In three-dimensional topology, a branch of mathematics, the **cyclic surgery theorem** states that, for a compact, connected, orientable, irreducible three-manifold *M* whose boundary is a torus *T*, if *M* is not a Seifert-fibered space and *r,s* are slopes on *T* such that their Dehn fillings have cyclic fundamental group, then the distance between *r* and *s* (the minimal number of times that two simple closed curves in *T* representing *r* and *s* must intersect) is at most 1. Consequently, there are at most three Dehn fillings of *M* with cyclic fundamental group. The theorem appeared in a 1987 paper written by Marc Culler, Cameron Gordon, John Luecke and Peter Shalen.^{ [1] }

In knot theory, a **figure-eight knot** is the unique knot with a crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the unknot and the trefoil knot. The figure-eight knot is a prime knot.

In mathematics, **Thurston's geometrization conjecture** states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries . In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.

**Max Wilhelm Dehn** was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. However, he was forced to retire in 1935 and eventually fled Germany in 1939 and emigrated to the United States.

**Geometric group theory** is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act.

In mathematics, a **3-manifold** is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

In mathematics, more precisely in topology and differential geometry, a **hyperbolic 3–manifold** is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to -1. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries.

In topology, a branch of mathematics, a **Dehn surgery**, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: *drilling* then *filling*.

**Peter B. Shalen** is an American mathematician, working primarily in low-dimensional topology. He is the "S" in JSJ decomposition.

In mathematics, **hyperbolic Dehn surgery** is an operation by which one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Hyperbolic Dehn surgery exists only in dimension three and is one which distinguishes hyperbolic geometry in three dimensions from other dimensions.

**Cameron Gordon** is a Professor and Sid W. Richardson Foundation Regents Chair in the Department of mathematics at the University of Texas at Austin, known for his work in knot theory. Among his notable results is his work with Marc Culler, John Luecke, and Peter Shalen on the cyclic surgery theorem. This was an important ingredient in his work with Luecke showing that knots were determined by their complement. Gordon was also involved in the resolution of the Smith conjecture.

**William "Bus" H. Jaco** is an American mathematician who is known for his role in the Jaco–Shalen–Johannson decomposition theorem and is currently Regents Professor and Grayce B. Kerr Chair at Oklahoma State University and Executive Director of the Initiative for Mathematics Learning by Inquiry.

In mathematics, the **Gordon–Luecke theorem** on knot complements states that if the complements of two tame knots are homeomorphic, then the knots are equivalent. In particular, any homeomorphism between knot complements must take a meridian to a meridian.

In mathematics, the **2π theorem** of Gromov and Thurston states a sufficient condition for Dehn filling on a cusped hyperbolic 3-manifold to result in a negatively curved 3-manifold.

In the mathematical area of geometric group theory, a **van Kampen diagram** is a planar diagram used to represent the fact that a particular word in the generators of a group given by a group presentation represents the identity element in that group.

In the mathematical subject of group theory, **small cancellation theory** studies groups given by group presentations satisfying **small cancellation conditions**, that is where defining relations have "small overlaps" with each other. Small cancellation conditions imply algebraic, geometric and algorithmic properties of the group. Finitely presented groups satisfying sufficiently strong small cancellation conditions are word hyperbolic and have word problem solvable by **Dehn's algorithm**. Small cancellation methods are also used for constructing Tarski monsters, and for solutions of Burnside's problem.

In the mathematical subject of geometric group theory, a **Dehn function**, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the *area* of a *relation* in that group in terms of the length of that relation. The growth type of the Dehn function is a quasi-isometry invariant of a finitely presented group. The Dehn function of a finitely presented group is also closely connected with non-deterministic algorithmic complexity of the word problem in groups. In particular, a finitely presented group has solvable word problem if and only if the Dehn function for a finite presentation of this group is recursive. The notion of a Dehn function is motivated by isoperimetric problems in geometry, such as the classic isoperimetric inequality for the Euclidean plane and, more generally, the notion of a filling area function that estimates the area of a minimal surface in a Riemannian manifold in terms of the length of the boundary curve of that surface.

**Mladen Bestvina** is a Croatian-American mathematician working in the area of geometric group theory. He is a Distinguished Professor in the Department of Mathematics at the University of Utah.

**Marc Edward Culler** is an American mathematician who works in geometric group theory and low-dimensional topology. A native Californian, Culler did his undergraduate work at the University of California at Santa Barbara and his graduate work at Berkeley where he graduated in 1978. He is now at the University of Illinois at Chicago. Culler is the son of Glen Jacob Culler who was an important early innovator in the development of the Internet.

**John Edwin Luecke** is an American mathematician who works in topology and knot theory. He got his Ph.D. in 1985 from the University of Texas at Austin and is now a professor in the department of mathematics at that institution.

In geometric group theory, the **Rips machine** is a method of studying the action of groups on **R**-trees. It was introduced in unpublished work of Eliyahu Rips in about 1991.

- ↑ M. Culler, C. Gordon, J. Luecke, P. Shalen (1987). Dehn surgery on knots. The Annals of Mathematics (
*Annals of Mathematics*) 125**(2)**: 237-300.

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