Theorems in group theory | Infinite group theory
In group theory, Higman's embedding theorem states that every finitely generated recursively presented group R can be embedded as a subgroup of some finitely presented group G. This is a result of Graham Higman from the 1960s. On the other hand, it is an easy theorem that every finitely generated subgroup of a finitely presented group is recursively presented, so the recursively presented finitely generated groups are (up to isomorphism) exactly the finitely generated subgroups of finitely presented groups. Since every countable group is a subgroup of a finitely generated group, the theorem can be restated for those groups. As a corollary, there is a universal finitely presented group that contains all finitely presented groups as subgroups (up to isomorphism); in fact, its finitely generated subgroups are exactly the finitely generated recursively presented groups (again, up to isomorphism). Higman's embedding theorem also implies the Novikov-Boone theorem (originally proved in the 1950s by other methods) about the existence of a finitely presented group with algorithmically undecidable word problem. Indeed, it is fairly easy to construct a finitely generated recursively presented group with undecidable word problem. Then any finitely presented group that contains this group as a subgroup will have undecidable word problem as well. The usual proof of the theorem uses a sequence of HNN extensions starting with R and ending with a group G which can be shown to have a finite presentation. (Wikipedia).
Marianna Russkikh (MIT) -- Dimers and embeddings
One of the main questions in the context of the universality and conformal invariance of a critical 2D lattice model is to find an embedding which geometrically encodes the weights of the model and that admits "nice" discretizations of Laplace and Cauchy-Riemann operators. We establish a c
From playlist Northeastern Probability Seminar 2020
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From playlist Mathematics
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Daniel CRISTOFARO GARDINER - Symplectic embeddings of products
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From playlist 2015 Summer School on Moduli Problems in Symplectic Geometry
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Beata Randrianantoanina: On a difference between two methods of low-distortion embeddings of...
Abstract: In a recent paper, the speaker and M.I. Ostrovskii developed a new metric embedding method based on the theory of equal-signs-additive (ESA) sequences developed by Brunel and Sucheston in 1970’s. This method was used to construct bilipschitz embeddings of diamond and Laakso graph
From playlist Analysis and its Applications
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From playlist 2017 Summer REU Presentations
Laurent Bartholdi - Imbeddings in groups of subexponential growth
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Nicholas Katz - Exponential sums and finite groups
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From playlist Dynamical Systems and Ordinary Differential Equations
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From playlist Algorithm Whiteboard
Alexander HULPKE - Computational group theory, cohomology of groups and topological methods 3
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From playlist École d'Été 2022 - Cohomology Geometry and Explicit Number Theory