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Density of systoles of hyperbolic manifolds
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We show that for each
n
≥
2
, the systoles of closed hyperbolic
n
-manifolds form a dense subset of
(
0
,
+
∞
)
. We also show that for any
n
≥
2
and any Salem number
λ
, there is a closed arithmetic hyperbolic
n
-manifold of systole
log
(
λ
)
. In particular, the Salem conjecture holds if and only if the systoles of closed arithmetic hyperbolic manifolds in some (any) dimension fail to be dense in
(
0
,
+
∞
)
.
Title: Density of systoles of hyperbolic manifolds
Description:
We show that for each
n
≥
2
, the systoles of closed hyperbolic
n
-manifolds form a dense subset of
(
0
,
+
∞
)
.
We also show that for any
n
≥
2
and any Salem number
λ
, there is a closed arithmetic hyperbolic
n
-manifold of systole
log
(
λ
)
.
In particular, the Salem conjecture holds if and only if the systoles of closed arithmetic hyperbolic manifolds in some (any) dimension fail to be dense in
(
0
,
+
∞
)
.
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