Property (T) was introduced by Kazhdan in 1967 in terms of unitary representations. Its most striking application is that this property is inherited by lattices, and implies compact generation and compact abelianization. Property (T) has very early on been linked to fixed point properties of group actions and the work of Guichardet and Delorme showed that a locally compact second countable group has property (T) if and only if any isometric action on a Hilbert space has a fixed point. It follows that a group with property (T) also has Serre's property FA that any action on a tree has a fixed point. It has been an active field of research to find objects on which an effective action gives an obstruction to property (T) (see for instance the work of Niblo-Reeves, Haglund-Paulin, Cherix-Martin-Valette or Chatterji-Drutu-Haglund) and of course, given Delorme and Guichardet's result on fixed point property for actions on a Hilbert space, Lp-spaces are natural objects to look at. (The class of all Banach space is too large as any group acts by isometries and without a fix point on some Banach space.) Fixed point properties for a group action on a class of objects is a measure of complication for the group as it prevents the use of the geometry of that class of objects to fit in the geometry of the group. For instance, the Baum-Connes conjecture for groups with property (T) was open until a few years ago when V. Lafforgue gave the first examples of property (T) groups satisfying that conjecture. Recently, several stronger versions of property (T) have been given, notably by V. Lafforgue to explain the limitations of his techniques to prove the Baum-Connes conjecture, but also by Gomez or by Fisher and Hitchman who twist definitions of property (T) to obtain stronger versions. This project has several parts, all of them of independent interest but part of a whole. 1. Introducing and studying, for a property (T) group G, a constant tau(G) indicating the class of Lp-spaces the group G cannot act on isometrically without a fix point; 2. Understand the relationship between the lack of action on any Lp-space with other existing stronger or weaker notions of property (T); 3. For a group G with property (T) and such that tau(G) is finite, to which extent do we recover some non-property (T) behaviour (like a proof to the Baum-Connes conjecture, or an action on a geometric object resembling a median space).