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 Collections: Mathematics