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Take any norm, $|cdot|$on $mathbbR^n,$ and consider the resulting norm on $SL_n(mathbbR)$:

$$|A|:= supAv.$$

Now take any left-invariant Riemannian metric, $g$, on $SL_n$. How do the geodesic balls, $B_g(I, r)$ around the identity matrix, $I$, compare with the metric balls, $B_(I,r)$ coming from $|cdot|$? In particular do there exist $c, C$ such that $$B_(I,cr)subset B_g(I, r) subset B_(I,Cr)$$ for all sufficiently small $r$? Or anything of the sort?

Here's what I see in Einsiedler-Ward:

Your norm on $SL_n(mathbbR)$ is induced by the operator norm on the vector space $M_n(mathbbR)$. Being a norm on a finite-dimensional vector space, it is equivalent to the euclidean norm $|cdot|$ on $M_n(mathbbR)$.

Now any left-invariant Riemannian metric $langle ,rangle$ on $TG=Gtimes mathfrakg$ is determined by its restriction to $TG_I=Itimes mathfrakg cong mathfrakg subset M_n(mathbbR)$. Without loss of generality, we can assume that the Riemannian metric restricted to $mathfrakg$ is induced by the euclidean norm on $M_n(mathbbR).$ Let $d$ be the distance on $G$ induced by the path integral formula of $langle , rangle$.

Let $B$ be a pre-compact neighbourhood of $I$ in $G$ where the local inverse ($log$) of the exponential map is defined. Assume $log(B)$ is a convex ball in $mathfrakg$. Let $B'$ be another pre-compact neighbourhood containing the closure of $B$.

Say $phi:[0,1] to B'$ joins $g_0, g_1 in B$. Then since the norm of $phi(t)$ is bounded, we get $c>0$ (independent of $g_0,g_1$) such that

$$L(phi):= intlangle Dphi(t), Dphi(t) rangle^1/2dt = int leftlangle DL^-1_phi(t)circ Dphi(t), DL^-1_phi(t)circ Dphi(t)rightrangle^1/2 dt = int |phi(t)^-1phi'(t)| dt
\ geq cint|phi'(t)|dt geq c| g_1-g_0|.$$

This shows that $c|g_1-g_0| leq d(g_0,g_1)$ if the infimum of path integrals is taken over paths which remain in $B'$. But since $dleft(B,(B')^cright)>0,$ we can assume that this estimate holds in general. Hence for all $g_0,g_1 in B$, we have

beginequation
c|g_1-g_0| leq d(g_0,g_1) qquad (1)
endequation

and it remains to show a reverse inequality.

Consider the path $phi:[0,1] to B$ given by $t mapsto expleft(log g_0 + t(log g_1-log g_0)right)$. This is well defined since we assumed $log(B)$ was a convex ball in $mathfrakg$. Then, since the norm of $phi(t)^-1$ is bounded, and since $d(exp)$ is bounded in $log(B)$ and since $log$ is Lipschitz (by the mean value theorem) in a neighbourhood of $I$,

$$d(g_0,g_1) leq intlangle Dphi(t), Dphi(t) rangle^1/2dt = int |phi(t)^-1phi'(t)|dt leq int C_1|phi'(t)|dt leq C_1C_2|g_1-g_0|$$

for some $C_1, C_2>0$ (independent of $g_0, g_1$). Hence for all $g_0,g_1in B$, we have

$$ d(g_0,g_1) leq C |g_0-g_1|. qquad (2)$$

I can prove the following. If $h$ denotes the metric on $SL(n,mathbbR)$ coming from the operator norm, and $h_1$ denotes a left-invariant metric on $SL(n,mathbbR)$, then, at $g in SL(n,mathbbR)$, and for any vector $x$ tangent to $SL(n,mathbbR)$ at $g$, we have:

$h(x,x) leq |g|^2 h_1(x,x)$

I just used left translations, and things like that. If interested, I can write some more details. Also, one can prove (using the equivalence of any 2 norms on a finite-dimensional vector space) that there is a $C>0$ such that:

$h_1(x,x) <= C |g^-1|^2 h(x,x)$.

Hence, if $|g|$ and $|g^-1|$ are bounded above by some constants, then the two metrics induce uniformly equivalent norms on the tangent spaces of that region. In particular, there exists a neighborhood of the identity in $SL(n,mathbbR)$ for which the two geodesic distances are equivalent.