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- W2066268779 abstract "If G is a finitely generated group and A is an algebraic group, then R A (G) = Hom (G, A) is an algebraic variety. Define the dimension sequence of G over A as P d (R A (G)) = (N d (R A (G)), …, N 0 (R A (G))), where N i (R A (G)) is the number of irreducible components of R A (G) of dimension i (0 ≤ i ≤ d) and d = Dim (R A (G)). We use this invariant in the study of groups and deduce various results. For instance, we prove the following: Theorem A.Let w be a nontrivial word in the commutator subgroup ofF n = 〈x 1 , …, x n 〉, and letG = 〈x 1 , …, x n ; w = 1〉. IfR SL(2, ℂ) (G)is an irreducible variety andV -1 = {ρ | ρ ∈ R SL(2, ℂ) (F n ), ρ(w) = -I} ≠ ∅, thenP d (R SL(2, ℂ) (G)) ≠ P d (R PSL(2, ℂ) (G)). Theorem B.Let w be a nontrivial word in the free group on{x 1 , …, x n }with even exponent sum on each generator and exponent sum not equal to zero on at least one generator. SupposeG = 〈x 1 , …, x n ; w = 1〉. IfR SL(2, ℂ) (G)is an irreducible variety, thenP d (R SL(2, ℂ) (G)) ≠ P d (R PSL(2, ℂ) (G)). We also show that if G = 〈x 1 , . ., x n , y; W = y p 〉, where p ≥ 1 and W is a word in F n = 〈x 1 , …, x n 〉, and A = PSL(2, ℂ), then Dim (R A (G)) = Max {3n, Dim (R A (G′)) +2 } ≤ 3n + 1 for G′ = 〈x 1 , …, x n ; W = 1〉. Another one of our results is that if G is a torus knot group with presentation 〈x, y; x p = y t 〉 then P d (R SL(2, ℂ) (G))≠P d (R PSL(2, ℂ) (G))." @default.
- W2066268779 created "2016-06-24" @default.
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- W2066268779 date "2011-06-01" @default.
- W2066268779 modified "2023-09-26" @default.
- W2066268779 title "ALGEBRO-GEOMETRIC INVARIANTS OF GROUPS (THE DIMENSION SEQUENCE OF A REPRESENTATION VARIETY)" @default.
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- W2066268779 doi "https://doi.org/10.1142/s0218196711006352" @default.
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