Abstract
Let π be the fundamental group of a Riemann surface of genus g ≥ 2. The group π has a well-known presentation, as the quotient of a free group on generators {a1, a2, . . . , ag, b1, b2, . . . , bg} by the one relation [a1, b1][a2, b2] ⋯ [ag, bg] = 1. This gives two inclusions F π, where F is the free group on g generators; we could map the generators to the a's, or to the b's. Call the images of these inclusions F1 ⊂ π and F2 ⊂ π. Given a connected, reductive group G over an algebraically closed field of characteristic 0, any representation π → G restricts to two representations f1 : F1 → G, f2 : F2 → G. We prove that on a Zariski open, dense subset of the space of pairs of representations {f1, f2}, there exists a representation f : π → G lifting them, up to (separate) conjugacy of f1 and f2. Ben Martin proved this theorem, with the hypothesis that the semisimple rank of G is > g. We remove the hypothesis.
| Original language | English |
|---|---|
| Pages (from-to) | 411-415 |
| Number of pages | 5 |
| Journal | Mathematical Research Letters |
| Volume | 7 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - 2000 |
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Neeman, A. (2000). An improvement on a theorem of Ben Martin. Mathematical Research Letters, 7(4), 411-415. https://doi.org/10.4310/MRL.2000.v7.n4.a7
Neeman, Amnon. / An improvement on a theorem of Ben Martin. In: Mathematical Research Letters. 2000 ; Vol. 7, No. 4. pp. 411-415.
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title = "An improvement on a theorem of Ben Martin",
abstract = "Let π be the fundamental group of a Riemann surface of genus g ≥ 2. The group π has a well-known presentation, as the quotient of a free group on generators {a1, a2, . . . , ag, b1, b2, . . . , bg} by the one relation [a1, b1][a2, b2] ⋯ [ag, bg] = 1. This gives two inclusions F π, where F is the free group on g generators; we could map the generators to the a's, or to the b's. Call the images of these inclusions F1 ⊂ π and F2 ⊂ π. Given a connected, reductive group G over an algebraically closed field of characteristic 0, any representation π → G restricts to two representations f1 : F1 → G, f2 : F2 → G. We prove that on a Zariski open, dense subset of the space of pairs of representations {f1, f2}, there exists a representation f : π → G lifting them, up to (separate) conjugacy of f1 and f2. Ben Martin proved this theorem, with the hypothesis that the semisimple rank of G is > g. We remove the hypothesis.",
author = "Amnon Neeman",
year = "2000",
doi = "10.4310/MRL.2000.v7.n4.a7",
language = "English",
volume = "7",
pages = "411--415",
journal = "Mathematical Research Letters",
issn = "1073-2780",
publisher = "International Press of Boston, Inc.",
number = "4",
}
Neeman, A 2000, 'An improvement on a theorem of Ben Martin', Mathematical Research Letters, vol. 7, no. 4, pp. 411-415. https://doi.org/10.4310/MRL.2000.v7.n4.a7
An improvement on a theorem of Ben Martin. / Neeman, Amnon.
In: Mathematical Research Letters, Vol. 7, No. 4, 2000, p. 411-415.
Research output: Contribution to journal › Article › peer-review
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AU - Neeman, Amnon
PY - 2000
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N2 - Let π be the fundamental group of a Riemann surface of genus g ≥ 2. The group π has a well-known presentation, as the quotient of a free group on generators {a1, a2, . . . , ag, b1, b2, . . . , bg} by the one relation [a1, b1][a2, b2] ⋯ [ag, bg] = 1. This gives two inclusions F π, where F is the free group on g generators; we could map the generators to the a's, or to the b's. Call the images of these inclusions F1 ⊂ π and F2 ⊂ π. Given a connected, reductive group G over an algebraically closed field of characteristic 0, any representation π → G restricts to two representations f1 : F1 → G, f2 : F2 → G. We prove that on a Zariski open, dense subset of the space of pairs of representations {f1, f2}, there exists a representation f : π → G lifting them, up to (separate) conjugacy of f1 and f2. Ben Martin proved this theorem, with the hypothesis that the semisimple rank of G is > g. We remove the hypothesis.
AB - Let π be the fundamental group of a Riemann surface of genus g ≥ 2. The group π has a well-known presentation, as the quotient of a free group on generators {a1, a2, . . . , ag, b1, b2, . . . , bg} by the one relation [a1, b1][a2, b2] ⋯ [ag, bg] = 1. This gives two inclusions F π, where F is the free group on g generators; we could map the generators to the a's, or to the b's. Call the images of these inclusions F1 ⊂ π and F2 ⊂ π. Given a connected, reductive group G over an algebraically closed field of characteristic 0, any representation π → G restricts to two representations f1 : F1 → G, f2 : F2 → G. We prove that on a Zariski open, dense subset of the space of pairs of representations {f1, f2}, there exists a representation f : π → G lifting them, up to (separate) conjugacy of f1 and f2. Ben Martin proved this theorem, with the hypothesis that the semisimple rank of G is > g. We remove the hypothesis.
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U2 - 10.4310/MRL.2000.v7.n4.a7
DO - 10.4310/MRL.2000.v7.n4.a7
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Neeman A. An improvement on a theorem of Ben Martin. Mathematical Research Letters. 2000;7(4):411-415. doi: 10.4310/MRL.2000.v7.n4.a7
