The word despertaban is the imperfect form of despertar in the third person plural.
See the full despertar conjugation

despertar

Sus sufrimientos no despertaban piedad en el corazón de los sacerdotes.
Their suffering awakened no pity in the hearts of the priests.
Y me despertaban cuando era momento de hacerlo.
And woke me when it was time to perform.
Y ellos sabían inmediatamente por qué se despertaban por la mañana.
They instantly knew why they woke up in the morning.
Tenían esta extraña mirada cuando nos despertaban.
They had this strange look when they woke us.
Te despertaban a las 6:00 a.m todas las mañanas con los martillazos.
You were woken up at 6:00 a.m. every morning by the hammering.
Los trenes me despertaban cada mañana.
The trains would wake me up every morning.
Almas despertaban de un sueño profundo.
Souls were awakening from a deep sleep.
A veces limpiaban a las 4 de la mañana y nos despertaban.
Often times they would clean up at 4am and wake us.
De día, me despertaban a las 4 para ordeñar las vacas.
I would get up at 4:00 a.m. To milk the cows.
Tan pronto como comenzaban a soñar, los despertaban y el sueño se detenía.
As soon as they started to dream, they were awakened and the dream stopped.
Y simplemente se despertaban llorando.
And they would just wake themselves crying.
Y que despertaban mucho mi curiosidad.
And that made me very curious.
Una mañana de verano del 2010 los londinenses se despertaban inexplicablemente con una extraña sensación de vértigo.
One summer morning in 2010 Londoners woke up with an inexplicable sensation of vertigo.
En un principio, notamos que las acciones para ampliar el círculo de participantes despertaban cierta resistencia.
In the beginning, we noticed that efforts to widen the circle of actors were met with some resistance.
No obstante, hay algunas situaciones, como las pertinentes a la intolerancia religiosa, que despertaban cierta preocupación.
Nevertheless, a number of situations, including for reasons related to religious intolerance, raised some concerns.
Con cada esfuerzo cobraba más energía y valor al ver el marcado interés que despertaban sus palabras.
With every effort he gained strength and courage as he saw the widespread interest excited by his words.
Este es el comienzo de Canadá y nadie podrá negar que levanta unas expectativas que no despertaban los mencionados anteriormente.
This is the beginning of Canada and nobody can deny that raises expectations that the aforementioned beginnings.
Cada vez que ocurría eso, los guardias la despertaban con agua fría, con el fin de poder aplicarle más descargas eléctricas.
Each time, the guards revived her with cold water, so that they could shock her again.
Llegó el punto donde ustedes dos se despertaban uno al otro, así que hicimos que él volviera a su propia cama.
It got to the point where you two were keeping each other up, so we finally made him go back to his own bed.
Al principio planteaba algunos problemas técnicos y los ordenadores entraban en reposo, pero luego no se 'despertaban' con suficiente rapidez.
He said there were technical problems to begin with, because the computers went into 'sleep mode' alright, but failed to 'wake up' again quickly enough.
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