On various rank conditions in infinite groups

In the current survey the authors consider some of the main theorems concerning groups satisfying certain rank con- ditions. They present these theorems starting with recently estab- lished results. This order of exposition is different, indeed opposite to chronological, but it allows them to present the main develop- ment of the theory. They illustrate the connections between the different ranks emphasizing, in particular, the connection between the special rank and the Hirsch–Zaitsev rank.