  Kazhdan constants of discrete groups are hard to compute and the actual
constants are known only for several classes of groups. By solving a
semidefinite programming problem by a computer, we obtain a lower bound of the
Kazhdan constant of a discrete group. Positive lower bounds imply that the
group has property (T). We study lattices on $\tilde{A}_2$-buildings in detail.
For $\tilde{A}_2$-groups, our numerical bounds look identical to the known
actual constants. That suggests that our approach is effective. For a family of
groups, $G_1, \cdots, G_4$, that are studied by Ronan, Tits and others, we
conjecture the spectral gap of the Laplacian is $(\sqrt 2-1)^2$ based on our
experimental results. For $\mathrm{SL}(3,\Bbb Z)$ and $\mathrm{SL}(4,\Bbb Z)$
we obtain lower bounds of the Kazhdan constants, 0.2155 and 0.3285,
respectively, which are better than any other known bounds. We also obtain
0.1710 as a lower bound of the Kazhdan constant of the Steinberg group
$\mathrm{St}_3(\Bbb Z)$.

Group Theory;Operator Algebras