Sur l'entropie volumique des groupes de pr\'esentation finie

Abstract

In the article \cite{BM1.22}, the minimum volume entropy is introduced for each group of finite presentation and there we study some of its general properties. Another concept of minimum volume entropy for geometrically finite groups has recently been defined in \cite{BC21}. In this paper, we present a comparative analysis of these two notions which coincide if the geometric dimension is equal to $1$; they are quite close in dimension $2$ but they differ radically in the other cases. Finally, we introduce a class of groups called soft groups, for which we calculate the minimum volume entropy defined in \cite{BM1.22}. This class contains several known groups, for example, generalized Baumslag-Solitare groups as well $SL(2,\mathbb{Z})$. The volume entropy defined in \cite{BC21}, in general, does not apply to soft groups.