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# 32 questions about loops and negative curvature

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$(M^n,g)$ is a compact $n$ dimensional manifold of negative curvature

1) with n>2 . let $\alpha$ be a simple closed geodesic loop in $M$

if $n=2$ then we know that the geodesic in the free homotopy class of $\alpha$ is simple . now suppose based at a point $n>2$ p$1) will the geodesic in the free homotopy class of alpha be simple ? 2) suppose$n>3$and let$\alpha$be a simple closed geodesic loop at a point$p$in$M$can$\alpha$be homotopic ( with respect to$p$) to a power of another closed curve at$p$3) suppose$n>3$let$p$be a point in$M$. Let$\Gamma$be subgroup of$\pi_1(M,p)$generated by {$\alpha \in \pi_1(M,p)$such that$ length(\alpha)= Systole(M,p)$} . suppose now$\Gamma$is cyclic does that necessary mean that there exist a unique$\alpha$such that$length(\alpha) = Systole(M,p)$2 deleted 5 characters in body; added 1 characters in body in all questions$(M^n,g)$is a compact$n$dimensional manifold of negative curvature 1) let$\alpha$be a simple closed geodesic loop in$M$if$n=2$then we know that the geodesic in the free homotopy class of$\alpha$is simple . i think if now suppose$n>3$then this n>2$ will stop being true can but can someone give me a counter example the geodesic in the free homotopy class of alpha be simple ?

2) suppose $n>3$ and let $\alpha$ be a simple closed geodesic loop at a point $p$ in $M$ can $\alpha$ be homotopic ( with respect to $p$ ) to a power of another closed curve at $p$

3) suppose $n>3$ let $p$ be a point in $M$ . Let $\Gamma$ be subgroup of $\pi_1(M,p)$ generated by {$\alpha \in \pi_1(M,p)$ such that $length(\alpha)= Systole(M,p)$} . suppose now $\Gamma$ is cyclic does that necessary mean that there exist a unique $\alpha$ such that $length(\alpha) = Systole(M,p)$

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