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Transition-metal-doped semiconductor GaAs crystals are used as model systems for spintronic research, as well as in other applications. To explore the structure and properties of such impurities, we extended the methodology of ultrasonic investigation of the Jahn-Teller effect in dielectric impurity-centers to the study of significantly different semiconductor impurities using the GaAs:Cu crystal as an example. Phase velocity and attenuation of ultrasound in this system were measured in the temperature interval of 1.9-80 K at 52 MHz and 156 MHz. The anomaly in the velocity and a peak of attenuation found for the longitudinal and slow shear waves indicate the presence of the Jahn-Teller effect with the e-type local distortions of the Cu_Ga4As impurity complex. The temperature dependence of the elastic modulus and relaxation time shows that below 5 K, the thermal activation mechanism of relaxation is possibly replaced by resonance type transitions. The main parameters of the Jahn-Teller effect, stabilization energy, minima positions and the barrier between them, frequency of pseudorotation of the distortions, and the tunneling splitting of the ground state energy, as well as the constant of exchange interaction between the two holes in Cu²⁺ centers and the concentration of the centers were estimated. © 2014 AIP Publishing LLC.