Atomic Properties of the Hawaiian Earring Group for HNN Extensions

Abstract

In 2011, while investigating fundamental groups of wild spaces, K.Eda [7] showed that the fundamental group of the Hawaiian earring (the Hawaiian earring group, in short) has the property that for any homomorphism h from it to a free product A*B, there exists a natural number N such that is contained in a conjugate subgroup to A or B. In the present article, we prove a corresponding property for certain HNN extensions and amalgamated free products. This allows us to show that some one-relator groups, including Baumslag–Solitar groups, are n-slender.

Key Words:

• Baumslag-Solitar groups
• HNN extension
• Hawaiian earring
• Infinitary words
• N-slender

2010 Mathematics Subject Classification:

• Primary: 03E75
• Secondary: 20E06

ACKNOWLEDGMENTS

I would like to thank to Prof. Katsuya Eda for giving helpful advice. I also thank the members of the seminar within the Eda Laboratory.

Notes

¹There is a straightforward generalization of slenderness to noncommutative groups G [11]. But it depends on only abelian subgroups of G. The property of n-slenderness is essentially different from it.

Communicated by M. Kambites.