  In the present paper we initiate the challenging task of building a
mathematically sound theory for Adaptive Virtual Element Methods (AVEMs). Among
the realm of polygonal meshes, we restrict our analysis to triangular meshes
with hanging nodes in 2d -- the simplest meshes with a systematic refinement
procedure that preserves shape regularity and optimal complexity. A major
challenge in the a posteriori error analysis of AVEMs is the presence of the
stabilization term, which is of the same order as the residual-type error
estimator but prevents the equivalence of the latter with the energy error.
Under the assumption that any chain of recursively created hanging nodes has
uniformly bounded length, we show that the stabilization term can be made
arbitrarily small relative to the error estimator provided the stabilization
parameter of the scheme is sufficiently large. This quantitative estimate leads
to stabilization-free upper and lower a posteriori bounds for the energy error.
This novel and crucial property of VEMs hinges on the largest subspace of
continuous piecewise linear functions and the delicate interplay between its
coarser scales and the finer ones of the VEM space. An important consequence
for piecewise constant data is a contraction property between consecutive loops
of AVEMs, which we also prove. Our results apply to $H^1$-conforming (lowest
order) VEMs of any kind, including the classical and enhanced VEMs.
