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


Lehmer’s conjecture says: There exists $m>1$ such that $M(p)\geq m$ for all noncyclotomic $P$.


Systole is a closed geodesic of the shortest length in Riemannian manifold.


Thanks.



What’s the relation between Lehmer’s Conjecture and Systole
The relationship between Lehmer's Conjecture and systoles is related to the following important conjecture regarding the geometry of arithmetic hyperbolic orbifolds:


*Short Geodesic Conjecture*: There is a universal, positive lower bound for the systole of an arithmetic hyperbolic 2- or 3-orbifold.


To illustrate what is going on, suppose that $M$ is an arithmetic hyperbolic $3$-manifold and set $\Gamma=\pi\_1(S)$. (I am going to assume going forward that $\Gamma$ is *derived from a quaternion algebra* in order to ignore some technicalities. Also, everything works in the same way in the case of arithmetic hyperbolic surfaces.) Let $\gamma\in\Gamma$ be a loxodromic or hyperbolic element, and write $\mathrm{tr}(\gamma)=u+u^{-1}$ where $|u|>1$. Then the length of the geodesic associated to $\gamma$ is either $\ln(M(p\_u))$ or $2\ln(M(p\_u))$ (depending on whether $\gamma$ is loxodromic or hyperbolic), where $p\_u$ is the minimal polynomial of $u$.


In particular, all of this shows that Lehmer's Conjecture implies the Short Geodesic Conjecture. In fact, the Salem Conjecture (i.e., Lehmer's Conjecture for Salem numbers) is equivalent to the Short Geodesic Conjecture in dimension $2$ and implied by the Short Geodesic Conjecture in dimension $3$.


All of this is described in quite a bit of depth in Chapter 12.3 of Maclachlan and Reid's *The arithmetic of hyperbolic $3$-manifolds*.


For an interesting discussion on the relationship between Lehmer's Conjecture and more general arithmetic manifolds, see the paper 


*Gelander, Tsachik*, [**Homotopy type and volume of locally symmetric manifolds**](http://dx.doi.org/10.1215/S0012-7094-04-12432-7), Duke Math. J. 124, No. 3, 459-515 (2004). [ZBL1076.53040](https://zbmath.org/?q=an:1076.53040),


especially Section 10.


