We use molecular dynamics computer simulations to investigate the local motion of the particles in a supercooled simple liquid. Using the concept of the distance matrix we find that the α−relaxation corresponds to a small number of crossings from one meta-basin to a neighboring one. Each crossing is very rapid and involves the collective motion of O(40) particles that form a relatively compact cluster, whereas string-like motions seem not to be relevant for these transitions. These compact clusters are thus candidates for the cooperatively rearranging regions proposed long times ago by Adam and Gibbs. PACS numbers: 61.20.Ja; 61.20c.Lc; 64.70.Pf In recent years significant progress has been made in our understanding of the relaxation dynamics of glass-forming liquids at intermediate and low temperatures. Sophisticated experiments and computer simulations have identified many of the salient features of this dynamics, and theoretical approaches have helped to rationalize them, at least to some extend [1,2,3,4]. Despite this progress, many of the most elementary questions have not been answered so far and among them is the nature of the motion of the particles in the α−relaxation regime at low temperatures. Experiments and simulations have demonstrated that this dynamics is quite heterogeneous and therefore can be used to explain the observed stretching of the time correlation functions [5,6,7,8,9,10,11]. This heterogeneous dynamics has been shown to be related to cooperative motion in which a small number of particles (a few percent) undergo a collective relaxation dynamics in that they move, often in a string-like fashion, by a distance that is comparable to the one between neighboring particles [9,10,12]. Since a qualitatively similar heterogeneous dynamics has also been found in simple lattice models that show a glassy dynamics and for which it is well known that the α−relaxation is intimately connected to the dynamical heterogeneities (DH), it has been proposed that these DH are crucial for the relaxation dynamics of all glass-forming systems [13]. However, since in these lattice models all elastic or quasi-elastic effects are completely neglected, it is not at all clear whether or not the DH are indeed the only relevant mechanism for the relaxation.Another approach to describe the relaxation dynamics on the time scale of the α−relaxation is by means of the so-called potential energy landscape (PEL) [14,15,16] (or more precisely the free energy landscape) and hence to describe the dynamics of the system by considering its trajectory in configuration space. This PEL is rugged due to the presence of barriers in the free energy and hence at low temperatures the resulting dynamics will be slow. Using the concept of the inherent structures, evidence has been given that (roughly speaking) the motion of the system in the PEL can be decomposed into two types of movements [17,18]: In the first type the system explores some minima which are locally connected to each other and are not separated by a significant barri...

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