We give a formula with explicit constants relating the subsurface projection of the end invariants of a hyperbolic 3-manifold diffeomorphic to and the length of the geodesic representative in of the multicurve . This makes effective and computable the large projections versus short curves relation proved by Minsky. We give an application to closed hyperbolic 3-manifolds fibering over the circle providing a geometric analog of the uniform projection bound in fibered faces of Minsky and Taylor.
We introduce the notion of an -ring with prelogarithmic structure, define logarithmic topological Hochschild homology and logarithmic topological cyclic homology in this context, and establish localization sequences for these theories. Our approach is based on Thom -algebras. It extends and strengthens our earlier work on the subject in several regards. Our examples include the fraction field of topological -theory, the existence of which was suggested by calculations by Ausoni and the first author. To illustrate the computational accessibility of log and log , we determine these for non-negative even periodic sphere spectra, with their canonical prelogarithmic structures.
Using factorization homology with coefficients in twisted commutative algebras (TCAs), we prove two flavors of higher representation stability for the cohomology of (generalized) configuration spaces of a scheme/topological space . First, we provide an iterative procedure to study higher representation stability using actions coming from the cohomology of and prove that all the modules involved are finitely generated over the corresponding TCAs. More quantitatively, we compute explicit bounds for the derived indecomposables in the sense of Galatius-Kupers-Randal-Williams. Second, when certain -operations on the cohomology of vanish, we prove that the cohomology of its configuration spaces forms a free module over a TCA built out of the configuration spaces of the affine space. This generalizes a result of Church-Ellenberg-Farb on the freeness of -modules arising from the cohomology of configuration spaces of open manifolds and, moreover, resolves the various conjectures of Miller-Wilson under these conditions.
Let and be knot manifolds and be the closed manifold obtained by gluing up and via . We show that if admits a co-oriented taut foliation, then identifies some CTF-detected rational boundary slopes of and , affirming a conjecture proposed in Boyer, McA Gordon, and Hu [Adv. Math. 495 (2026), 110956].
We define a new notion of splitting complexity for a group along a non-trivial integral character . If is a one-ended coherent right-angled Artin group, we show that the splitting complexity along an epimorphism equals the -Euler characteristic of the kernel of . This allows us to define a Thurston-type semi-norm which measures the splitting complexity of integral characters. Our main tool is Friedl-L & uuml;ck's -polytope which we compute for all groups in this class. We also compute the -Betti numbers of all kernels of characters and show that the -homology of any finitely generated subgroup of is concentrated in the first dimension, which may be of independent interest.
