Interim Austin Police Chief Joseph Chacon said Broderick was suspected in Sunday's deaths of two women and one man, all of whom Broderick knew.
Police initially told Austin residents who lived nearby to shelter in place, but they lifted that request later Sunday when they said it was unclear whether Broderick was still in the city.
Chacon said that Broderick was a former deputy with the Travis County sheriff's office, which is based in Austin.
Travis County District Attorney Jose Garza said in a statement that Broderick had been arrested for sexual assault of a child last June and was released on $50,000 bond. He said that the district attorney's office on Sunday filed a motion to revoke that bond.