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GENERIC PROPERTIES OF FINITELY PRESENTED GROUPS AND HOWSON'S
 

Summary: GENERIC PROPERTIES OF FINITELY
PRESENTED GROUPS AND HOWSON'S
THEOREM.
G. N. Arzhantseva
Department of Mathematics and Mechanics, Moscow State University,
Moscow 119899, Russia. E­mail: arjantse@nw.math.msu.su
0. Introduction. Let Xm = fx \Sigma1
1
; : : : ; x \Sigma1
m g be an alphabet with m – 2.
For fixed n one can consider all group presentations
G = hx 1 ; : : : ; xm j r 1
= 1; : : : ; r n = 1i; (1)
where r 1 ; : : : ; r n are cyclically reduced words in the alphabet Xm of length
j r i jŸ t. Let N = N(m; n; t) be the number of all such presentations (1) nd
NP be the number of presentations among them such that the group G has a
property P. A property P of m ­ generated groups is said to be generic if for
any n the ratio NP =N tends to 1 as t ! 1. We shall say that the genericity
is exponential if this ratio tends to 1 faster than some function 1 \Gamma exp(ct),
where c ! 0. Further, we will consider the exponential genericity only. The

  

Source: Arzhantseva, Goulnara N. - Section de Mathématiques, Université de Genève

 

Collections: Mathematics