Topological restrictions on relatively Anosov representations

We obtain restrictions on which groups can admit relatively Anosov representations into specified target Lie groups, by examining the topology of possible Bowditch boundaries and how they interact with the Anosov limit maps. For instance, we prove that, up to finite index, any group admitting a relatively Anosov representation into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper S sans-serif upper L Subscript 3 Baseline left-parenthesis double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">S</mml:mi> <mml:mi mathvariant="sans-serif">L</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msub> <mml:mo stretchy="false">(</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">\mathsf {SL}_3(\mathbb {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a free group or surface group, and any group admitting a relatively <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Anosov representation into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper S sans-serif p Subscript 2 m Baseline left-parenthesis double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">S</mml:mi> <mml:mi mathvariant="sans-serif">p</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</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">\mathsf {Sp}_{2m}(\mathbb {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an odd integer between 1 and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , is a surface group or a free product of nilpotent groups. We also obtain a characterization of groups admitting relatively 1-Anosov representations into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper S sans-serif upper L Subscript 4 Baseline left-parenthesis double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">S</mml:mi> <mml:mi mathvariant="sans-serif">L</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msub> <mml:mo stretchy="false">(</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">\mathsf {SL}_4(\mathbb {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , general bounds on the dimension of the Bowditch boundary of groups admitting relatively Anosov representations into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper S sans-serif upper L Subscript d Baseline left-parenthesis double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">S</mml:mi> <mml:mi mathvariant="sans-serif">L</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy="false">(</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">\mathsf {SL}_d(\mathbb {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , statements relating spheres in the Bowditch boundary to the (non-)existence of relatively Anosov representations, and a characterization of groups of cohomological dimension at least <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d minus 1"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> admitting relatively 1-Anosov representations into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper S sans-serif upper L Subscript d Baseline left-parenthesis double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">S</mml:mi> <mml:mi mathvariant="sans-serif">L</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msub> <mml:mo stretchy="false">(</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">\mathsf {SL}_d(\mathbb {R})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>