By Benjamin Fine
A survey of one-relator items of cyclics or teams with a unmarried defining relation, extending the algebraic examine of Fuchsian teams to the extra common context of one-relator items and similar staff theoretical concerns. It presents a self-contained account of yes average generalizations of discrete teams.
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Additional info for Algebraic Generalizations of Discrete Groups: A Path to Combinatorial Group Theory Through One-Relator Products
Since the scheme QQMaps(Y, T, E1 ) is proper, the map QQMaps(Y, T, E1 ) → QQMaps(Y, T, E2 ) is proper. Combined with the previous statement, we obtain that it is a closed embedding. 9, we obtain that QMaps(Y, T, E1 ) → QMaps(Y, T, E2 ) is a closed embedding as well. To prove the last assertion, we claim that for any normal scheme S, the map Hom(S, QMaps(Y, T, E1 )) → Hom(S, QMaps(Y, T, E2 )) 28 Alexander Braverman, Michael Finkelberg, and Dennis Gaitsgory is in fact a bijection. Indeed, let us consider the map between the cones C(T; E1 ) → C(T; E2 ).
It is easy to show that QQMaps(Y, P(E)) is a union of open subschemes, each of which is a projective limit of schemes of ﬁnite type. However, QQMaps(Y, P(E)) itself is generally not of ﬁnite type. If now T is a closed subscheme on P(E), we deﬁne QQMaps(Y, T; E) as the corresponding closed subscheme of QQMaps(Y, P(E)). As before, the open subscheme Uhlenbeck Spaces via Afﬁne Lie Algebras 35 QMaps(Y, T; E) ⊂ QQMaps(Y, T; E) is deﬁned by the condition that L is a line bundle. 11. Let T be a projective scheme of ﬁnite type embedded into P(E), where E is a pro-ﬁnite-dimensional vector space.
Ek ) to be the closed subscheme in i QMapsai (Y, P(Ei )), deﬁned by the condition that for each n-tuple n1 , . . ,nk → OY×S ⊗ Symn1 (E∗1 ) ⊗ · · · ⊗ Symnk (E∗k ) 30 Alexander Braverman, Michael Finkelberg, and Dennis Gaitsgory ⊗n1 k → (L−1 ⊗ · · · ⊗ (L−1 )⊗n k 1 ) vanishes. For Y integral, this condition is equivalent to demanding that the generic point of Y gets mapped to T, just as in the deﬁnition of QMapsa (Y, T; E). Alternatively, let C(T; E1 , . . , Ek ) ⊂ E1 × · · · × Ek be the afﬁne cone over T.
Algebraic Generalizations of Discrete Groups: A Path to Combinatorial Group Theory Through One-Relator Products by Benjamin Fine