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Geometric group theory grew out of combinatorial group theory that largely studied properties of discrete groups via analyzing group presentations, which describe groups as quotients of free groups; this field was first systematically studied by Walther von Dyck, student of Felix Klein, in the early 1880s, [2] while an early form is found in the 1856 icosian calculus of William Rowan Hamilton ...
For compact surfaces with boundary, the only extra information needed is the number of boundary components. [1] This result is presented informally at the start of the book, as the first of its six chapters.
This is well-defined because the intersection of a cycle and a boundary consists of an even number of points (by definition of a cycle and a boundary). If M {\displaystyle M} is oriented, analogously (i.e. counting intersections with signs) one defines the intersection form on the 2 {\displaystyle 2} nd homology group
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set.
A discrete group G of isometries of hyperbolic space is called geometrically finite if it has a fundamental domain C that is convex, geometrically finite, and exact (every face is the intersection of C and gC for some g ∈ G) (Ratcliffe 1994, 12.4).
The following Application Integrated Resources (AICs) define boundary models that are constraints of the generic geometric and topological capabilities: ISO 10303-511 Topologically bounded surface , definition of an advanced face , that is a bounded surface where the surface is of type elementary (plane, cylindrical, conical, spherical or ...
The basic concept of a Caccioppoli set was first introduced by the Italian mathematician Renato Caccioppoli in the paper (Caccioppoli 1927): considering a plane set or a surface defined on an open set in the plane, he defined their measure or area as the total variation in the sense of Tonelli of their defining functions, i.e. of their parametric equations, provided this quantity was bounded.
In general relativity, the Gibbons–Hawking–York boundary term is a term that needs to be added to the Einstein–Hilbert action when the underlying spacetime manifold has a boundary. The Einstein–Hilbert action is the basis for the most elementary variational principle from which the field equations of general relativity can be defined.