  We obtain solutions of Einstein's equations describing gravitational field
outside a noncanonical global monopole with cosmological constant. In
particular, we consider two models of k-monopoles: the Dirac-Born-Infeld (DBI)
and the power-law types, and study their corresponding exterior gravitational
fields. For each model we found two types of solutions. The first of which are
global k-monopole black hole with conical global topology. These are
generalizations of the Barriola-Vilenkin solution of global monopole. The
appearance of noncanonical kinetic terms does not modify the critical
symmetry-breaking scale, $\eta_{crit}$, but it does affect the corresponding
horizon(s). The second type of solution is compactification, whose topology is
a product of two $2$-dimensional spaces with constant curvatures; ${\mathcal
Y}_4\rightarrow {\mathcal Z}_2\times S^2$, with ${\mathcal Y}, {\mathcal Z}$
can be de Sitter, Minkowski, or Anti-de Sitter, and $S^2$ is the $2$-sphere. We
investigate all possible compactifications and show that the nonlinearity of
kinetic terms opens up new channels which are otherwise non-existent. For
$\Lambda=0$ four-dimensional geometry, we conjecture that these
compactification channels are their (possible) non-static super-critical
states, right before they undergo topological inflation.
