A199044 The number of identity elements of length n in Z*Z^2.

1, 0, 6, 0, 74, 0, 1140, 0, 19562, 0, 357756, 0, 6824684, 0, 134166696, 0, 2697855082, 0, 55213424556, 0, 1146078241284, 0, 24067465856088, 0, 510351502965548, 0, 10911807871502232, 0, 234970037988773560, 0, 5091149074269149520, 0, 110912377099411850090, 0

Offset: 0

Author

Nick Loughlin, Nov 02 2011

Keywords

• nonn
• word

Comments

Z*Z^2 is the free product of the free group on one letter (say, x) and the free abelian group on two letters (say, y and z).

Viewed as the quotient of the free group F on three letters {x,y,z} by the normal subgroup generated by the commutator [y,z], the sequence gives the number of words in F of length n that are sent to the identity in Z*Z^2 under the quotient map.

Note that odd-numbered terms are zero.

Examples

			The identity from the free group F maps to the identity in Z*Z^2, and is the only word of length zero in F, so a(0)=1.
The group Z*Z^2 maps onto the direct product C_2^3, the group of exponent 2 with 8 elements. Therefore no elements of odd length are sent to the identity and thus a(2i-1)=0 for all positive integers i.
The only word of length zero is the empty word, which vacuously represents the identity. Therefore, a_0=1.
For n=2, there are a_2=6 identities; each is a (positive or negative) generator x,y, or z, followed or preceded by its inverse. We have the words x*x^-1, y*y^-1, z*z^-1, plus the reverse of each.
		

References

• Derek F. Holt, Sarah Rees, Claas E. Röver, and Richard M. Thomas, Groups with Context-Free Co-Word Problem, J. London Math. Soc. (2005) 71 (3): 643-657. doi: 10.1112/S002461070500654X
• Brough, Tara Rose, Groups with poly-context-free word problem, PhD thesis (2010), University of Warwick.

Links

• Nick Loughlin, Table of n, a(n) for n = 0..881

Extensions

Edited by Max Alekseyev, Jan 24 2012

Edited by Nick Loughlin, Mar 12 2012