Thurston norm, polytopes and splitting complexity

Abstract: We show that if $G$ is a finitely generated torsion-free group satisfying the Strong Atiyah Conjecture with vanishing first $L<sup>{2}$-Betti number, then the map that assigns to each surjective integral character the first $L<sup>2$-Betti number of the kernel extends to a seminorm on the first cohomology group of $G$ with real coefficients. We call this seminorm the Thurston norm. Moreover, we show that this norm is induced by a polytope in the first homology group with real coefficients. We also generalize this result to higher $L<sup>{2}$-Betti numbers of the kernels, thereby confirming a conjecture of Friedl, Lück and Tillmann. In the case where $G$ is either a free-by-cyclic group or the fundamental group of an admissible $3$-manifold, we show that the Thurston norm of $G$ admits a combinatorial interpretation that relates it to the splitting complexity of the character. This confirms a conjecture of Gardam and Kielak. As an application, we show that there exists an algorithm to compute the Bieri--Neumann--Strebel invariant of free-by-cyclic groups, and discuss connections to the isomorphism problem in free-by-cyclic groups.

• The paper extends the Thurston norm to finitely generated groups by connecting L2-homology with polytope-induced seminorms.
• It demonstrates that the Thurston norm is uniquely determined by a symmetric polytope, resolving conjectures and enabling algorithmic computation of BNS invariants.
• The authors reveal that splitting complexity correlates with L2-Betti numbers and subgroup rigidity, offering new insights into free-by-cyclic groups and 3-manifolds.

Thurston Norm, Polytope Structures, and Splitting Complexity in Finitely Generated Groups

Introduction and Background

This paper establishes a unified framework connecting the Thurston norm, polytopal geometry, and splitting complexity in the study of finitely generated groups, focusing on groups satisfying analytic properties such as the Strong Atiyah Conjecture and vanishing $L^2$-Betti numbers. Building on the classical Thurston norm for 3-manifolds, which measures the complexity of embedded surfaces dual to cohomology classes, the authors generalize this concept to a broader class of groups via $L^2$-homology and provide a detailed polytope-theoretic characterization. The analysis culminates in new results on the computation of Bieri–Neumann–Strebel (BNS) invariants and significant advances on algorithmic problems for free-by-cyclic groups.

Thurston Norm and Its Extension via $L^2$-Betti Numbers

For an admissible 3-manifold group $G = \pi_1(M)$, Thurston's geometric seminorm on $H^1(M; \mathbb{R})$ is classically given by the minimal complexity of surfaces dual to cohomology classes. Friedl–Lück previously observed that for such $G$, the Thurston norm can be recovered numerically as

$\|\phi\|_T = \alpha_\phi \, b_1^{(2)}(\ker \phi)$

where $b_1^{(2)}$ is the first $L^2$-Betti number and $\alpha_\phi$ is a scaling factor determined by $L^2$0.

The main innovation of this work is in extending the definition of the Thurston norm to highly non-manifold settings. Specifically, the authors prove:

If $L^2$1 is finitely generated, torsion-free, satisfies the Strong Atiyah Conjecture, and has vanishing first $L^2$2-Betti number, then the map assigning to each surjective character $L^2$3 the $L^2$4-Betti number of $L^2$5 extends uniquely to a seminorm on $L^2$6. This seminorm is the Thurston norm of $L^2$7. Moreover, this norm is induced by a polytope in $L^2$8.

This result explicitly connects the algebraic structure of $L^2$9 with convex geometry: the Thurston norm is realized as

$L^2$0

for a certain polytope $L^2$1. The authors further generalize this to higher $L^2$2-Betti numbers, thereby verifying a conjecture of Friedl–Lück–Tillmann.

Polytope Structures and Their Homological Meaning

The authors systematically construct the required polytopal objects for general crossed products and skew group rings using algebraic $L^2$3-theory and Ore localizations. The key technical insights include:

• Identification of the Newton polytope associated to modules over crossed products, and its role in translating algebraic data (rank, complexity, etc.) to convex geometry.
• Proving that for any module $L^2$4 in a suitable category, the Minkowski sum $L^2$5 is an integral polytope, with the Thurston norm function being subadditive and continuous on $L^2$6 (i.e., convex).
• Establishing that the Thurston norm is fully determined by a single symmetric polytope, resolving questions about uniqueness and the geometric structure underlying splittings.

Splitting Complexity and Combinatorial Interpretations

Splitting complexity measures the minimal $L^2$7-Betti number arising in splittings dual to an integral character. For $L^2$8 of type $L^2$9 with trivial $G = \pi_1(M)$0, the authors formulate and resolve the following:

Does the $G = \pi_1(M)$1-Betti number of the kernel of a character coincide with the minimal $G = \pi_1(M)$2-Betti number of edge groups in all admissible HNN splittings dual to the character?

Through a combination of advanced techniques in von Neumann algebras, Sylvester rank theory, and the construction of universal skew field localizations, the authors affirmatively resolve this question for all torsion-free virtually free-by-cyclic groups and admissible 3-manifold groups. The proof employs a critical reduction to subgroup rigidity phenomena (so-called $G = \pi_1(M)$3-subgroup rigidity), and establishes that for these cases, $G = \pi_1(M)$4-compressed subgroups are always $G = \pi_1(M)$5-independent.

Algorithmic Applications: Computation of BNS Invariants

As an application of the preceding structure theory, the paper delivers a strong constructive result:

There is an algorithm to compute the Bieri–Neumann–Strebel invariant for (finitely generated free)-by-cyclic groups.

This is achieved by leveraging the combinatorial structure of the Thurston norm ball (encoded as a polytope), combinatorial group theory, and deep properties of HNN splittings. The BNS invariant, detecting finite generation properties of kernels of characters, is shown to coincide precisely with regions of the Thurston ball corresponding to faces dual to fibered classes—an insight due to Bieri, Neumann, and Strebel, and here rendered algorithmic by the polytope machinery.

The work also discusses reductions and algorithmic approaches to the isomorphism problem for free-by-cyclic groups, relating it to the still open conjugacy problem in $G = \pi_1(M)$6.

Implications and Further Directions

The results of this paper have significant consequences in both geometric group theory and low-dimensional topology:

• Unification of geometric and algebraic approaches: By embedding the Thurston norm in the context of group rings and operator algebras, the work opens avenues for analyzing manifolds and their fundamental groups via purely algebraic invariants.
• Extension of polytope techniques: The translation of deep topological phenomena into convex geometric language paves the way for further applications in representation theory, $G = \pi_1(M)$7-invariants, and decision problems.
• Algorithmic geometry of groups: The explicit algorithms for BNS invariants and fibered face detection suggest that geometric group invariants may be more tractable than previously believed for substantial classes of groups, provided analytic control (Atiyah conjectures, $G = \pi_1(M)$8-acyclicity) holds.

The techniques and reductions involving subgroup rigidity, the interplay between $G = \pi_1(M)$9-compressed and $H^1(M; \mathbb{R})$0-independent subgroups, and the use of universal skew field localizations offer powerful models for potential generalization, possibly even outside virtually free-by-cyclic or 3-manifold contexts.

Conclusion

This paper offers a comprehensive structure theory bridging the Thurston norm, convex polytopes, and splitting complexity for a large class of finitely generated groups. The identification of the Thurston norm with polytope-induced seminorms underlines an intimate connection between analytic invariants arising from $H^1(M; \mathbb{R})$1-homology and combinatorial group theory, with substantial ramifications for both theoretical understanding and computational approaches to invariants like the BNS invariant. This work lays a robust and flexible foundation for further exploration of analytic, geometric, and algorithmic phenomena in group theory and topology (2606.31774).