Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 2 10.wxyz/CV72P2 http://www.austms.org.au/Publ/Bulletin/V72P2/ Francisco F. Lasheras 14 February 2006 2006 2 14 187 196 722-5066-Lasheras-2005 10.wxyz/C2005V72P2p187 http://www.austms.org.au/Publ/Bulletin/V72P2/722-5066-Lasheras/ In this paper, we show that any ascending HNN-extension of a finitely presented group is properly 3-realisable. We recall that a finitely presented group $G$ is said to be properly 3-realisable if there exists a compact 2-polyhedron $K$ with $\pi _1(K) \cong G$ and whose universal cover $\widetilde {K}$ has the proper homotopy type of a (PL) 3-manifold (with boundary). 57M07, 57M10, 57M20 MR2183402 02246383 This work was partially supported by the project MTM 2004-01865. R. Ayala, M. Cárdenas, F.F. Lasheras and A. Quintero Proc. Amer. Math. Soc. MR2111954 R. Ayala, M. C\'ardenas, F.F. Lasheras and A. Quintero; Properly 3-realizable groups, \textit{Proc. Amer. Math. Soc.} \textbf{133} (2005), pp.~1527--1535. H-J. Baues and A. Quintero K-monographs in Mathematics 6 Kluwer Academic Publishers MR1848146 H-J. Baues and A. Quintero; \textit{Infinite homotopy theory}, K-monographs in Mathematics 6 (Kluwer Academic Publishers, Dordrecht, 2001). R. Brown and R. Geoghegan Invent. Math. MR752825 R. Brown and R. Geoghegan; An infinite-dimensional torsion-free $FP\infty $ group, \textit{Invent. Math.} \textbf{77} (1984), pp.~367--381. J.W. Cannon, W.J. Floyd and W.R. Parry Enseig. Math. MR1426438 J.W. Cannon, W.J. Floyd and W.R. Parry; Introductory notes on Richard Thompson's groups, \textit{Enseig. Math.} \textbf{42} (1996), pp.~215--256. M. Cárdenas, F.F. Lasheras, F. Muro and A. Quintero Topology Appl. M. C\'ardenas, F.F. Lasheras, F. Muro and A. Quintero; Proper L-S category, fundamental pro-groups and 2-dimensional proper co-$H$-spaces, \textit{Topology Appl.} (to appear). M. Cárdenas, F.F. Lasheras and R. Roy Bull. Austral. Math. Soc. MR2094287 M. C\'ardenas, F.F. Lasheras and R. Roy; Direct products and properly $3$-realisable groups, \textit{Bull. Austral. Math. Soc.} \textbf{70} (2004), pp.~199--206. M.J. Dunwoody Invent. Math. MR807066 M.J. Dunwoody; The accessibility of finitely presented groups, \textit{Invent. Math.} \textbf{81} (1985), pp.~449--457. H. Federer and B. Jonsson Trans. Amer. Math. Soc. MR32643 H. Federer and B. Jonsson; Some properties of free groups, \textit{Trans. Amer. Math. Soc.} \textbf{68} (1950), pp.~1--27. R. Geoghegan (book in preparation) R. Geoghegan; \textit{Topological methods in group theory}, (book in preparation). R. Geoghegan and M. Mihalik J. Pure Appl. Algebra MR787167 R. Geoghegan and M. Mihalik; Free abelian cohomology of groups and ends of universal covers, \textit{J. Pure Appl. Algebra} \textbf{36} (1985), pp.~123--137. R. Geoghegan and M. Mihalik Topology MR1396771 R. Geoghegan and M. Mihalik; The fundamental group at infinity, \textit{Topology} \textbf{35} (1996), pp.~655--669. R.C. Lyndon and P.E. Schupp Berlin, Heidelberg, New York MR577064 R.C. Lyndon and P.E. Schupp; \textit{Combinatorial group theory} (Berlin, Heidelberg, New York, 1977). S. Mardešic and J. Segal North-Holland MR676973 S. Marde\vsic and J. Segal; \textit{Shape theory} (North-Holland, Amsterdam, New York, 1982). M. Mihalik J. Pure Appl. Algebra MR777262 M. Mihalik; Ends of groups with the integers as quotient, \textit{J. Pure Appl. Algebra} \textbf{35} (1985), pp.~305--320. P. Scott and C.T.C. Wall Homological group theory London Math. Soc. Lecture 36 Notes Cambridge Univ. Press MR564422 P. Scott and C.T.C. Wall; Topological methods in group theory, in \textit{Homological group theory}, London Math. Soc. Lecture 36 Notes (Cambridge Univ. Press, Cambridge, 1979), pp.~137--204.