Exact ground states of two-dimensional Ising spin glasses with Gaussian and bimodal (+/- J) distributions of the disorder are calculated using a ``matching''algorithm, which allows large system sizes of up to N=480^2 spins to be investigated. We study domain walls induced by two rather different types of boundary-condition changes, and, in each case, analyze the system-size dependence of an appropriately defined ``defect energy'', which we denote by Delta E. For Gaussian disorder, we find a power-law behavior Delta E ~ L^theta, with theta=-0.266(2) and theta=-0.282(2) for the two types of boundary condition changes. These results are in reasonable agreement with each other, allowing for small systematic effects. They also agree well with earlier work on smaller sizes. The negative value indicates that two dimensions is below the lower critical dimension d_c. For the +/- J model, we obtain a different result, namely the domain-wall energy saturates at a nonzero value for L -> infinity, so \theta = 0, indicating that the lower critical dimension for the +/- J model is exactly d_c=2.
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