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 basic context of one-relator items and similar staff theoretical issues. It offers a self-contained account of yes usual generalizations of discrete teams.
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Extra info for Algebraic Generalizations of Discrete Groups: A Path to Combinatorial Group Theory Through One-Relator Products
Both assertions follow from this equality (the second since St(p) is open in Endr(V(p))) . 23) b) implies that the dimension of 0(p) , the 2 Zariski closure of 0(p) , is n - dinL (Endr(V(p)) since this latter is the dimension of GL (k)/St(p) . 21). p(p ) if and only if The map 0 ( P 1 ) meets p is surjective, and 0(p2) . ) = p(p«) if and only if p. = T • P2 for some T € GL (k) , or Px(y)T = Tp2(y) y € T . ) = T-modules. It S (T) are in one-one correspondence with the isomorphism classes of simple representations.
Let y= (p(p]L),.. 5). 2). and definitions of derivatives. If each p. is simple and scheme non- singular, scheme and variety tangent spaces coincide so the first assertions also hold for representation varieties. 2. 19). p = p-@«»«©p Then (Df) F p. , i = 1,. . , r , satisfy the hypothesis of is scheme non-singular. 1). Moreover, we will assume k = C (the complex numbers) and work in the analytic topology. This means that R (F) -* S (F) m m is locally trivial, which is more con- venient to handle than local triviality in the etale topology.
N C = Q x.. 16), we consider the map algebra So if : we have x?. 16) implies: M then the automorphism orbit of Aut(M ) are the image of the n p P is closed. k-points of GL n Now the k- under the map 18 LUBOTZKY & MAGID Inn : GL -* Aut(M ) (since every k-automorphism of M (k) is inner Cc-R, Thm. 62, p. 17. Let A be a finitely generated k-algebra and p €RS(r)(A) . Then: 0(p)(k) , the k-points of the n GL (k) orbit of n p , is closed in R (D (k)xSpec(A) (k) . In particular, if n the GL (k) orbit of n p € R S (D(k) n p is closed in R (D (k) .
Algebraic Generalizations of Discrete Groups: A Path to Combinatorial Group Theory Through One-Relator Products by Benjamin Fine