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Register a new userFree groups, covering spaces and Artins theorem
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Gopala Krishna Srinivasan Dr
Publication date
2019
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and research's language is
English
Authors
Gopala Krishna Srinivasan
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In this expository note we provide a proof of Artins theorem which states that the commutator subgroup of a free group on two generators is not finitely generated. The proof employs the infinite grid as in two other proofs in the literature mentioned in the note but takes a somewhat different approach which seems to be of didactic value.
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In real Hilbert spaces, this paper generalizes the orthogonal groups $mathrm{O}(n)$ in two ways. One way is by finite multiplications of a family of operators from reflections which results in a group denoted as $Theta(kappa)$, the other is by considering the automorphism group of the Hilbert space denoted as $O(kappa)$. We also try to research the algebraic relationship between the two generalizations and their relationship to the stable~orthogonal~group~$mathrm{O}=varinjlimmathrm{O}(n)$ in terms of topology. In this paper we mainly show that : (a) $Theta(kappa)$ is a topological and normal subgroup of $O(kappa)$; (b) $O^{(n)}(kappa) to O^{(n+1)}(kappa) stackrel{pi}{to} S^{kappa}$ is a fibre bundle where $O^{(n)}(kappa)$ is a subgroup of $O(kappa)$ and $S^{kappa}$ is a generalized sphere.
We incorporate covers of quasisplit reductive groups into the Langlands program, defining an L-group associated to such a cover. We work with all covers that arise from extensions of quasisplit reductive groups by $mathbf{K}_2$ -- the class studied by Brylinski and Deligne. We use this L-group to parameterize genuine irreducible representations in many contexts, including covers of split tori, unramified representations, and discrete series for double covers of semisimple groups over $mathbb R$. An appendix surveys torsors and gerbes on the etale site, as they are used in the construction of the L-group.
We extend the notions of quasi-monomial groups and almost monomial groups, in the framework of supercharacter theories, and we study their connection with Artins conjecture regarding the holomorphy of Artin $L$-functions.
In 1992, Osamu Kakimizu defined a complex that has become known as the Kakimizu complex of a knot. Vertices correspond to isotopy classes of minimal genus Seifert surfaces of the knot. Higher dimensional simplices correspond to collections of such classes of Seifert surfaces that admit disjoint representatives. We show that this complex is simply connected.
General acceptance of a mathematical proposition $P$ as a theorem requires convincing evidence that a proof of $P$ exists. But what constitutes convincing evidence? I will argue that, given the types of evidence that are currently accepted as convincing, it is inconsistent to deny similar acceptance to the evidence provided for the existence of proofs by certain randomized computations.
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