Expansions of the real field by discrete subgroups of 𝐺𝑙_{𝑛}(ℂ)

<p>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ<! -- Γ --></mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an infinite discrete subgroup of Gl<inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="Subscript n Baseline left-parenthesis double-struck upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi /> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">_n(\mathbb {C})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Then either <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="left-parenthesis double-struck upper R comma greater-than comma plus comma dot comma normal upper Gamma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>></mml:mo> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo>,</mml:mo> <mml:mo>⋅<! -- ⋅ --></mml:mo> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">Γ<! -- Γ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbb {R},>,+,\cdot ,\Gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is interdefinable with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="left-parenthesis double-struck upper R comma greater-than comma plus comma dot comma lamda Superscript double-struck upper Z Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>></mml:mo> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo>,</mml:mo> <mml:mo>⋅<! -- ⋅ --></mml:mo> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>λ<! -- λ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbb {R},>,+,\cdot , \lambda ^{\mathbb {Z}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some real number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="lamda"> <mml:semantics> <mml:mi>λ<! -- λ --></mml:mi> <mml:annotation encoding="application/x-tex">\lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="left-parenthesis double-struck upper R comma greater-than comma plus comma dot comma normal upper Gamma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mo>></mml:mo> <mml:mo>,</mml:mo> <mml:mo>+</mml:mo> <mml:mo>,</mml:mo> <mml:mo>⋅<! -- ⋅ --></mml:mo> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">Γ<! -- Γ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbb {R},>,+,\cdot ,\Gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defines the set of integers. When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ<! -- Γ --></mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not virtually abelian, the second case holds. </p>