  The role of the isoscalar hyperon Lambda in probing the density dependence of
the nuclear symmetry energy is studied in multi-Lambda hypernuclei,
hyperon-rich matter and neutron stars in relativistic models. Relationships
between the properties of three types of objects and the neutron thickness in
208Pb are established with respect to the isoscalar-isovector coupling that
modifies the density dependence of the symmetry energy. The exotic isotopes far
from the neutron drip line can be stabilized by filling in considerable Lambda
hyperons. The difference of the binding energy of multi-Lambda hypernuclei from
different models is attributed to different symmetry energies. The isovector
potential together with the neutron thickness in multi-Lambda hypernuclei
investigated is very sensitive to the isoscalar-isovector coupling. The large
sensitivity of the Lambda hyperon fraction to the isoscalar-isovector coupling
occurs at about 2-3 rho_0 in beta equilibrated hyperon-rich matter. In neutron
stars with hyperonization, an on-off effect with respect to the
isoscalar-isovector coupling exists for the neutron star radius.
