The twin group Tn is a Coxeter group generated by n- 1 involutions and the pure twin group PTn is the kernel of the natural surjection of Tn onto the symmetric group on n letters. In this paper, we investigate structural aspects of twin and pure twin groups. We prove that the twin group Tn decomposes into a free product with amalgamation for n> 4. It is shown that the pure twin group PTn is free for n= 3 , 4 , and not free for n≥ 6. We determine a generating set for PTn, and give an upper bound for its rank. We also construct a natural faithful representation of T4 into Aut (F7). In the end, we propose virtual and welded analogues of these groups and some directions for future work.
CITATION STYLE
Bardakov, V., Singh, M., & Vesnin, A. (2019). Structural aspects of twin and pure twin groups. Geometriae Dedicata, 203(1), 135–154. https://doi.org/10.1007/s10711-019-00429-1
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