  For various finitely presented groups, including right angled Artin groups
and free by cyclic groups, we investigate what is the smallest dimension of a
faithful linear representation. This is done both over C and over fields of
positive characteristic. In particular we show that Gersten's free by cyclic
group has no faithful linear representation of dimension 4 or less over C, but
has no faithful linear representation of any dimension over fields of positive
characteristic.
