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We show that every sequence of torsion-free arithmetic congruence lattices in$\mathrm{PGL}(2,\mathbb R)$ or $\mathrm{PGL}(2,\mathbb C)$ satisfies a strongquantitative version of the Limit Multiplicity property. We deduce that for$R>0$ in certain range, growing linearly in the degree of the invariant tracefield, the volume of the $R$-thin part of any congruence arithmetic hyperbolicsurface or congruence arithmetic hyperbolic $3$-manifold $M$ is of order atmost $\mathrm{Vol}(M)^{11/12}$. As an application we prove Gelander'sconjecture on homotopy type of arithmetic hyperbolic $3$-manifolds: We showthat there are constants $A,B$ such that every such manifold $M$ is homotopyequivalent to a simplicial complex with at most $A\mathrm{Vol}(M)$ vertices,all of degrees bounded by $B$.

Strong Limit Multiplicity for arithmetic hyperbolic surfaces and $3$-manifolds