Different
notions of
discreteness
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Different notions of discreteness and the
modified Poincaré exponent
David Simmons
Ohio State University
Different
notions of
discreteness
David
Simmons
Discreteness
1 Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
2 Modified Poincaré exponent
3 Discreteness and the modified Poincaré exponent
References
Different
notions of
discreteness
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
“DSU ’14”
T. Das, D. S. Simmons, and M. Urbański, Geometry and
dynamics in Gromov hyperbolic metric spaces: with an
emphasis on non-proper settings,
http://arxiv.org/abs/1409.2155, preprint 2014.
Different notions of discreteness
Different
notions of
discreteness
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Let X be a metric space, and let G ≤ Isom(X ).
Definition (DSU ’14)
We say that G is strongly discrete (SD) if for all ρ > 0,
#{g ∈ G : d(o, g (o)) ≤ ρ} < ∞.
Different notions of discreteness
Different
notions of
discreteness
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Let X be a metric space, and let G ≤ Isom(X ).
Definition (DSU ’14)
We say that G is strongly discrete (SD) if for all ρ > 0,
#{g ∈ G : d(o, g (o)) ≤ ρ} < ∞.
G is moderately discrete (MD) if for every x ∈ X , there
exists an open U 3 x with
#(g ∈ G : g (U) ∩ U 6= ) < +∞.
Different notions of discreteness
Different
notions of
discreteness
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Let X be a metric space, and let G ≤ Isom(X ).
Definition (DSU ’14)
We say that G is strongly discrete (SD) if for all ρ > 0,
#{g ∈ G : d(o, g (o)) ≤ ρ} < ∞.
G is moderately discrete (MD) if for every x ∈ X , there
exists an open U 3 x with
#(g ∈ G : g (U) ∩ U 6= ) < +∞.
G is weakly discrete (WD) if for every x ∈ X , there exists
an open U 3 x with
g (U) ∩ U 6= ⇒ g (x) = x.
Different notions of discreteness
Different
notions of
discreteness
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Definition (Continued)
Different notions of discreteness
Different
notions of
discreteness
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Definition (Continued)
G is COT-parametrically discrete (COT-PD) if it is
discrete as a subset of Isom(X ) equipped with the
compact-open topology (COT).
Different notions of discreteness
Different
notions of
discreteness
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Definition (Continued)
G is COT-parametrically discrete (COT-PD) if it is
discrete as a subset of Isom(X ) equipped with the
compact-open topology (COT).
G is properly discontinuous (PrD) if for every x ∈ X , there
exists an open U 3 x with
g (U) ∩ U 6= ⇒ g = id.
Different notions of discreteness
Different
notions of
discreteness
Definition (Continued)
David
Simmons
G is COT-parametrically discrete (COT-PD) if it is
discrete as a subset of Isom(X ) equipped with the
compact-open topology (COT).
Discreteness
Modified
Poincaré
exponent
G is properly discontinuous (PrD) if for every x ∈ X , there
exists an open U 3 x with
Discreteness
and the
modified
Poincaré
exponent
g (U) ∩ U 6= ⇒ g = id.
Fact
If X is a finite-dimensional manifold, then the first four notions
of discreteness are equivalent. Moreover,
PrD ⇔ discrete + torsion-free.
Relations between different notions of discreteness
Different
notions of
discreteness
General
metric
space
Proper
metric
space
Finite
dimensional
manifold
Infinite
dimensional
ROSSONCT1
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
SD
PrD
SD
↑
PrD
SD
↑
PrD
SD
→
%
MD
→
&
↔
MD
↔
↔
MD
↔
→
%
MD
→
PrD
WD
COT-PD
COT-PD
↓
WD
WD
l
COT-PD
WD
↓
COT-PD
All implications not listed have counterexamples.
1
rank one symmetric space of noncompact type
The Poincaré exponent
Different
notions of
discreteness
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Fix b > 1. For each s ≥ 0, the Poincaré series for G with
exponent s is the series
X
Σs (G ) :=
b −sd(o,g (o)) .
g ∈G
The Poincaré exponent of G is the number
δ(G ) := inf{s ≥ 0 : Σs (G ) < ∞}.
The Poincaré exponent
Different
notions of
discreteness
David
Simmons
Fix b > 1. For each s ≥ 0, the Poincaré series for G with
exponent s is the series
X
Σs (G ) :=
b −sd(o,g (o)) .
g ∈G
Discreteness
Modified
Poincaré
exponent
The Poincaré exponent of G is the number
δ(G ) := inf{s ≥ 0 : Σs (G ) < ∞}.
Discreteness
and the
modified
Poincaré
exponent
Observation
If δ(G ) < ∞, then G is strongly discrete.
The Poincaré exponent
Different
notions of
discreteness
David
Simmons
Fix b > 1. For each s ≥ 0, the Poincaré series for G with
exponent s is the series
X
Σs (G ) :=
b −sd(o,g (o)) .
g ∈G
Discreteness
Modified
Poincaré
exponent
The Poincaré exponent of G is the number
δ(G ) := inf{s ≥ 0 : Σs (G ) < ∞}.
Discreteness
and the
modified
Poincaré
exponent
Observation
If δ(G ) < ∞, then G is strongly discrete.
Question
How to “measure the quantity that δ(G ) measures” for groups
which are not strongly discrete?
The modified Poincaré exponent
Different
notions of
discreteness
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
For each s ≥ 0 and A ⊆ X , write
X
Σs (A) =
b −sd(o,x)
x∈A
δ(A) = inf{s ≥ 0 : Σs (A) < ∞}.
Definition (DSU ’14)
The modified Poincaré exponent is the number
e ) = lim δ(Aρ ),
δ(G
ρ→∞
where for each ρ > 0, Aρ is a maximal ρ-separated subset of
G (o).
The modified Poincaré exponent
Different
notions of
discreteness
David
Simmons
For each s ≥ 0 and A ⊆ X , write
X
Σs (A) =
b −sd(o,x)
x∈A
δ(A) = inf{s ≥ 0 : Σs (A) < ∞}.
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Definition (DSU ’14)
The modified Poincaré exponent is the number
e ) = lim δ(Aρ ),
δ(G
ρ→∞
where for each ρ > 0, Aρ is a maximal ρ-separated subset of
G (o).
Observation
e ) = δ(G ).
If G is strongly discrete, then δ(G
Geometric significance of the modified Poincaré
exponent
Different
notions of
discreteness
David
Simmons
Let X be a hyperbolic metric space, let ∂X denote its Gromov
boundary, and let Λr (G ) ⊆ ∂X be the radial limit set of G .
Theorem (DSU ’14)
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Suppose G is nonelementary. Then the Hausdorff dimension of
e ).
Λr (G ) with respect to the visual metric is δ(G
Geometric significance of the modified Poincaré
exponent
Different
notions of
discreteness
David
Simmons
Let X be a hyperbolic metric space, let ∂X denote its Gromov
boundary, and let Λr (G ) ⊆ ∂X be the radial limit set of G .
Theorem (DSU ’14)
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Suppose G is nonelementary. Then the Hausdorff dimension of
e ).
Λr (G ) with respect to the visual metric is δ(G
Corollary (Special cases by Bishop–Jones ’97, Paulin ’97)
If G is strongly discrete and nonelementary, then the Hausdorff
dimension of Λr (G ) with respect to the visual metric is δ(G ).
Geometric significance of the modified Poincaré
exponent
Different
notions of
discreteness
David
Simmons
Let X be a hyperbolic metric space, let ∂X denote its Gromov
boundary, and let Λr (G ) ⊆ ∂X be the radial limit set of G .
Theorem (DSU ’14)
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Suppose G is nonelementary. Then the Hausdorff dimension of
e ).
Λr (G ) with respect to the visual metric is δ(G
Corollary (Special cases by Bishop–Jones ’97, Paulin ’97)
If G is strongly discrete and nonelementary, then the Hausdorff
dimension of Λr (G ) with respect to the visual metric is δ(G ).
Open Question
e ) if G is an
What is the geometric significance of δ(G
elementary group?
Examples of modified Poincaré exponents
Different
notions of
discreteness
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Example
Let G ≤ Isom(X ) be a locally compact, metrically proper group
and for each s ≥ 0 let
Z
b −sd(o,g (o)) dg ,
Is =
G
where dg denotes Haar measure. Then
e ) = inf{s ≥ 0 : Is < ∞}.
δ(G
Examples of modified Poincaré exponents
Different
notions of
discreteness
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Example
Let G ≤ Isom(X ) be a locally compact, metrically proper group
and for each s ≥ 0 let
Z
b −sd(o,g (o)) dg ,
Is =
G
where dg denotes Haar measure. Then
e ) = inf{s ≥ 0 : Is < ∞}.
δ(G
Example (Special case)
If X is d-dimensional hyperbolic space, then
e
δ(Isom(X
)) = d − 1.
Examples of modified Poincaré exponents
Different
notions of
discreteness
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Example
Let X be an infinite-dimensional symmetric space or an
e
infinitely regular tree. Then δ(Isom(X
)) = ∞.
Examples of modified Poincaré exponents
Different
notions of
discreteness
David
Simmons
Example
Let X be an infinite-dimensional symmetric space or an
e
infinitely regular tree. Then δ(Isom(X
)) = ∞.
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Example
Let G ≤ Isom(X ) be a countable group whose closure is locally
compact and metrically proper, let Y ⊇ X , and let
Φ : G → Isom(Y ) be a homomorphism such that
Φ(g )(x) = g (x) for all g ∈ G and x ∈ X . Then
e
e ), but the closure of Φ(G ) may not be locally
δ(Φ(G
)) = δ(G
compact.
Examples of modified Poincaré exponents
Different
notions of
discreteness
David
Simmons
Example
Let X be an infinite-dimensional symmetric space or an
e
infinitely regular tree. Then δ(Isom(X
)) = ∞.
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Example
Let G ≤ Isom(X ) be a countable group whose closure is locally
compact and metrically proper, let Y ⊇ X , and let
Φ : G → Isom(Y ) be a homomorphism such that
Φ(g )(x) = g (x) for all g ∈ G and x ∈ X . Then
e
e ), but the closure of Φ(G ) may not be locally
δ(Φ(G
)) = δ(G
compact.
Open Question
e ) < ∞ “come from” locally compact
Do all groups with δ(G
metrically proper groups?
Discreteness and the modified Poincaré exponent
Different
notions of
discreteness
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
e ) = δ(G ).
We call G Poincaré regular if δ(G
Observation
e ) < ∞ = δ(G ).
If G is Poincaré irregular, then δ(G
Discreteness and the modified Poincaré exponent
Different
notions of
discreteness
David
Simmons
e ) = δ(G ).
We call G Poincaré regular if δ(G
Observation
e ) < ∞ = δ(G ).
If G is Poincaré irregular, then δ(G
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Theorem (DSU ’14)
Suppose that one of the following holds:
G is strongly discrete,
G is moderately discrete and X is a CAT(-1) space,
G is weakly discrete and X is a ROSSONCT,
G is COT-parametrically discrete, X is a ROSSONCT, and
G does not preserve any totally geodesic subspace of X .
Then G is Poincaré regular.
Again, all implications not listed have counterexamples.
Idea of the proof
Different
notions of
discreteness
If G is a Poincaré irregular group, then there exists ρ > 0 such
that the set
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Gρ = {g ∈ G : d(o, g (o)) ≤ ρ}
is infinite, but for all τ > 0, there exists a finite ρ-net for Gτ .
Idea of the proof
Different
notions of
discreteness
If G is a Poincaré irregular group, then there exists ρ > 0 such
that the set
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Gρ = {g ∈ G : d(o, g (o)) ≤ ρ}
is infinite, but for all τ > 0, there exists a finite ρ-net for Gτ .
Suppose X is a CAT(-1) space, and let ∂X be the Gromov
boundary of X . For all ξ ∈ ∂X , Gρ (ξ) is precompact.
Idea of the proof
Different
notions of
discreteness
If G is a Poincaré irregular group, then there exists ρ > 0 such
that the set
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Gρ = {g ∈ G : d(o, g (o)) ≤ ρ}
is infinite, but for all τ > 0, there exists a finite ρ-net for Gτ .
Suppose X is a CAT(-1) space, and let ∂X be the Gromov
boundary of X . For all ξ ∈ ∂X , Gρ (ξ) is precompact. Find a
sequence gn ∈ Gρ such that gn (ξ1 ) → η1 , gn (ξ2 ) → η2 for some
ξ1 , ξ2 ∈ ∂X distinct.
Idea of the proof
Different
notions of
discreteness
If G is a Poincaré irregular group, then there exists ρ > 0 such
that the set
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Gρ = {g ∈ G : d(o, g (o)) ≤ ρ}
is infinite, but for all τ > 0, there exists a finite ρ-net for Gτ .
Suppose X is a CAT(-1) space, and let ∂X be the Gromov
boundary of X . For all ξ ∈ ∂X , Gρ (ξ) is precompact. Find a
sequence gn ∈ Gρ such that gn (ξ1 ) → η1 , gn (ξ2 ) → η2 for some
ξ1 , ξ2 ∈ ∂X distinct. If x ∈ [ξ1 , ξ2 ], then gn (x) → y for some
y ∈ [η1 , η2 ].
Idea of the proof
Different
notions of
discreteness
If G is a Poincaré irregular group, then there exists ρ > 0 such
that the set
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Gρ = {g ∈ G : d(o, g (o)) ≤ ρ}
is infinite, but for all τ > 0, there exists a finite ρ-net for Gτ .
Suppose X is a CAT(-1) space, and let ∂X be the Gromov
boundary of X . For all ξ ∈ ∂X , Gρ (ξ) is precompact. Find a
sequence gn ∈ Gρ such that gn (ξ1 ) → η1 , gn (ξ2 ) → η2 for some
ξ1 , ξ2 ∈ ∂X distinct. If x ∈ [ξ1 , ξ2 ], then gn (x) → y for some
y ∈ [η1 , η2 ]. Thus G is not moderately discrete.
Idea of the proof
Different
notions of
discreteness
If G is a Poincaré irregular group, then there exists ρ > 0 such
that the set
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Gρ = {g ∈ G : d(o, g (o)) ≤ ρ}
is infinite, but for all τ > 0, there exists a finite ρ-net for Gτ .
Suppose X is a CAT(-1) space, and let ∂X be the Gromov
boundary of X . For all ξ ∈ ∂X , Gρ (ξ) is precompact. Find a
sequence gn ∈ Gρ such that gn (ξ1 ) → η1 , gn (ξ2 ) → η2 for some
ξ1 , ξ2 ∈ ∂X distinct. If x ∈ [ξ1 , ξ2 ], then gn (x) → y for some
y ∈ [η1 , η2 ]. Thus G is not moderately discrete.
Suppose that X is a ROSSONCT and that G does not preserve
any totally geodesic subspace.
Idea of the proof
Different
notions of
discreteness
If G is a Poincaré irregular group, then there exists ρ > 0 such
that the set
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Gρ = {g ∈ G : d(o, g (o)) ≤ ρ}
is infinite, but for all τ > 0, there exists a finite ρ-net for Gτ .
Suppose X is a CAT(-1) space, and let ∂X be the Gromov
boundary of X . For all ξ ∈ ∂X , Gρ (ξ) is precompact. Find a
sequence gn ∈ Gρ such that gn (ξ1 ) → η1 , gn (ξ2 ) → η2 for some
ξ1 , ξ2 ∈ ∂X distinct. If x ∈ [ξ1 , ξ2 ], then gn (x) → y for some
y ∈ [η1 , η2 ]. Thus G is not moderately discrete.
Suppose that X is a ROSSONCT and that G does not preserve
any totally geodesic subspace. Find a sequence gn ∈ Gρ such
± for a dense sequence ξ ∈ ∂X .
that gn± (ξm ) → ηm
m
Idea of the proof
Different
notions of
discreteness
If G is a Poincaré irregular group, then there exists ρ > 0 such
that the set
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Gρ = {g ∈ G : d(o, g (o)) ≤ ρ}
is infinite, but for all τ > 0, there exists a finite ρ-net for Gτ .
Suppose X is a CAT(-1) space, and let ∂X be the Gromov
boundary of X . For all ξ ∈ ∂X , Gρ (ξ) is precompact. Find a
sequence gn ∈ Gρ such that gn (ξ1 ) → η1 , gn (ξ2 ) → η2 for some
ξ1 , ξ2 ∈ ∂X distinct. If x ∈ [ξ1 , ξ2 ], then gn (x) → y for some
y ∈ [η1 , η2 ]. Thus G is not moderately discrete.
Suppose that X is a ROSSONCT and that G does not preserve
any totally geodesic subspace. Find a sequence gn ∈ Gρ such
± for a dense sequence ξ ∈ ∂X . Then
that gn± (ξm ) → ηm
m
gn → g in the compact-open topology for some g ∈ G .
Idea of the proof
Different
notions of
discreteness
If G is a Poincaré irregular group, then there exists ρ > 0 such
that the set
David
Simmons
Discreteness
Modified
Poincaré
exponent
Discreteness
and the
modified
Poincaré
exponent
Gρ = {g ∈ G : d(o, g (o)) ≤ ρ}
is infinite, but for all τ > 0, there exists a finite ρ-net for Gτ .
Suppose X is a CAT(-1) space, and let ∂X be the Gromov
boundary of X . For all ξ ∈ ∂X , Gρ (ξ) is precompact. Find a
sequence gn ∈ Gρ such that gn (ξ1 ) → η1 , gn (ξ2 ) → η2 for some
ξ1 , ξ2 ∈ ∂X distinct. If x ∈ [ξ1 , ξ2 ], then gn (x) → y for some
y ∈ [η1 , η2 ]. Thus G is not moderately discrete.
Suppose that X is a ROSSONCT and that G does not preserve
any totally geodesic subspace. Find a sequence gn ∈ Gρ such
± for a dense sequence ξ ∈ ∂X . Then
that gn± (ξm ) → ηm
m
gn → g in the compact-open topology for some g ∈ G . Thus G
is not COT-parametrically discrete.
Open question
Different
notions of
discreteness
David
Simmons
Let H∞ denote infinite-dimensional hyperbolic space, and let P
denote the stabilizer of a distinguished point of ∂H∞ in
Isom(H∞ ). Note that P ≡ Isom(`2 ).
Discreteness
Theorem
Modified
Poincaré
exponent
If G ≤ P is isomorphic to the integer Heisenberg group, then
δ(G ) ≥ 2. All exponents 2 < δ < ∞ are possible.
Discreteness
and the
modified
Poincaré
exponent
Question
Are the exponents δ = 2 and/or δ = ∞ possible?
Open question
Different
notions of
discreteness
David
Simmons
Let H∞ denote infinite-dimensional hyperbolic space, and let P
denote the stabilizer of a distinguished point of ∂H∞ in
Isom(H∞ ). Note that P ≡ Isom(`2 ).
Discreteness
Theorem
Modified
Poincaré
exponent
If G ≤ P is isomorphic to the integer Heisenberg group, then
δ(G ) ≥ 2. All exponents 2 < δ < ∞ are possible.
Discreteness
and the
modified
Poincaré
exponent
Question
Are the exponents δ = 2 and/or δ = ∞ possible?
Partial negative result:
Theorem (Pauls ’01)
There is no quasi-isometric embedding from the integer
Heisenberg group to `2 .