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Jul 10, 2014 at 10:04 comment added user75691 Thanks a lot for your answer. I will check these lecture notes.
Jul 10, 2014 at 9:45 comment added Seirios @user75691: See also Sageev's lecture notes math.utah.edu/pcmi12/lecture_notes/sageev.pdf
Jul 9, 2014 at 18:02 comment added user75691 This is looks like useful for me, I will read it. I Really thank you for your time.
Jul 9, 2014 at 17:53 comment added Khalid Bou-Rabee Also, check out this very nice paper of Agol, Groves, and Manning: arxiv.org/pdf/1405.0726.pdf . He discusses general graph of groups in Section 3.
Jul 9, 2014 at 17:50 comment added user75691 Thanks a lot again for your answer. I will see again this paper in more detail.
Jul 9, 2014 at 17:42 comment added Khalid Bou-Rabee Ok :) A remark: the canonical completion works in the context of virtually special groups and it is not hard to see that free products of virtually special groups are virtually special.
Jul 9, 2014 at 17:25 comment added user75691 Thanks a lot for your answer. Yes I understand that it is not possible to expect that the algebraic property can be generalised so much, but this is that I am actually asking is some notion of 'covering' using some space that involves the graph of groups decomposition of a free product(or the Bass-Serre tree) and somehow related to finite index subgroups. I am not sure of course, if this possible.
Jul 9, 2014 at 16:53 history edited Khalid Bou-Rabee CC BY-SA 3.0
added more details and a couple of references.
Jul 9, 2014 at 16:16 history answered Khalid Bou-Rabee CC BY-SA 3.0