PreprintCC BY-NC-ND 4.0HAL
Representations of the modular group of the twice-punctured torus and Burau representations
Abstract
We study representations from the pure mapping class group of the twice-punctured torus PMod(S_{1,2}) to SL(3, C) with a particular focus on those coming from the reduced Burau representations. We exploit connections between PMod(S_{1,2}), the braid group on four strands B_4 , and the fundamental group of the once-punctured torus π_1(S_{1,1}) to review and reprove results in multiple directions, including: a uniqueness result for extensions from irreducible representations on π_1(S_{1,1}) to PMod(S_{1,2}), a classification of representations from PMod(S_{1,2}) to SL(3, C), and a survey on specific parameters for Burau representations with geometric meaning.
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BibTeX
@misc{falbel2026representations,
title = {{Representations of the modular group of the twice-punctured torus and Burau representations}},
author = {Elisha Falbel and Grace Garden and Antonin Guilloux},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://hal.sorbonne-universite.fr/hal-05839040v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Representations of the modular group of the twice-punctured torus and Burau representations
AU - Elisha Falbel
AU - Grace Garden
AU - Antonin Guilloux
PY - 2026
DA - 2026/10/26
PB - HAL
UR - https://hal.sorbonne-universite.fr/hal-05839040v1
AB - We study representations from the pure mapping class group of the twice-punctured torus PMod(S_{1,2}) to SL(3, C) with a particular focus on those coming from the reduced Burau representations. We exploit connections between PMod(S_{1,2}), the braid group on four strands B_4 , and the fundamental group of the once-punctured torus π_1(S_{1,1}) to review and reprove results in multiple directions, including: a uniqueness result for extensions from irreducible representations on π_1(S_{1,1}) to PMod(S_{1,2}), a classification of representations from PMod(S_{1,2}) to SL(3, C), and a survey on specific parameters for Burau representations with geometric meaning.
ER -