Fundamental Groups of Real Arrangements and Torsion in the Lower Central Series Quotients

ABSTRACT

By using computer assistance, we prove that the fundamental group of the complement of a real complexified line arrangement is not determined by its intersection lattice, providing a counter-example for a problem of Falk and Randell. We also deduce that the torsion of the lower central series quotients is not combinatorially determined, which gives a negative answer to a question of Suciu.

Keywords:

• line arrangements
• fundamental groups
• lower central series

2010 AMS SUBJECT CLASSIFICATION:

• 52B30
• 14F35
• 32Q55
• 54F65
• 14N20
• 32S22

Acknowledgments

The authors would like to thank the referee for its valuable comments and suggestions which helped to improve the manuscript.

Funding

The second and the third named authors are supported by postdoctoral grants #2017/15369-0 and #2016/14580-7, respectively, by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP). The first and third named authors are partially supported by MTM2016-76868-C2-2-P.

Additional information

Funding

The second and the third named authors are supported by postdoctoral grants #2017/15369-0 and #2016/14580-7, respectively, by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP). The first and third named authors are partially supported by MTM2016-76868-C2-2-P.