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I am currently studying the Heisenberg group from the Riemannian geometry point of view, particularly focusing on its Gromov boundary and more generally its metric properties.

I would like to know about the hottest research themes on the topics, as I am about to approach my PHD studies and am writing my research project.

Any help, reference, hint is welcome. Thank you all.

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Of late, I've seen more work on the sub-Riemannian geometry of the Heisenberg group than on the Riemannian geometry. Two good texts on this topic are: An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem by Capogna, Danielli, Pauls, Tyson Birkhauser, 2007 Geometric Analysis on the Heisenberg Group and Its Generalizations by Calin, Chang, Greiner, AMS/IP, 2007 I don't know if this counts as particularly "hot," but I have found these to be interesting approaches. –  Brendan Foreman Jul 25 '12 at 10:53

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The hottest new information about nilpotent of class 2 groups is in Stefan Wenger's papers about Dehn functions: Wenger, Stefan Nilpotent groups without exactly polynomial Dehn function.J. Topol. 4 (2011), no. 1, 141–160. There are still many open questions there, and the topic is very interesting because it is on the border of analysis and algebra.

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