Dehn Filling Operations in 3-Manifolds

• Dehn filling operations are defined for 3-manifolds with torus boundaries by attaching solid tori along prescribed slopes, systematically altering their topology.
• They provide key insights into the classification of hyperbolic and relatively hyperbolic groups by controlling exceptional phenomena and preserving geometric invariants.
• Extensions to higher dimensions and convex projective geometry, along with algorithmic computation of quantum invariants, demonstrate their broad applications in topology and geometric group theory.

Dehn filling operations are a central method in 3-manifold topology, geometric group theory, and higher-dimensional geometric structures, enabling the controlled modification of manifolds and groups by replacing torus boundary components with solid tori. They have deep implications for the classification of 3-manifolds, the structure of hyperbolic and relatively hyperbolic groups, orderability of groups, topological quantum invariants, and the geometry of projective manifolds. This article provides a comprehensive account of Dehn filling, including rigorous definitions, exceptional phenomena, group-theoretic and geometric consequences, algorithmic aspects, and extensions to convex projective geometry.

1. Foundational Definitions and Algebraic Framework

A classical Dehn filling operation is defined for a compact, orientable 3-manifold $M$ with a torus boundary component $T$. Fixing a pair of oriented generators $(\mu, \lambda)$ representing the meridian and longitude in $H_1(T; \mathbb{Z})$, a slope is an isotopy class of primitive curves $p\mu + q\lambda$. Dehn filling $M$ along slope $\alpha = p/q$ attaches a solid torus to $T$, identifying its meridional disk with the curve corresponding to $\alpha$; the resulting manifold is denoted $M(\alpha)$ (Schmalian, 7 Mar 2025).

The notion extends to groups: for $T$0 relatively hyperbolic with peripheral subgroups $T$1, a Dehn filling is defined by choosing normal subgroups $T$2, forming the normal closure $T$3, and taking the quotient $T$4 (Antolín et al., 2015). The geometry and topology of these quotients depend crucially on the choice of "long" (deep) or "sufficiently generic" fillings, which avoid a finite set of exceptional configurations or group elements.

The distance between two slopes $T$5 and $T$6 on a torus is defined as $T$7, a key invariant in classifying and bounding exceptional phenomena under Dehn filling (Boyer et al., 2011).

2. Exceptional Dehn Fillings: Classification and Boundaries

Most Dehn fillings of a cusped hyperbolic 3-manifold yield hyperbolic, irreducible manifolds, but particular "exceptional slopes" produce reducible, toroidal, or Seifert fibered structures, or other non-hyperbolic behaviors. For hyperbolic knot manifolds with an essential once-punctured torus of boundary slope $T$8, the maximal distance $T$9 for a Seifert-fibered filling $(\mu, \lambda)$0 is 5, and the classification of large-distance exceptional fillings is sharp—these occur only in specific cases linked to the Whitehead link exterior and certain of its fillings (Boyer et al., 2011).

For any exceptional Dehn filling slope $(\mu, \lambda)$1 (with $(\mu, \lambda)$2 reducible, toroidal, or Seifert fibered), $(\mu, \lambda)$3, with detailed classifications for $(\mu, \lambda)$4, including explicit families and prism manifold fillings. For hyperbolic genus one knot exteriors, the possible exceptional slopes and the structure of the resulting manifolds are further constrained, with exact bounds on the slopes corresponding to either toroidal or small Seifert fibered outcomes.

The preservation of isotopy classes of essential surfaces is also largely stable under generic Dehn fillings: after filling along slopes at least distance 2 from a finite set $(\mu, \lambda)$5 of exceptional slopes, the set of essential surfaces in $(\mu, \lambda)$6 injects bijectively into those of $(\mu, \lambda)$7 (Bachman et al., 2010). Nontrivial identification or inessentiality only arises for non-generic fillings corresponding to $(\mu, \lambda)$8 and its first neighbors in the Farey graph.

3. Dehn Filling in Geometric Group Theory

Dehn filling has profound group-theoretic analogues, particularly for relatively hyperbolic groups. For $(\mu, \lambda)$9 relatively hyperbolic relative to $H_1(T; \mathbb{Z})$0, Dehn filling by quotienting out deep, finite-index normal subgroups $H_1(T; \mathbb{Z})$1 yields quotients $H_1(T; \mathbb{Z})$2 that remain relatively hyperbolic with respect to the images $H_1(T; \mathbb{Z})$3, provided the filling kernels avoid a finite forbidden set $H_1(T; \mathbb{Z})$4 (Antolín et al., 2015, Groves et al., 2016).

A fundamental feature is the stability of the Bowditch and Gromov boundaries: in toral relatively hyperbolic groups ($H_1(T; \mathbb{Z})$5), generic fillings with $H_1(T; \mathbb{Z})$6 produce hyperbolic quotients whose boundaries remain $H_1(T; \mathbb{Z})$7, and this underpins connections to the Cannon conjecture in 3-manifold groups (Groves et al., 2016).

For relatively quasiconvex subgroups $H_1(T; \mathbb{Z})$8, the new "$H_1(T; \mathbb{Z})$9-wide" filling condition provides control over the image and quasiconvexity of $p\mu + q\lambda$0 under filling, generalizing previous results that required fullness and enabling the extension of Wise's virtual fibering and specialness program (Groves et al., 2017). Under $p\mu + q\lambda$1-wide and sufficiently long peripherally finite fillings, relative (and ordinary) height is non-increasing, critical for applications in the structure of groups and separability of double cosets.

The normal closure of filling kernels decomposes as a free product, and preimages of infinite order elements in $p\mu + q\lambda$2 likewise split as a free product, a structure leveraged in proofs of the Farrell–Jones conjecture for groups hyperbolic relative to residually finite peripheries (Antolín et al., 2015).

4. Geometric and Topological Applications

Dehn filling modifies the geometry of hyperbolic 3-manifolds and their invariants. For a one-cusped manifold with an unknotting tunnel, Dehn filling yields a dual tunnel whose length can be explicitly estimated in terms of the length of a shortest longitude, with long slopes giving unbounded tunnel lengths. For generic fillings, tunnels are isotopic to geodesics, and their canonicity as edges in the canonical cell decomposition is understood (Cooper et al., 2011). Dehn filling extends to more complex geometric settings, including convex projective 4-manifolds, via continuous deformation of holonomy and dihedral angles in Coxeter polytopes, yielding the first higher-dimensional closed convex projective examples with controlled cusp fillings (Lee et al., 2019, Choi et al., 2016).

Explicit bilipschitz bounds on the change in metric during drilling and filling lead to uniform control of Margulis numbers under long Dehn fillings, enabling algorithmic searches for exceptional manifolds with small Margulis constants and reducing the cosmetic surgery conjecture to finite computation for each knot complement (Futer et al., 2019).

Heegaard splittings are also stable under explicit geometric control: for a genus $p\mu + q\lambda$3 Heegaard splitting, if each slope and shortest longitude filled has length exceeding computable bounds, then every genus $p\mu + q\lambda$4 splitting of the filled manifold arises from the original, with manifolds and surfaces isotopic in the appropriate sense (Futer et al., 2012).

5. Quantum, Topological, and Algorithmic Structures

Recent quantum-topological invariants, particularly the 3D-index, permit the detection of exceptional Dehn filling slopes purely from combinatorial data associated with an ideal triangulation, bypassing the need for solving complex geometric equations (Gang, 2018, Celoria et al., 11 Sep 2025). The transformation of the 3D-index under Dehn filling, given by the Gang–Yonekura formula, is rigorously established, with explicit algebraic forms involving Q-normal surfaces and layerings of solid tori. This index is algorithmically computable, topologically invariant, and its vanishing or divergence directly corresponds to exceptional fillings. Certified code now rigorously enumerates these invariants for families of hyperbolic manifolds and surgeries (Celoria et al., 11 Sep 2025).

Algorithmic methods for recognizing parent-child relationships in Dehn fillings are established: whether a manifold $p\mu + q\lambda$5 arises by Dehn filling another manifold $p\mu + q\lambda$6 on specified slopes reduces to deciding solvability of systems of linear and mono-quadratic Diophantine equations, a decidable problem with explicit procedures and implementations (Schmalian, 7 Mar 2025).

The algebraic varieties encoding representations (PSL$p\mu + q\lambda$7 or its universal cover) of the fundamental group under boundary conditions characterize intervals of left-orderable Dehn fillings, via the structure of the translation and holonomy extension loci in cohomology, with interval structures read off from the intersections with appropriately parameterized lines (Culler et al., 2016, Gao, 2018).

6. Generalizations to Convex Projective Geometry and Dimensions Beyond 3

Dehn filling has been extended to convex projective geometry in dimensions 4–6. Here, generalized Dehn filling is realized via the deformation of Coxeter orbifold polytopes, replacing ideal (cuspidal) vertices with faces of specific dihedral angles. These fillings yield properly convex real projective structures on orbifolds (and occasionally manifolds), with the resulting domains typically being non-strictly convex and divisible by the action of the fundamental group (Choi et al., 2016, Lee et al., 2019). This demonstrates a direct geometric link between classical hyperbolic Dehn filling and the production of higher-dimensional convex projective structures, bypassing Mostow's rigidity beyond dimension 3.

7. Structural and Methodological Innovations

Proof strategies of classification and exceptional distance bounds exploit a variety of advanced methodologies:

• Character variety and Culler–Shalen theory to extract essential surfaces from representation varieties and deduce geometric constraints (Boyer et al., 2011).
• Involution and branch-set analysis, exploiting symmetries to restrict admissible fillings in cases involving once-punctured tori and small Seifert fiberings.
• Combinatorial arguments using explicit diagrams and braid presentations to enumerate exceptional cases.
• Stability and convergence techniques for boundaries under group-theoretic filling, employing spiderwebs, truncated quotients, and covering constructions to track Gromov or Bowditch boundaries through sequences of fillings (Groves et al., 2016).
• Use of Ptolemy equations and symplectic reductions to algorithmically compute A-polynomials and their transformation under Dehn filling, exposing the algebraic structure tied to cluster algebras (Howie et al., 2020).

The interplay between these topological, geometric, algebraic, and algorithmic principles defines the modern landscape of Dehn filling operations, with key implications from the structure of 3-manifold topology to geometric group theory and quantum invariants. Recent advancements demonstrate the effectiveness of combining combinatorial, quantum, and geometric methods for the classification, computation, and recognition of manifolds and invariants under Dehn filling.