First-order sentences in random groups I: Universal sentences

We prove that a random group, in Gromov’s density model with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo><</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mn>16</m:mn> </m:mrow> </m:mrow> </m:math> d<1/16 , satisfies a universal sentence 𝜎 (in the language of groups) if and only if 𝜎 is true in a non-abelian free group.