.00337 2
• Listing status: New submission
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)
• Comments: 36 pages

The marked Schottky space records, up to conjugacy, all actions of a free group of fixed rank as a Schottky group on hyperbolic space of fixed dimension. In dimension three it is the classical Schottky space covering the moduli space of Riemann surfaces, studied complex-analytically. In higher dimensions each generator gains a rotational parameter, a special orthogonal transformation of the directions normal to its axis, with no classical analogue. Our main theorem treats the borderline dimension, twice the rank: there a dense open part of the space has fundamental group a product of cyclic groups of order two, one per generator, yet the whole space is simply connected, since each such loop contracts through the most degenerate configurations. As a consequence, any two Schottky groups of the same rank in this borderline dimension are quasiconformally isotopic, partially answering a question of Kapovich. We also show that a rotationally symmetric core is a strong deformation retract in every dimension, that this dense open part is homotopy equivalent to a product of special orthogonal groups, and that the analogous locus one dimension below has two connected components.

2. Quandle homology and relative group homology

• Authors: Ayumu Inoue
• arXiv: 2607.00701 3
• Listing status: New submission
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)
• Comments: 16 pages

We introduce a chain map from quandle homology to relative group homology, and construct several quandle cocycles through the chain map. We also relate this chain map to triangulations of Seifert (hyper)surfaces of 1- and 2-dimensional links.

3. Gromov boundary of the Grand Arc graph

• Authors: Carolyn Abbott, Assaf Bar-Natan, Arya Vadnere
• arXiv: 2607.00957 4
• Listing status: New submission
• Subjects: Geometric Topology (math.GT)
• Comments: 54 pages, 15 figures

We describe a dense subset of the Gromov boundary of the grand arc graph of an infinite-type surface as a space of geodesic laminations, analogous to Klarreich's description of the Gromov boundary of the curve complex. After showing that the grand arc graph satisfies a bounded geodesic image theorem, we also prove that the boundary is not compact.

4. Small Sets of Generators for Handlebody Groups

• Authors: Tülin Altunöz, Celal Can Bellek, Emir Gül, Mehmetcik Pamuk, Oğuz Yıldız
• arXiv: 2607.01003 5
• Listing status: New submission
• Subjects: Geometric Topology (math.GT)

The mapping class group of a -dimensional handlebody of genus , denoted by , is a fundamental object of study in geometric topology. Building upon the initial generators introduced by Suzuki and their explicit formulation by Takahashi, Wajnryb established that is generated by exactly five elements for . Motivated by recent minimality results in related subgroups we investigate further reductions to this generating set. Through the use of the relations in Wajnryb's presentation, we show that for , the handlebody group is generated by three elements, and for , is generated by four elements, reducing Wajnryb's generating set of five elements by two and one respectively.

5. Homology manifolds via six functor formalisms

• Authors: Markus Land, Marco Volpe
• arXiv: 2607.00236 6
• Listing status: Cross submission from math.AT
• Subjects: Algebraic Topology (math.AT); Category Theory (math.CT); Geometric Topology (math.GT)
• Comments: 45 pages

We study homology manifolds through the eyes of the six functor formalism of spectral sheaves on locally compact Hausdorff spaces. As main results, we characterize cohomologically smooth objects by adapting an argument of Scholze, deduce that any hypercomplete locally compact ANR homology manifold is cohomologically smooth, show that compact ANR homology manifolds are Poincaré duality complexes whose Spivak tangent fibration identifies with the dualizing sheaf of , and prove a generalization of Wilder's monotone mapping theorem about cell-like maps. Moreover, we introduce the notion of homotopy manifolds for which we prove an unstable analog of Wilder's orientability conjecture and show that hypercomplete ANR homology manifolds are homotopy manifolds. As a consequence, we show that for a compact -dimensional ANR homology manifold, the Spivak tangent fibration of its associated Poincaré duality complex canonically destabilizes to a pointed -fibration. Finally, we introduce homotopy manifolds with conical singularities, a generalization of Cohen's triangulated homotopy manifolds, and show that such objects are in fact topological manifolds, generalizing a result of Siebenmann.

Along the way, we obtain comparisons between sheaf and singular cohomology and between the shape and the weak homotopy type of a topological space, explore the relation between various notions of cohomological dimension and hypercompleteness, and study six functor formalisms satisfying the Künneth formula.

6. Actions of lattices in -arithmetic groups on manifolds

• Authors: Segev Gonen Cohen
• arXiv: 2607.00697 7
• Listing status: Cross submission from math.DS
• Subjects: Dynamical Systems (math.DS); Group Theory (math.GR); Geometric Topology (math.GT)
• Comments: Comments welcome!

We prove that an action by diffeomorphisms of a lattice in a simple -adic group on a compact manifold is finite, provided the dimension is less than the rank. We extend this statement to lattices in totally disconnected -arithmetic groups, where the critical dimension is the maximal rank of the simple factors. This uses the machinery developed by Brown, Fisher, and Hurtado.

7. The Singular Source of Vineyard Monodromy

• Authors: Erin W. Chambers, Christopher Fillmore, Shankha Shubhra Mukherjee, Rohit Roy, Elizabeth Stephenson, Mathijs Wintraecken
• arXiv: 2607.01046 8
• Listing status: Cross submission from cs.CG
• Subjects: Computational Geometry (cs.CG); Differential Geometry (math.DG); Geometric Topology (math.GT)

Vineyards, or time-varying families of persistence diagrams, are widely used in topological data analysis (TDA) pipelines to track how topological features change and evolve as a parameter varies. When the parameter traces a closed loop, a vineyard can exhibit monodromy: diagram points permute over the course of a full traversal, which obstructs feature tracking and can complicate downstream analysis of such data. Chambers et al. considered the periodic vineyards that arise from the radial persistence transform, which maps the manifold to a family of persistence diagrams, where each diagram fixes a base point and considers the filtration that is based on Euclidean distance to that point, and showed that monodromy and knotting can occur. Other recent work by Arya et al. considers geometric conditions that exclude monodromy in two dimensions, in an effort to better understand when this effect happens. That said, understanding when and why monodromy occurs is a fundamental open problem with direct practical consequences for many data analysis pipelines.

In this work, we study this question for 1-manifolds in , using a surprising connection with tools from singularity theory, and provide a classification for the causes of monodromy in vineyards. More precisely, we prove that the vineyard of a sufficiently small loop cannot exhibit monodromy unless it contains a specific singularity of the distance function. The central geometric object in our analysis is the symmetry set, which is the locus of centers of spheres tangent in more than one point to the manifold; this object classifies singularities of the distance function, and in our setting, dictates precisely when monodromy occurs. This characterization opens the door to the development of algorithmic criteria for detecting and utilizing (or avoiding) monodromy in TDA pipelines.

8. Coarse geometry of homeomorphism groups: Classifying countable Stone spaces

• Authors: George Domat, Hannah Hoganson, Robert Alonzo Lyman
• arXiv: 2607.01196 9
• Listing status: Cross submission from math.GR
• Subjects: Group Theory (math.GR); General Topology (math.GN); Geometric Topology (math.GT)
• Comments: 21 pages, 4 figures

Towards developing the tools of geometric group theory for non-locally compact topological groups, we give one of the first complete classifications of a family of such groups up to coarse equivalence, and when possible, up to quasi-isometry. In a previous paper, we placed the homeomorphism groups of countable Stone spaces into three classes: coarsely bounded, unbounded yet generated by a coarsely bounded set, and unbounded but not generated by any coarsely bounded set. Now we show that these are the coarse equivalence classes: Any two groups within one of these classes are in fact coarsely equivalent.

Furthermore, we show that groups in the second class are quasi-isometric to the Hamming cube, the space comprising infinite binary sequences with finitely many nonzero entries equipped with the Hamming distance. As part of the proof, we show that infinite Hamming graphs over finite alphabets are all bi-Lipschitz equivalent.