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1.
1)laying low

2)running from the feds
She broke the law, now she's on the lam.
by Marcus November 23, 2004
 
2.
To be running away from the authorities. This term dates back to the 1920's and was a part of urban slang.
"I pinched the hooch from some drugstore cowboy of a raggle-muffin and now im on the lam so you better scram or ill knock you off. Kepeesh!"
by Brian R. Turner September 26, 2006
 
3.
"On the lam" or "on the run" often refers to fugitives. The Thesaurus of American Slang proclaim that lam, lamister, and "on the lam" — all referring to a hasty departure — were common in thieves' slang before the turn of the twentieth century. A "lamister", or, "lamster," is a slang term for fugitive.
Hey, Johnny! Have you seen Mugsy?" "Naw, he's on the lam for that caper he pulled in the Bronx
by antkneep June 28, 2011
 
4.
To be in hiding. LAM means "Leave a Message," meaning you are not home or in town, i.e. on the run.
"When the gangs decided to go to war, Jamal went on the LAM to avoid getting killed."
by jjp54 February 03, 2010