On manifolds corresponding to cyclically presented groups
by
A. Yu. Vesnin
Sobolev Institute of Mathematics, Novosibirsk, Russia

A group G is said to be cyclically presented if for some n and w it has the presentation G = G_n (w) = <x₁, ... , x_n  |  w,  \eta(w),  ...,  \eta^n-1 (w) >, where \eta: F_n --> F_n is an automorphism of the free group F_n = <x₁, ... , x_n > of rank n given by \eta(x_i) = x_i+1, i=1, ..., n-1, \eta(x_n) = x₁, and w in F_n is a cyclically reduced word.

Obviously, \eta induces an automorphism \Phi: G_n(w) --> G_n(w) given by \Phi(x_i) = x_i+1, i=1, ..., n-1 with \Phi(x_n)=x₁. Let us define the natural HNN-extension G_n (w)   =  { G_n (w), t  |  t^-1 g t  =  \Phi(g),  g in G_n (w) } .

The polynomial associated with the cyclically presented group G_n(w) is given by f_w (t)   =  \sum_i=1ⁿ a_i t^i-1, where a_i is the exponent sum of x_i in w, 1 <= i <= n. We will say that the word w is admissable if | f_w (1) | = 1.

Let K^l denotes a l-dimensional knot in the (l+ 2)-sphere. The connection between high-dimensional knots and cyclically presented groups is described in [1].

Theorem. Let G_n (w) be the natural HNN extension of a cyclically presented group G_n(w). Assume that the abelianization G_n(w)^ab is finite. Then G_n (w) is isomorphic to a l-knot group \pi(K^l), l >= 3, if and only if the word w is admissable.

The research was particially supported by Russian Foudation for Basic Research (grant no. 98-01-00699).

Reference

[1] A. Szscepanski, A. Vesnin, HNN extensions of cyclically presented groups, Universitat Bielefeld Preprint Series, no. 00-002, (2000), 12 pp.

Date received: January 26, 2000