TY - JOUR

T1 - Structural aspects of twin and pure twin groups

AU - Bardakov, Valeriy

AU - Singh, Mahender

AU - Vesnin, Andrei

PY - 2019/12/1

Y1 - 2019/12/1

N2 - The twin group T n is a Coxeter group generated by n- 1 involutions and the pure twin group PT n is the kernel of the natural surjection of T n 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 T n decomposes into a free product with amalgamation for n> 4. It is shown that the pure twin group PT n is free for n= 3 , 4 , and not free for n≥ 6. We determine a generating set for PT n , and give an upper bound for its rank. We also construct a natural faithful representation of T 4 into Aut (F 7 ). In the end, we propose virtual and welded analogues of these groups and some directions for future work.

AB - The twin group T n is a Coxeter group generated by n- 1 involutions and the pure twin group PT n is the kernel of the natural surjection of T n 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 T n decomposes into a free product with amalgamation for n> 4. It is shown that the pure twin group PT n is free for n= 3 , 4 , and not free for n≥ 6. We determine a generating set for PT n , and give an upper bound for its rank. We also construct a natural faithful representation of T 4 into Aut (F 7 ). In the end, we propose virtual and welded analogues of these groups and some directions for future work.

KW - Coxeter group

KW - Doodle

KW - Eilenberg–Maclane space

KW - Free group

KW - Hyperbolic plane

KW - Pure twin group

KW - Twin group

UR - http://www.scopus.com/inward/record.url?scp=85062792034&partnerID=8YFLogxK

U2 - 10.1007/s10711-019-00429-1

DO - 10.1007/s10711-019-00429-1

M3 - Article

AN - SCOPUS:85062792034

VL - 203

SP - 135

EP - 154

JO - Geometriae Dedicata

JF - Geometriae Dedicata

SN - 0046-5755

IS - 1

ER -