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We show that the counts of galaxy clusters in future deep cluster surveys canplace strong constraints on the matter density, Omega_m, the vacuum energydensity, Omega_L, and the normalization of the matter power spectrum, sigma_8.Degeneracies between these parameters are different from those in studies ofeither high--redshift type Ia Supernovae (SNe), or cosmic microwave background(CMB) anisotropies. Using a mass threshold for cluster detection expected to betypical for upcoming SZE surveys, we find that constraints on Omega_m andsigma_8 at the level of roughly 5% or better can be expected, assuming redshiftinformation is known at least to z=0.5 and in the absence of significantsystematic errors. Without information past this redshift, Omega_L isconstrained to 25%. With complete redshift information, deep (M_{lim}=10^{14}h^{-1}{M_sun}), relatively small solid angle (roughly 12 {deg}^2)surveys can further constrain Omega_L to an accuracy of 15%, while large solidangle surveys with ground-based large-format bolometer arrays could measureOmega_L to a precision of 4% or better.Comment: 5 pages, 3 figures, to appear in ApJL, minor revision

Constraints on Ω[TINF][CLC][ITAL]m[/ITAL][/CLC][/TINF], Ω[TINF]Λ[/TINF], and σ[TINF]8[/TINF] from Galaxy Cluster Redshift Distributions