math.GT daily digest: 9 new submissions for 8 July 2026

1. Convex projective manifolds, symmetric spaces and geometric decompositions

• Authors: Stefano Riolo, Andrea Seppi, Leone Slavich
• arXiv: 2607.05662 2
• Subjects: Geometric Topology (math.GT)
• Comments: 42 pages, 2 figures

Abstract: We prove that if a closed, indecomposable, properly convex real projective -manifold is geometric or admits a geometric decomposition in the sense of Thurston, then every piece is real hyperbolic. This extends a theorem of Benoist to dimension four. Moreover, we build orientable (non-hyperbolic) -manifolds of the above type, with arbitrary positive, even, Euler characteristic. Along the way, we characterise the compact locally symmetric spaces that virtually support properly convex real projective structures.

2. Simultaneous universal circles and continuous extension

• Authors: Sérgio R. Fenley, Michael P. Landry, Samuel J. Taylor
• arXiv: 2607.05703 3
• Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)
• Comments: 7 pages

Abstract: Fenley proved that any foliation almost transverse to a quasigeodesic pseudo-Anosov flow in a closed atoroidal 3-manifold has the continuous extension property, meaning the inclusions of leaves into the universal cover continuously extend to their ideal boundaries. This article gives an alternate proof of an upgraded version of this: the associated Cannon-Thurston map for the flow, constructed by Frankel and Fenley, organizes all of the leafwise continuous extensions. The proof uses the fact that the boundary of the flowspace is naturally a universal circle for the foliation.

3. Rigidity of maps between configuration spaces

• Authors: Rodrigo De Pool, Peter Huxford, Daniel Minahan, Jeroen Schillewaert
• arXiv: 2607.05826 4
• Subjects: Geometric Topology (math.GT); Algebraic Geometry (math.AG); Group Theory (math.GR)
• Comments: 63 pages, 18 figures

Abstract: Let and . Let be a homomorphism of braid groups. We prove that if the image of is irreducible and not cyclic, then and agrees with an automorphism modulo the center . This resolves in the affirmative a conjecture of Chen, Kordek, and Margalit. It also provides a partial resolution to a problem on the K3 problem list. As a consequence, we prove that every holomorphic map for and is affine equivalent to either a constant map or the identity map. This resolves a conjecture of Farb for .

4. Triangulating Spun 2-Knot Complements

• Authors: Rhuaidi Antonio Burke, Benjamin Burton, Arunima Ray, Jonathan Spreer, Finn Thompson, Orion Zymaris
• arXiv: 2607.05923 5
• Subjects: Geometric Topology (math.GT)
• Comments: 15 pages, 4 figures, comments welcome

Abstract: A -knot is an embedding of a -sphere into the -sphere. Similar to the case of embedding circles into the -sphere, this allows the -sphere to be knotted. In this short paper, we present an algorithm to generate triangulations of the exteriors of -knots obtained by spinning -knots. We give an implementation of the algorithm in \emph{Regina} and present triangulations of exteriors obtained from all -knots with up to eight crossings.

5. On the mathematics table problem

• Authors: Xiao-Song Yang, Xuan Zhou
• arXiv: 2607.05945 6
• Subjects: Geometric Topology (math.GT)

Abstract: In this paper we study the mathematical table problem from a geometric-topological point of view. We prove a zero-existence theorem on a cylinder, which gives a new proof of Fenn's square-table theorem under Fenn's boundary conditions, and establish a variant under different boundary conditions. We also prove that every square table admits a horizontal placement on saddle surfaces. Finally, we show that almost every level set of a smooth Fenn graph contains a rectangle similar to any prescribed rectangle and an orientation-preserving similar copy of every prescribed cyclic quadrilateral.

6. Closed geodesics in homology classes on random hyperbolic surfaces of large genus

• Authors: Zeev Rudnick
• arXiv: 2607.06263 7
• Subjects: Geometric Topology (math.GT); Number Theory (math.NT)

Abstract: We study the distribution of closed geodesics in homology classes on random hyperbolic surfaces of large genus. Viewing the surface as a random point in moduli space equipped with the Weil--Petersson probability measure, we investigate the fluctuations of the weighted counting function of closed geodesics in homology classes modulo . We show that, in the large genus limit, the variance is asymptotic to for every modulus , with an exceptional factor of two when . This contrasts with Hooley's conjecture for primes in arithmetic progressions, where the variance is expected to be . We suggest an explanation for this discrepancy, by comparing our result with the corresponding theory for function fields over a finite field.

7. Dynamics and geometry of character varieties for surface groups

• Authors: David Chasteen-Boyd, Sara Maloni, Suzanne Schlich
• arXiv: 2607.06473 8
• Subjects: Geometric Topology (math.GT)
• Comments: 64 pages, 4 figures; survey to appear as a chapter in the book "In the Tradition of Thurston IV'' (K. Ohshika and A. Papadopoulos ed.)

Abstract: The problem of classifying geometric structures on manifolds is very much related to the discussion of the automorphism groups actions on character varieties, which are spaces of equivalence classes of representations. In this chapter we survey some results on this topic, mostly focusing on representations of surface groups (both in the orientable and non-orientable cases) and free groups. An important principle in the study of the dynamics on character varieties for surface groups is the following dichotomy: when the target group is compact, has nontrivial homotopy type, and the action of the mapping class group is chaotic; whereas when the target group is non-compact, contains contractible sets on which the mapping class group acts properly. We will expand on this dichotomy in various cases. After introducing the necessary background, we will discuss representations into and , discussing the number of connected components, the geometric properties (Bowditch question), the dynamics (Goldman conjecture) and some components with an `exotic' behaviour (Deroin-Tholozan representations). We will also underline how the theory for representations of fundamental groups of orientable closed hyperbolizable surfaces needs to be adapted when one considers surfaces with punctures or non-orientable surfaces. We will then discuss representations into compact groups, where we will discuss mostly ergodicity results in various settings, and some non-ergodicity results at the end. Thirdly, we will consider representations in . We will discuss convex-cocompact representations, primitive-stable and Bowditch representations and their relationship. Finally, we will describe how some of the results mentioned can be generalized for representations into higher-rank Lie groups.

8. The Thick Part of the -Hitchin-Riemann Moduli Space has Infinite Volume

• Authors: Huiseong An, Heejae Kim, Mingook Kim, Seungjin Lee, Hongtaek Jung
• arXiv: 2607.06554 9
• Subjects: Geometric Topology (math.GT)
• Comments: 8 pages

Abstract: We prove that the thick part of the -Hitchin-Riemann moduli space has infinite total Atiyah--Bott--Goldman volume for . This result stands in contrast to Mumford's compactness criterion. To achieve this result, we employ Goldman flows and internal sequences to find an infinite series of subsets of identical volume, the images of which in the Hitchin-Riemann moduli space are all mutually disjoint and sit in the thick part.

9. Homology fiber bundles of varieties, that are not topological fiber bundles

• Authors: Maurício Corrêa, János Kollár
• arXiv: 2607.05603 10
• Subjects: Algebraic Geometry (math.AG); Complex Variables (math.CV); Geometric Topology (math.GT)

Abstract: We construct flat, projective morphisms that are -homology fiber bundles, have a smooth total space, but are not smooth. This disproves one of the conjectures of the second author and Fernández de Bobadilla.