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Max is a human-size conversational agent that employs synthetic speech, gesture, gaze, and facial display to act in cooperative construction tasks taking place in immersive virtual reality. In the mixed-initiative dialogs involved in our research scenario, turn-taking abilities and dialog competences play a crucial role for Max to appear as a convincing multimodal communication partner. The way how they rely on Max's perception of the user and, in special, how turn-taking signals are handled in the agent's cognitive architecture is the focus of this paper.