One can define the convergence of a sequence $(\Lambda\_k)\_k$ of full rank lattices as folow : $(\Lambda\_k)\underset{k\rightarrow +\infty}{\longrightarrow} \Lambda \iff \forall k\in \mathbb{N} ,\exists B\_k\in (\mathbb{R}^n)^n \text{ a $\mathbb{Z}$-basis of $\Lambda\_k$}, \exists B\in (\mathbb{R}^n)^n \text{ a $\mathbb{Z}$-basis of $\Lambda$}, \underset{k\rightarrow + \infty}{\lim}B\_k = B $.
Also, we can identify the set of all full rank lattices with the quotient $GL\_n(\mathbb{R})/GL\_n(\mathbb{Z})$ because if $B\_1,B\_2$ are $\mathbb{Z}$-basis of a lattice $\Lambda$, if $A\_1,A\_2$ are the $(n\times n )$ matrix with vectors of $B\_i$ as columns ($i\in \{1,2\}$), it exists a matrix $P\in GL\_n(\mathbb{Z})$ such that $A\_1 = PA\_2$.


Thus I'm looking for a distance $d$ (if it exists) for $GL\_n(\mathbb{R})/GL\_n(\mathbb{Z})$ such that $(\Lambda\_k)\underset{k\rightarrow +\infty}{\longrightarrow} \Lambda \iff d(\Lambda\_k,\Lambda)\underset{k\rightarrow +\infty}{\longrightarrow} 0$.


A guess could be $d(X,Y) = \max(\underset{(A,B)\in X\times Y , P\in GL\_n(\mathbb{Z})}{\sup} \Vert AB^{-1} - P\Vert,\underset{(A,B)\in X\times Y , P\in GL\_n(\mathbb{Z})}{\sup} \Vert BA^{-1} - P\Vert)$ for $ (X,Y)\in (GL\_n(\mathbb{R})/GL\_n(\mathbb{Z}))^2$ because $X,Y$ should be close close if $X \approx GL\_n(\mathbb{Z})\cdot Y = Y$. But the "$\sup$" is not even well defined...


Thank you.



Distance for $GL_n(\mathbb{R})/GL_n(\mathbb{Z})$
Fixing any right invariant metric $D$ on $G=\text{GL}\_n(\mathbb{R})$ you get a metric $d$ on $G/\Gamma$ ($\Gamma=\text{GL}\_n(\mathbb{Z})$) be letting $d(g\Gamma,h\Gamma)=D(g,h\Gamma)$. If $D$ generates the standard topology on $G$ then (as you can check) $d$ will generate the standard topology on $G/\Gamma$.
A way to get a right invariant metric on $G$ is by fixing an inner product on its Lie algebra (eg the Hilbert-Schmidt/Frobenius/$L^2$ norm on $M\_n(\mathbb{R})$) and right-shifting it around, thus getting a right invariant Riemannian metric.


