Mann wrote he will donate the equivalent of one week’s pay to charities to be determined in the near future.
Mann said he's not admitting guilt to the city of Atlanta ordinance violations of indecency and obstruction.
An incident report said a police officer saw Mann exposing himself in Piedmont Park on May 6, and then Mann ran away when the officer identified himself and turned his flashlight on Mann.
“The mere fact of placing myself in a position to be arrested is sufficient reason for this self-imposed discipline,” Mann wrote. “I cannot, in good faith, fail to take responsibility for the negative and unwanted criticism brought to this great agency and the county, and I apologize to each of you. You deserve a leader who takes responsibility for his actions.”
Besides the charges, Mann is also being investigated by a panel appointed by Gov. Nathan Deal. Based on the panel's findings, the governor could suspend him for up to 90 days, order an additional investigation by the GBI and ask the DeKalb District Attorney to initiate proceedings to remove the sheriff from office.
Mann and his attorney are asking a judge to stop the investigation, saying Mann isn't accused of any misconduct in his official capacity as sheriff.
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