Type: Article
Publication Date: 2021-04-14
Citations: 5
DOI: https://doi.org/10.1090/tran/8453
We show that a collar lemma holds for Anosov representations of fundamental groups of surfaces into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper L left-parenthesis n comma double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">SL(n,\mathbb {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that satisfy partial hyperconvexity properties inspired from Labourie’s work. This is the case for several open sets of Anosov representations not contained in higher rank Teichmüller spaces, as well as for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Theta"> <mml:semantics> <mml:mi mathvariant="normal">Θ<!-- Θ --></mml:mi> <mml:annotation encoding="application/x-tex">\Theta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-positive representations into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper O left-parenthesis p comma q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">SO(p,q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than-or-equal-to 4"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p\geq 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We moreover show that ‘positivity properties’ known for Hitchin representations, such as being positively ratioed and having positive eigenvalue ratios, also hold for partially hyperconvex representations.