Finite generation of fundamental groups for manifolds with nonnegative Ricci curvature whose universal cover is almost 𝑘-polar at infinity

Abstract In this article, we prove that the fundamental group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>π</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \pi_{1}(M) of a complete open manifold 𝑀 with nonnegative Ricci curvature is finitely generated, under the condition that the Riemannian universal cover <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover accent="true"> <m:mi>M</m:mi> <m:mo>̃</m:mo> </m:mover> </m:math> \tilde{M} satisfies an “almost 𝑘-polar at infinity” condition. Additionally, such <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>π</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>M</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \pi_{1}(M) is virtually abelian. Furthermore, we demonstrate that the base point of any tangent cone at infinity of such a manifold is nearly a pole. In the case where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover accent="true"> <m:mi>M</m:mi> <m:mo>̃</m:mo> </m:mover> </m:math> \tilde{M} exhibits almost maximal Euclidean volume growth, we prove that 𝑀 deformation retracts to a closed submanifold 𝐹 which is diffeomorphic to a flat manifold, provided 𝑀 is not simply connected.