Optimum solution of Hanging Problem

From: Ashutosh (ashu_at_iitk.ac.in)
Date: 02/27/05 

Date: Sun, 27 Feb 2005 16:00:04 +0000 (UTC)

The hanging problem takes the following form in the realm of algebraic
topology:
Given the free group G(n) generated by n letters a(1), a(2),..., a(n).
Find a word w(a(1),..., a(n)) in G(n) such that:
1) w(a(1),..., a(n)) != e [the identity],
2) w(a(1),..., a(k-1) = e, a(k+1), ..., a(n)) = e for k = 1, 2,..., n.
The existence of w is easily verified by induction. My problem is to
optimise the "length" of w, i.e. finding the smallest "length" of such
a word in G(n). By "length" of a word we mean the sum of the moduli of
exponents of each of the a(i) in w when w is written in product form.