## Homotopy type and volume of locally symmetric manifolds.(English)Zbl 1076.53040

The work provides partial answers (cf. Theorem 1.5) to the following conjecture: For any symmetric space of noncompact type $$S$$, there are constants $$\alpha(S), d(S)$$, such that any irreducible $$S$$-manifold $$\,M=\Gamma\setminus S\,$$ (which is assumed also to be arithmetic in the case $$\,\text{dim}S=3\,$$) is homotopically equivalent to a simplicial complex with at most $$\alpha(S)\,\text{vol}(M)$$ vertices, all of them of valence at most $$d(S)$$. For example, the conjecture holds for noncompact arithmetic $$S$$-manifolds. The results imply quantitative versions for some classical finiteness statements. The author provides a linear bound on the size of a minimal presentation of the fundamental group in terms of the volume and is able to estimate the number of $$S$$-manifolds with bounded volume.
The basic idea behind the main result of Theorem 1.5 is to construct, inside each $$S$$-manifold $$M$$, a submanifold $$M'$$, which is similar enough to $$M$$ and for which one can construct a triangulation of size $$\,c\cdot\text{vol}(M)\,$$. For this construction, one requires a lower bound on the injectivity radius of $$M'$$ (independent of $$M$$) and some bounds on the geometry of the boundary of $$M'$$.

### MathOverflow Questions:

What’s the relation between Lehmer’s Conjecture and Systole

### MSC:

 53C20 Global Riemannian geometry, including pinching 22E40 Discrete subgroups of Lie groups
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### References:

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