Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume
In this paper, we give a systolic inequality for a quotient space of a Carnot group Γ∖ with Popp's volume. Namely, we show the existence of a positive constant C such that the systole of Γ∖ is less than Cvol(Γ∖ )¹ᐟQ, where Q is the Hausdorff dimension. Moreover, the constant depends only on t...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2022 |
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| Цитувати: | Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume. Kenshiro Tashiro. SIGMA 18 (2022), 058, 16 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859811435145592832 |
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| author | Tashiro, Kenshiro |
| author_facet | Tashiro, Kenshiro |
| citation_txt | Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume. Kenshiro Tashiro. SIGMA 18 (2022), 058, 16 pages |
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| description | In this paper, we give a systolic inequality for a quotient space of a Carnot group Γ∖ with Popp's volume. Namely, we show the existence of a positive constant C such that the systole of Γ∖ is less than Cvol(Γ∖ )¹ᐟQ, where Q is the Hausdorff dimension. Moreover, the constant depends only on the dimension of the grading of the Lie algebra = ⨁ ᵢ. To prove this fact, the scalar product on introduced in the definition of Popp's volume plays a key role.
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| first_indexed | 2026-03-16T09:46:39Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 058, 16 pages
Systolic Inequalities for Compact Quotients
of Carnot Groups with Popp’s Volume
Kenshiro TASHIRO
Department of Mathematics, Tohoku University, Sendai Miyagi 980-8578, Japan
E-mail: kenshiro.tashiro.b2@tohoku.ac.jp
URL: https://sites.google.com/view/kenshiro-tashiro-homepage/home
Received February 10, 2022, in final form July 28, 2022; Published online August 02, 2022
https://doi.org/10.3842/SIGMA.2022.058
Abstract. In this paper, we give a systolic inequality for a quotient space of a Carnot
group Γ\G with Popp’s volume. Namely we show the existence of a positive constant C
such that the systole of Γ\G is less than Cvol(Γ\G)
1
Q , where Q is the Hausdorff dimension.
Moreover, the constant depends only on the dimension of the grading of the Lie algebra
g =
⊕
Vi. To prove this fact, the scalar product on G introduced in the definition of Popp’s
volume plays a key role.
Key words: sub-Riemannian geometry; Carnot groups; Popp’s volume; systole
2020 Mathematics Subject Classification: 53C17; 26B15; 22E25
1 Introduction
A sub-Riemannian manifold is a triple (M,E, g), where M is a smooth manifold, E is a sub-
bundle of the tangent bundle TM , and g is a metric on E. If E = TM , then it is an ordinary
Riemannian manifold. Geometry and analysis on sub-Riemannian manifolds have been actively
studied in relation to differential operator theory, geometric group theory, control theory and
optimal transport theory. In this paper, we study systolic inequalities on compact quotient
spaces of Carnot groups.
We recall systolic inequalities on Riemannian manifolds. On a Riemannian manifold (M, g),
the systole sys(M, g) is defined by
sys(M, g) = inf{l(c) | c : non-contractible closed curve},
where l(c) denotes the length of c. IfM is closed, then the minimum exists. A systolic inequality
asserts that for a large class of closed Riemannian manifolds, there is a constant C > 0 such
that
sys(M, g) ≤ C · vol(M, g)
1
d ,
where d = dimM and vol(M, g) =
∫
M dµg is the total volume with respect to the natural volume
form dµg. A constant C may depend on the topological type such as the dimension or the genus
of the surface.
As an example, let us consider a flat torus Γ\Ed, where Γ is a lattice in the d-dimensional
Euclidean space Ed. Then its systole sys
(
Γ\Ed
)
is equal to the minimum of the length of
a straight segment that connects two points in the lattice Γ, and the total volume is equal to
the volume of the fundamental domain of Γ.
The systolic inequality on flat tori is obtained in the following procedure, for example see [7,
Section 1]. Let p : Ed → Γ\Ed be the covering map. Trivially there is a unique positive number
R0 > 0 such that the volume of the R0-ball of Ed centered at 0, say B(R0), is equal to the
mailto:kenshiro.tashiro.b2@tohoku.ac.jp
https://sites.google.com/view/kenshiro-tashiro-homepage/home
https://doi.org/10.3842/SIGMA.2022.058
2 K. Tashiro
total volume of the flat torus Γ\Ed, i.e., vol(B(R0)) = vol
(
Γ\Ed
)
. Then the restriction of the
covering map p to the R0-ball is not injective. This implies that there is a non-contractible
closed curve in Γ\Ed such that its length is less than or equal to 2R0. Hence the systole of the
flat manifold Γ\Ed is bounded above by
sys
(
Γ\Ed
)
≤ 2R0 = 2ω
− 1
d
d vol
(
Γ\Ed
) 1
d ,
where ωd is the volume of the unit ball of the Euclidean space Ed.
Such systolic inequalities are proved for surfaces such as non-flat 2-dimensional torus (its
proof is not in the literature, but mentioned in [12]), projective space [12], and higher genus
ones [4, 6, 9]. For d-dimensional Riemannian manifolds with d > 2, Gromov showed the existence
of the systolic constant Cd for so called essential manifolds [6].
We give systolic inequalities for compact quotient spaces of Carnot groups. Let (G,V1, ⟨·, ·⟩1)
be a Carnot group, and g =
⊕k
i=1 Vi its grading of the Lie algebra (see Definition 2.1). We call
a discrete subgroup Γ of a simply connected nilpotent Lie group G a lattice if it is cocompact
and discrete.
Since Γ acts isometrically on G from the left, we can define the quotient sub-Riemannian
metric on Γ\G. The systole on the quotient space Γ\G is defined from this quotient metric.
We denote by vol(Γ\G) the integral of Popp’s volume form (Definition 2.3). Popp’s volume
is the left-invariant volume form defined by the scalar product ⟨·, ·⟩g on the Lie algebra g.
We describe its construction and properties in Section 2.
Set di = dimVi and Q =
∑k
i=1 idi. Notice that Q is equal to the Hausdorff dimension of the
Carnot group with respect to any left-invariant homogeneous geodesic distance on it [10] (its
corrected proof can be found in [5]).
As we saw in flat tori case, the lower bound estimate of the volume of the unit ball can
be expected to show the desired systolic inequality. The volume of the unit ball of a Carnot
group is more complicated than the Euclidean one. In [8], Hassannezhad–Kokarev estimated
the volume of the unit ball of corank 1 Carnot groups, and gave an upper bound of the volume
which depends on the dimension. It seems hard to apply their method to general Carnot groups.
We indirectly compute a lower bound of the volume to obtain the following proposition, which
implies the main theorem.
Main Proposition. There exists a constant D = D(d1, . . . , dk) > 0, independent of the choice
of a scalar product ⟨·, ·⟩1, such that for any k-step Carnot group (G,V1, ⟨·, ·⟩1) with dimVi = di,
the volume of the unit ball vol(Bcc(1)) is greater than D.
The main proposition shall be restated as Theorem 3.2 in the body text.
Let ⟨·, ·⟩i be the scalar product on Vi which will be defined in the construction of Popp’s
volume, Section 2. This scalar product on a Carnot group plays a key role to prove the main
proposition. Roughly speaking, the main idea is to construct the di-dimensional balls Bdi(ϵi) ⊂
(Vi, ⟨·, ·⟩i), i = 1, . . . , k, of radius ϵi centered at 0 such that
1) the numbers ϵ1, . . . , ϵk depend only on d1, . . . , dk,
2) their product Bd1(ϵ1)× · · · ×Bdk(ϵk) is contained in the unit ball Bcc(1) via the identifi-
cation exp: g ≃ G.
Remark 1.1. In fact, the radius ϵ1, . . . , ϵk will be chosen so that they depend only on d1
and k, while the volume of the balls Bd1(ϵ1), . . . , B
dk(ϵk) need the information of the dimensions
d1, . . . , dk.
The Ball–Box theorem asserts a similar statement. Let M be a sub-Riemannian manifold
andBsR(x,R) theR-ball centered at x. Fix a relatively compact chart (U, (x1, . . . , xn)) around x.
Systolic Inequalities for Compact Quotients of Carnot Groups with Popp’s Volume 3
The Ball–Box theorem claims the existence of positive constants c and C, such that for all R > 0
with BsR(x,R) ⊂ U ,
Boxd1(cR)× · · · × Boxdk
(
cRk
)
⊂ BsR(x,R) ⊂ Boxd1(CR)× · · · × Boxdk
(
CRk
)
,
where Boxdi(r) is the di-dimensional box of size r centered at x in the chart, and d1, . . . , dk
are determined by the nilpotentization at x. The constants c and C depend on the choice of
a chart. Thus it is a priori impossible to control the constants c and C by topological/geometric
invariants of M .
The second claim just before Remark 1.1 asserts that if we equip a Carnot group G with the
global coordinates induced from the scalar products ⟨·, ·⟩i’s on
⊕
Vi = g ≃ G, then the unit ball
contains the direct product of balls with their radius controlled by d1, k. Therefore the second
claim can be regarded as a quantitative version of (a part of) the Ball–Box theorem.
The main proposition implies the main theorem of this paper.
Theorem 1.2. There is a positive constant C = C(d1, . . . , dk), independent of the choice of
a scalar product ⟨·, ·⟩1, such that for any Carnot group (G,V1, ⟨·, ·⟩1) with the grading dimVi = di,
and any lattice Γ < G,
sys(Γ\G) ≤ C · vol(Γ\G)
1
Q .
Proof. Let Bcc(R) be the ball in (G,V1, ⟨·, ·⟩1) of radius R centered at the identity element e.
It is well known that the volume of the R-ball satisfies
vol(Bcc(R)) = vol(Bcc(1))R
Q
under a Haar volume, see (2.1). On the other hand, there is a positive number R0 > 0 such
that vol(Bcc(R0)) = vol(Γ\G). It implies that the systole of Γ\G is less than or equal to 2R0.
Combined with the main proposition,
sys(Γ\G) ≤ 2R0 = 2vol(Bcc(1))
− 1
Q vol(Γ\G)
1
Q ≤ 2D
− 1
Q vol(Γ\G)
1
Q . ■
An example of a Carnot group is the Euclidean space Ed. Thus Theorem 1.2 is a generalization
of the systolic inequality for flat tori.
Remark 1.3. We can write a constant C by using the Hausdorff dimension Q since the numbers
d1, . . . , dk can be controlled by Q.
2 Carnot groups
Let G be a simply connected nilpotent Lie group, and g its Lie algebra. The nilpotent Lie
algebra g is said to be graded if g has a direct sum decomposition g =
⊕k
i=1 Vi such that
[Vi, Vj ] ⊂ Vi+j . We will identify the Lie algebra g to the tangent space TeG, and call a left-
invariant vector field v ∈ g a horizontal vector if it is in V1.
Let ⟨·, ·⟩1 be an scalar product on V1. Then we can define a sub-Riemannian structure on G as
follows. For all g ∈ G, we define a fiber-wise scalar product ⟨·, ·⟩g on Lg∗V1 by ⟨Lg∗v1, Lg∗v2⟩g =
⟨v1, v2⟩ for any couple of horizontal vectors v1, v2 ∈ V1. Then the sub-bundle E =
⊔
g∈G Lg∗V1
and the inner metric ⟨·, ·⟩g defines a left-invariant sub-Riemannian metric.
Definition 2.1 (Carnot group). A triple (G,V1, ⟨·, ·⟩) is called a Carnot group.
4 K. Tashiro
We say that an absolutely continuous curve c : [a, b] → G is horizontal if the derivative ċ(t)
is in Lc(t)∗V1 for a.e. t ∈ [a, b]. The length of a horizontal curve is given by the integral l(c) =∫ b
a ∥ċ(t)∥c(t)dt, and the distance of two points in the sub-Riemannian manifold (G,V1, ⟨·, ·⟩) is
given by the infimum of the length of horizontal curves joining these points. We call it the
Carnot–Carathéodory distance and denote by dcc. We also denote by Bcc(R) the ball in G of
radius R > 0 centered at the identity element e.
Dilation. For a positive number t > 0 and Xi ∈ Vi, i = 1, . . . , k, define the Lie algebra
isomorphism δt : g → g by
δt
(
k∑
j=1
Xj
)
=
k∑
j=1
tjXj .
This isomorphism δt is called the dilation.
Let exp: g → G be the exponential map. Since the groupG is simply connected and nilpotent,
the exponential map is a diffeomorphism. So we will identify group elements in G to vectors in g
via the exponential map. Moreover we can regard the dilation δt as the Lie group automorphism.
The origin of the name dilation comes from the following property. Let c be a length min-
imizing curve joining two points x, y ∈ G. Since the derivative of the curve c is in the left
translation of V1 a.e., the length of the curve δt ◦ c is equal to t · length(c). It implies that
dcc(δt(x), δt(y)) = tdcc(x, y). Moreover, it also implies that the volume of the ball of radius R
satisfies
vol(Bcc(R)) = vol(Bcc(1))R
Q, (2.1)
with respect to a Haar volume.
Popp’s volume. Popp’s volume is introduced by Montgomery in [11] as a generalization of
a Riemannian canonical volume form dµg =
√
det(g) dx1 ∧ · · · ∧dxd. The qualitative properties
of Popp’s volume has been actively studied. By the result of Barilari–Rizzi [3], Popp’s volume is
characterized as an (local) isometric invariant volume on equiregular sub-Riemannian manifolds.
Agrachev–Barilari–Boscain exhibited a condition for Popp’s volume being a scalar multiple of
the spherical Hausdorff volume in [1]. The qualitative study of Popp’s volume on a Carnot
group is not interesting since it is a Haar volume. Our interest is in the quantitative properties
of Popp’s volume.
Popp’s volume is constructed as follows. For i ≥ 2, we inductively define the linear map
ϕi : V
⊗i
1 → Vi by
ϕ2(X1 ⊗X2) = [X1, X2],
ϕi(X1 ⊗ · · · ⊗Xi) = [X1, ϕi−1(X2 ⊗ · · · ⊗Xi)], i ≥ 3.
Clearly the linear map ϕi is surjective. We will shortly write
[X1, . . . , Xi] := ϕi(X1 ⊗ · · · ⊗Xi).
Recall that on the tensor product space V ⊗i
1 , we can define the canonical scalar product by
⟨u1 ⊗ · · · ⊗ ui, v1 ⊗ · · · ⊗ vi⟩⊗i =
i∏
j=1
⟨uj , vj⟩1.
From the scalar product space
(
V ⊗i
1 , ⟨·, ·⟩⊗i
)
, we set an scalar product on Vi by using the following
lemma.
Systolic Inequalities for Compact Quotients of Carnot Groups with Popp’s Volume 5
Lemma 2.2 ([2, Lemma 20.3]). Let E be an scalar product space, V a vector space, and f :
E → V a surjective linear map. Then f induces an scalar product on V such that the norm of
v ∈ V is
∥v∥V = min{∥u∥E | f(u) = v}.
Applying the above lemma to the surjective linear map ϕi, we obtain the scalar product ⟨·, ·⟩i
on each Vi for i = 2, . . . , k, and on their direct sum g =
⊕
Vi. We denote by ⟨·, ·⟩g the induced
scalar product on g.
Definition 2.3. Popp’s volume on a Carnot group (G,V1, ⟨·, ·⟩1) is the left-invariant volume
form induced from the scalar product ⟨·, ·⟩g on g ≃ TeG.
The Baker–Campbell–Hausdorff formula. Roughly speaking, the Baker–Campbell–
Hausdorff formula (BCH formula) describes the solution for Z to the equation Z = log(exp(X1)×
exp(X2)) = X1 · X2. It links the Lie group product on G and the Lie bracket on g by using
combinatorial coefficients. Namely, for each positive integer p ∈ N and multi-index (i1, . . . , ip) ∈
{1, 2}p, there is a constant c(i1,...,ip) such that
X1 ·X2 = X1 +X2 +
1
2
[X1, X2] +
∑
p≥3
∑
(i1,...,ip)∈{1,2}p
c(i1,...,ip)[Xi1 , . . . , Xip ].
In this paper, we will use the following formulae which are obtained by applying the BCH
formula several times. Set Ipj = {1, . . . , j}p for positive integers j, p ∈ N:
� For each (i1, . . . , ip)∈Ip2k, there is a constant α(i1,...,ip) such that for any vectorsX1, . . . , X2k
with Xi, Xi+k ∈ Vi, i = 1, . . . , k,(
k∑
i=1
Xi
)(
k∑
i=1
Xi+k
)
=
2k∑
i=1
Xi +
∑
p≥2
∑
(i1,...,ip)∈Ip2k
α(i1,...,ip)[Xi1 , . . . , Xip ]. (2.2)
� For each (n1, . . . , np) ∈ IpN , there is a constant β(n1,...,np) such that for any vectors X1, . . . ,
XN ∈ g,
X1 · · ·XN =
N∑
n=1
Xn +
∑
p≥2
∑
(n1,...,np)∈IpN
β(n1,...,np)[Xn1 , . . . , Xnp ]. (2.3)
� For each (i1, . . . , ip) ∈ Ipj , there is a constant γ(i1,...,ip) such that for any vectors X1, . . . ,
Xj ∈ g,
[X1, . . . , Xj ]c = [X1, . . . , Xj ] +
∑
p≥j+1
∑
(i1,...,ip)∈Ipj
γ(i1,...,ip)[Xi1 , . . . , Xip ]. (2.4)
Here we write the commutator of the group by [x, y]c = xyx−1y−1, and define the map ψn:
Gn → G by
ψ2(x1, x2) = [x1, x2]c,
ψn(x1, . . . , xn) = [x1, ψn−1(x2, . . . , xn)]c, i ≥ 3.
We shortly wrote
[x1, . . . , xn]c := ψn(x1, . . . , xn).
Equation (2.4) is proved by induction on j.
6 K. Tashiro
Remark 2.4.
� The constants α(i1,...,ip), β(n1,...,np) and γ(i1,...,ip) depend only on the choice of indices.
� The commutator [·, ·]c acts on a group G, while the Lie bracket [·, ·] acts on its Lie algebra g.
A set of horizontal vectors. Let {Xni}n=1,...,N,i=1,...,j be a set of (multi-indexed) horizontal
vectors, and X = {{Xni}n=1,...,N,i=1,...,j | N, j ∈ N} a family of those sets. We always assume
j ≤ k for k-step Carnot groups. Sometimes we simply write {Xni}. Throughout the paper, a set
of horizontal vectors plays a central role.
From a given set of horizontal vectors, we introduce the following three notions.
Definition 2.5. Let {Xni}n=1,...,N, i=1,...,j be a set of horizontal vectors.
1. Define the group element y({Xni}) in G by
y({Xni}) =
N∏
n=1
[Xn1, . . . , Xnj ]c.
2. Define the vector Y ({Xni}) in g by
Y ({Xni}) =
N∑
n=1
[Xn1, . . . , Xnj ].
3. Define the function dcom : X → R by
dcom({Xni}) =
N∑
n=1
j∑
i=1
∥Xni∥1.
We call it the combinatorial distance function.
Remark 2.6. For a given set of horizontal vectors {Xni}, the group element y({Xni}) coincides
with the vector Y ({Xni}) via the exponential map if G is abelian or 2-step.
We denote by dcc(X) the Carnot–Carathéodory distance from the identity element e to
a group element X ∈ g ≃ G. The function dcom gives an upper bound of the Carnot–
Carathéodory distance dcc in the following sense.
Lemma 2.7. If a Carnot group G is k-step, then for any set of horizontal vectors {Xni}, we
have
dcc(y({Xni})) ≤ 2k−1dcom({Xni}).
Proof. By the triangle inequality and the left-invariance of the Carnot–Carathéodory distance,
we have
dcc(y({Xni}n=1,...,N,i=1,...,j)) = dcc
(
N∏
n=1
[Xn1, . . . , Xnj ]c
)
≤
N∑
n=1
dcc([Xn1, . . . , Xnj ]c). (2.5)
Systolic Inequalities for Compact Quotients of Carnot Groups with Popp’s Volume 7
To compute [Xn1, . . . , Xnj ]c, we observe the case j = 3. By the definition,
[Xn1, Xn2, Xn3]c = [Xn1, [Xn2, Xn3]c]c
= Xn1 · [Xn2, Xn3]c ·X−1
n1 · [Xn2,Xn3 ]
−1
c
= Xn1 ·
(
Xn2 ·Xn3 ·X−1
n2 ·X−1
n3
)
·X−1
n1 ·
(
Xn3 ·Xn2 ·X−1
n3 ·X−1
n2
)
.
The final term is the product of two X±1
n1 , four X±1
n2 and four X±1
n3 . In the same way, the
group element [Xn1, . . . Xnj ] is the product of X±1
n1 , . . . , X
±1
nj . For i = 1, . . . , j − 1, the group
element X±1
ni appears 2i times in this product. For i = j, the group element Xnj appears 2j−1
times in this product. Therefore the triangle inequality shows that
dcc([Xn1, . . . , Xnj ]c) ≤
j−1∑
i=1
2idcc(Xni) + 2j−1dcc(Xnj). (2.6)
Since vectors Xni are horizontal, we have dcc(Xni) ≤ ∥Xni∥1. Moreover, since i ≤ j ≤ k, we
have
dcc([Xn1, . . . , Xnj ]c) ≤
N∑
n=1
[
j−1∑
i=1
2idcc(Xni) + 2j−1dcc(Xnj)
]
≤
N∑
n=1
[
j−1∑
i=1
2i∥Xni∥1 + 2j−1∥Xnj∥1
]
≤ 2k−1
N∑
n=1
j∑
i=1
∥Xni∥1
= 2k−1dcom({Xni}). ■
Remark 2.8. It is difficult to replace y({Xi}) with Y ({Xi}) in Lemma 2.7. Indeed, we use
the triangle inequality to prove the inequality (2.5) and (2.6). The triangle inequality is applied
to the product a sequence of group elements Xni. However, the triangle inequality cannot be
applied to the Lie algebraic sum. This is why we introduce the group element y({Xni}).
3 Proof of the main theorem
3.1 2-step case
We start from 2-step Carnot groups. Recall that here g ≃ G via the exponential map, and each
layer Vi ⊂ g is equipped with the scalar product ⟨·, ·⟩i defined in the definition of Popp’s volume.
Denote by Bdi(R) the ball centered at 0 of radius R of the inner metric space (Vi, ⟨·, ·⟩i). Via the
identification g ≃ G, we identify the direct product of balls Bd1(R1) × Bd2(R2) ⊂ V1 ⊕ V2 = g
to a subset in G.
Theorem 3.1. There exists positive constants ϵ1 and ϵ2, which depend only on d1, such that
for any 2-step Carnot group (G,V1, ⟨·, ·⟩1) with di = dimVi,
Bd1(ϵ1)×Bd2(ϵ2) ⊂ Bcc(1).
In particular, the volume of the unit ball is greater than ϵd11 ϵ
d2
2 ωd1ωd2.
Proof. Let Z2 be a given vector in V2, and put ν = ∥Z2∥2.
8 K. Tashiro
Consider {X1, . . . , Xd1} an orthonormal (for ⟨·, ·⟩1) basis of V1. Choose u ∈ ϕ−1
2 (Z2) ⊂ V1⊗V1
so that ν = ∥Z2∥2 = ∥u∥⊗2, where ∥·∥⊗2 is the norm on V1⊗V1 induced by ⟨·, ·⟩⊗2. By definition
ν = ∥Z2∥2 = ∥u∥⊗2, with
u =
d1∑
s,t=1
αstXs ⊗Xt =
d1∑
s,t=1
(
α1
stXs
)
⊗
(
α2
stXt
)
=:
d21∑
n=1
Xn1 ⊗Xn2,
where α1
st and α2
st are chosen in such a way that
∥∥α1
stXs
∥∥
1
=
∥∥α2
stXt
∥∥
1
, and where we rename
the multi-index (s1, s2) ∈ {1, . . . , d1}2 the index n ∈
{
1, . . . , d21
}
. In this way, we obtain a set of
horizontal vectors {Xni}n=1,...,d21,i=1,2 such that the following three properties hold:
Z2 =
d21∑
n=1
[Xn1, Xn2], (3.1)
ν =
√√√√√ d21∑
n,m=1
⟨Xn1, Xm1⟩1⟨Xn2, Xm2⟩1 =
√√√√ d21∑
n=1
∥Xn1∥21∥Xn2∥21, (3.2)
∥Xn1∥1 = ∥Xn2∥1 for all n = 1, . . . , d21. (3.3)
Indeed, equation (3.1) holds since u∈ϕ−1
2 (Z2). Since {X1, . . . , Xd1} is orthonormal in (V1, ⟨·, ·⟩1),
{Xn1 ⊗ Xn2}n=1,...,d21
is an orthogonal subset in (V1 ⊗ V1, ⟨·, ·⟩⊗2), which implies (3.2). Equa-
tion (3.3) holds by the choice of α1
st and α
2
st.
By equation (3.3),
dcom({Xni}) =
d21∑
n=1
∥Xn1∥1 + ∥Xn2∥1 = 2
d21∑
n=1
∥Xn1∥1,
ν =
√√√√ d21∑
n=1
∥Xn1∥21∥Xn2∥21 =
√√√√ d21∑
n=1
∥Xn1∥41.
Combined with the above two equalities and Hölder’s inequality, we obtain an upper bound of
dcom({Xni}) by
dcom({Xni}) = 2
d21∑
n=1
∥Xn1∥1 ≤ 2d
3
2
1
4
√√√√ d21∑
n=1
∥Xn1∥41 = 2d
3
2
1
√
ν. (3.4)
Set ϵ1 =
1
2 and ϵ2 =
1
64d31
. We show that if a vector Z = Z1+Z2 (Zi ∈ Vi) satisfies ∥Zi∥i ≤ ϵi,
then dcc(Z) ≤ 1.
Note that Z1 · Z2 = Z1 + Z2 since Z2 is in the center of G. As we see in Remark 2.6,
Z2 = Y ({Xni}) = y ({Xni}). Hence by Lemma 2.7 and the inequality (3.4),
dcc(Z) = dcc(Z1 · Z2) = dcc(Z1 · Z2)
≤ dcc(Z1) + dcc(Z2) ≤ ∥Z1∥1 + 2dcom({Xni}) ≤
1
2
+
1
2
= 1.
By using the volume of the unit ball in the d-dimensional Euclidean space ωd, we have
vol(Bcc(1)) ≥ vol
(
Bd1(ϵ1)×Bd2(ϵ2)
)
=
1
2d1+8d2
1
d3d21
ωd1ωd2 . ■
Systolic Inequalities for Compact Quotients of Carnot Groups with Popp’s Volume 9
3.2 Higher step cases
For a k-step Carnot group, the following theorem implies the main Theorem 1.2.
Theorem 3.2. There exist positive constants ϵ1, . . . , ϵk, which depend only on d1 and k, such
that for any k-step Carnot group (G,V1, ⟨·, ·⟩1) with di = dimVi,
k∏
i=1
Bdi(ϵi) ⊂ Bcc(1).
In particular, the volume of the unit ball is greater than
∏k
i=1 ϵ
di
i ωdi.
Compared from the 2-step case, its difficulty is the non-coincidence of y({Xni}) and Y ({Xni}).
We will compute its difference by using the BCH formula.
Let Zj be a given vector in Vj , and put νj = ∥Zj∥j . Consider {X1, . . . , Xd1} an orthonormal
(for ⟨·, ·⟩1) basis of V1. Choose uj ∈ ϕ−1
j (Z2) ⊂ V ⊗j
1 so that νj = ∥Zj∥j = ∥uj∥⊗j , where ∥ · ∥⊗j
is the norm on V ⊗j
1 induced by ⟨·, ·, ⟩⊗j . By definition νj = ∥Zj∥j = ∥uj∥⊗j , with
uj =
d1∑
s1,...,sj=1
αs1...sjXs1 ⊗ · · · ⊗Xsj =
d1∑
s1,...,sj=1
(
α1
s1...sjXs1
)
⊗ · · · ⊗
(
αj
s1...sjXsj
)
=:
dj1∑
n=1
Xn1 ⊗ · · · ⊗Xnj ,
where αi
s1...sj , i = 1, . . . , j, are chosen in such a way that
∥∥α1
s1...sjXs1
∥∥
1
= · · · =
∥∥αj
s1...sjXsj
∥∥
1
,
and where we rename the multi-index (s1, . . . , sj) ∈ {1, . . . , d1}j the index n ∈
{
1, . . . , dj1
}
.
In this way, we obtain a set of horizontal vectors {Xni}n=1,...,dj1, i=1,...,j
such that the following
three properties holds:
Zj =
dj1∑
n=1
[Xn1, . . . , Xnj ], (3.5)
νj =
√√√√√ dj1∑
n,m=1
⟨Xn1, Xm1⟩1 · · · ⟨Xnj , Ynj⟩1 =
√√√√√ dj1∑
n=1
∥Xn1∥21 · · · ∥Xnj∥21, (3.6)
∥Xn1∥1 = · · · = ∥Xnj∥1 for all n = 1, . . . , dj1. (3.7)
Definition 3.3. We say that a set of horizontal vectors {Xni} is adjusted to Zj if the three
conditions (3.5), (3.6) and (3.7) hold.
Let us consider the difference between the group element y({Xni}) and the vector Y ({Xni}) of
a set of horizontal vectors adjusted to Zj . Define the map Pl : g → Vl to be the linear projection.
By the equation (2.4),
Pl(y({Xni})) =
0, l ≤ j − 1,
Zj = Y ({Xni}), l = j,
a possibly non-zero vector, l ≥ j + 1.
We label a possibly non-zero vector Pl(y({Xni})), l ≥ j + 1, as follows.
10 K. Tashiro
Definition 3.4. Denote by Al({Xni}) the image of y({Xni}) by Pl. We call the vector Al({Xni})
the l-error vector of {Xni}.
Sometimes we simply write Al. We will give an upper bound of ∥Al∥l later in Lemma 3.6.
Let us start from the preparation.
Lemma 3.5. For any two vectors Zp ∈ Vp and Zq ∈ Vq, we have
∥[Zp, Zq]∥p+q ≤ 2p∧q∥Zp∥p∥Zq∥q,
where p ∧ q = min{p, q}.
Proof. By the skew-symmetry of the Lie bracket, we can assume p ≤ q. Let
{
X
(p)
ni
}(
resp.
{
X
(q)
mj
})
be a set of horizontal vectors adjusted to Zp (resp. Zq). From the bi-linearity of
the Lie bracket and the subadditivity of the norm ∥ · ∥p+q, we have
∥[Zp, Zq]∥p+q =
∥∥∥∥[ dp1∑
n=1
[
X
(p)
n1 , . . . , X
(p)
np
]
,
dq1∑
m=1
[
X
(q)
m1, . . . , X
(q)
mq
]]∥∥∥∥
p+q
≤
dp1∑
n=1
dq1∑
m=1
∥∥[[X(p)
n1 , . . . , X
(p)
np
]
,
[
X
(q)
m1, . . . , X
(q)
mq
]]∥∥
p+q
. (3.8)
By applying the Jacobi identity [[X,Y ], Z] = [X, [Y, Z]]−[Y, [X,Z]] several times, we can rewrite[[
X
(p)
n1 , . . . , X
(p)
np
]
,
[
X
(q)
m1, . . . , X
(q)
mq
]]
=
∑
σ∈S
ϵσ
[
X
(p)
nσ(1),
[
X
(p)
nσ(2), . . . ,
[
X
(p)
nσ(p),
[
X
(q)
m1, . . . , X
(q)
mq
]]
, . . .
]]
,
where S is the subset of the symmetric group of degree p consisting of σ such that if σ(a) = p,
then
σ(1) < σ(2) < · · · < σ(a) > σ(a+ 1) > · · · > σ(p),
and ϵσ ∈ {±1}. Since the size of the subset S is 2p, we can compute an upper bound of (3.8) by
dp1∑
n=1
dq1∑
m=1
∥∥∥∥∑
σ∈S
ϵσ
[
X
(p)
nσ(1),
[
X
(p)
nσ(2), . . . ,
[
X
(p)
nσ(p),
[
X
(q)
m1, . . . , X
(q)
mq
]]
, . . .
]]∥∥∥∥
p+q
≤
dp1∑
n=1
dq1∑
m=1
2p
p∏
i=1
∥∥X(p)
ni
∥∥
1
q∏
j=1
∥∥X(q)
mj
∥∥
1
= 2p
( dp1∑
n=1
p∏
i=1
∥∥X(p)
ni
∥∥
1
)( dq1∑
m=1
q∏
j=1
∥∥X(q)
mj
∥∥
1
)
≤ 2p∥Zp∥p∥Zq∥q. ■
Next we will control an upper bound of the ∥Al({Xni})∥l.
Lemma 3.6. For j = 1, . . . , k, there is a positive constant θj = θj(d1, j, k) such that for any
vector Zj ∈ Vj, νj = ∥Zj∥j, and any set of horizontal vectors {Xni} adjusted to Zj,
∥Al({Xni})∥l ≤ θjν
l
j
j .
Systolic Inequalities for Compact Quotients of Carnot Groups with Popp’s Volume 11
Proof. For each n = 1, . . . , dj1, let Un = y({Xni}i=1,...,j). By the equation (2.4),
Un = [Xn1, . . . , Xnj ] +
∑
m≥j+1
∑
(i1,...,im)∈Imj
γ(i1,...,im)[Xni1 , . . . , Xnim ]. (3.9)
By using the equation (2.3), the product of the group elements Un is written by
dj1∏
n=1
Un = Zj +
∑
q≥2
∑
(n1,...,nq)∈Iq
d
j
1
β(n1,...,nq)[Un1 , . . . , Unq ].
Notice that the group element
∏dj1
n=1 Un coincides with y({Xni}) = Zj +
∑k
l=j+1Al from its
definition. Thus we can explicitly write Al by
Al =
∑
q≥2
∑
(n1,...,nq)∈Iq
d
j
1
∑
m1+···+mq=l
β(n1,...,nq)[Pm1(Un1), . . . , Pmq(Unq)].
From (3.9), we can compute ∥Pm(Un)∥m by
∥Pm(Un)∥m ≤
0, m = 1, . . . , j − 1,
∥Xn1∥j1, m = j,∑
(i1,...,im)∈Imj
|γ(i1,...,im)|∥Xn1∥m1 , m = j + 1, . . . , k.
In any case, ∥Pm(Un)∥m is less than or equal to γ̃ν
m
j
j , where γ̃=max
{
1,
∑
(i1,...,im)∈Imj
|γ(i1,...,im)|
}
.
By the subadditivity of the norm ∥ · ∥m and Lemma 3.5, we have
∥Al∥l ≤
∑
q≥2
∑
(n1,...,nq)∈Iq
d
j
1
∑
m1+···+mq=l
2l|β(n1,...,nq)|∥Pm1(Un1)∥m1 · · · ∥Pmq(Unq)∥mq
≤
∑
q≥2
∑
(n1,...,nq)∈Iq
d
j
1
∑
m1+···+mq=l
2l|β(n1,...,nq)|
(
γ̃ν
m1
j
j
)
· · ·
(
γ̃ν
mq
j
j
)
=
∑
q≥2
∑
(n1,...,nq)∈Iq
d
j
1
∑
m1+···+mq=l
2l|β(n1,...,nq)|γ̃
qν
l
j
j
≤
∑
q≥2
∑
(n1,...,nq)∈Iq
d
j
1
(
l + q − 1
q − 1
)
2l|β(n1,...,nq)|γ̃
qν
l
j
j
≤
∑
q≥2
∑
(n1,...,nq)∈Iq
d
j
1
(
l + q − 1
q − 1
)
2lβ̃γ̃qν
l
j
j
≤
∑
q≥2
djq1
(
l + q − 1
q − 1
)
2lβ̃γ̃qν
l
j
j ≤
∑
q≥2
djk1
(
l + q − 1
q − 1
)
2lβ̃γ̃kν
l
j
j
≤ djk1 2l+k−12lβ̃γ̃kν
l
j
j ≤ 8kdjk1 β̃γ̃
kν
l
j
j ,
where β̃ = max
{
|β(n1,...,nq)| | (n1, . . . , nq) ∈ Iq
dj1
, q = 2, . . . , k
}
.
Notice that the constants β̃ and γ̃ depend only on the constants β(n1,...,nq) and γ(i1,...,im) in
the BCH formulas (2.3) and (2.4). Since m and q are not greater than k, they ultimately depend
on d1, j and k. Hence we have obtained a desired constant θj by letting θj = 8kdjk1 β̃γ̃
k. ■
12 K. Tashiro
The norm ∥ · ∥j controls the combinatorial distance dcom as follows.
Lemma 3.7. For any Zj ∈ Vj with νj = ∥Zj∥j and any set of horizontal vector {Xni} adjusted
to Zj, there is an upper bound of the combinatorial distance given by
dcom({Xni}) ≤ jd
2j−1
2
1 ν
1
j
j .
Proof. By the assumption (3.7) we obtain
dcom({Xni}) = j
dj1∑
n=1
∥Xn1∥1.
By Hölder’s inequality,
j
dj1∑
n=1
∥Xn1∥1 ≤ jd
2j−1
2
1
2j
√√√√√ dj1∑
n=1
∥Xn1∥2j1 .
By the equality (3.6),
jd
2j−1
2
1
2j
√√√√√ dj1∑
n=1
∥Xn1∥2j1 = jd
2j−1
2
1 ν
1
j
j .
The above three inequalities prove the lemma. ■
We have considered a set of vectors adjusted to a vector Zj in each layer Vj . We will introduce
a similar notion for a vector Z in the whole Lie algebra g.
Let Z =
∑k
j=1 Zj be a vector in g =
⊕k
j=1 Vj , Zj ∈ Vj . We will inductively define a set of
vectors
{
X
(j)
ni
}
n=1,...,dj1, i=1,...,j
for j = 1, . . . , k as follows. For Z1 ∈ V1, define a set of vectors{
X
(1)
ni
}
= {Z1, 0, . . . , 0}. Let
{
X
(2)
ni
}
be a set of horizontal vectors adjusted to Z2. Then for
a couple of sets of horizontal vectors
({
X
(1)
ni
}
,
{
X
(2)
ni
})
, there are vectors B
(2)
l
({
X
(1)
ni
}
,
{
X
(2)
ni
})
∈ Vl for l > 2 such that
y
({
X
(1)
ni
})
· y
({
X
(2)
ni
})
= Z1 + Z2 +
k∑
l=3
B
(2)
l
({
X
(1)
ni
}
,
{
X
(2)
ni
})
.
Next let
{
X
(3)
ni
}
be a set of horizontal vectors adjusted to Z3 − B
(2)
3 . Then for a triple of sets
of horizontal vectors
({
X
(1)
ni
}
,
{
X
(2)
ni
}
,
{
X
(3)
ni
})
, there are vectors B
(3)
l
({
X
(1)
ni
}
,
{
X
(2)
ni
}
,
{
X
(3)
ni
})
∈ Vl for l > 3 such that
3∏
j=1
y
({
X
(j)
ni
})
=
3∑
l=1
Zl +
k∑
l=4
B
(3)
l
({
X
(1)
ni
}
,
{
X
(2)
ni
}
,
{
X
(3)
ni
})
.
In this way, we can inductively define a set of horizontal vectors
{
X
(j)
ni
}
and error vectors
B
(j)
l
({
X
(1)
ni
}
, . . . ,
{
X
(j)
ni
})
. We will summarize this argument in the following definition.
Definition 3.8. For a vector
∑k
j=1 Zj ∈
⊕
Vj , sets of horizontal vectors
{
X
(1)
ni
}
, . . . ,
{
X
(k)
ni
}
and vectors B
(j)
l
({
X
(1)
ni
}
, . . . ,
{
X
(j)
ni
})
∈ Vl, j = 1, . . . , k, are inductively defined by
Systolic Inequalities for Compact Quotients of Carnot Groups with Popp’s Volume 13
(X1)
{
X
(1)
ni
}
= {Z1, 0, . . . , 0},
(B1) B
(1)
l
({
X
(1)
ni
})
= 0 for l = 1, . . . , k,
(Xj)
{
X
(j+1)
ni
}
is a set of horizontal vectors adjusted to Zj+1 −B
(j)
j+1
({
X
(1)
ni
}
, . . . ,
{
X
(j)
ni
})
,
(Bj) B
(j)
l
({
X
(1)
ni
}
, . . . ,
{
X
(j)
ni
})
= Pl
(∏j
m=1 y
({
X
(m)
ni
}))
for l = j + 1, . . . , k.
We call
({
X
(1)
ni
}
, . . . ,
{
X
(k)
ni
})
a k-tuple of sets of horizontal vectors adjusted to Z, and call
B
(j)
l
({
X
(1)
ni
}
, . . . ,
{
X
(j)
ni
})
an (l, j)-error vector.
We will simply write B
(j)
l . Since g is k-step, the error vectors B
(k)
l vanishes and
Z =
k∑
j=1
Zj =
k∏
j=1
y
({
X
(j)
ni
})
.
Remark 3.9. Since the choice of a set of horizontal vectors adjusted to Zj ∈ Vj is not unique,
the choice of error vectors is not unique too.
The norm of error vectors
∥∥B(j)
l
∥∥
l
can be controlled as follows.
Lemma 3.10. For Z =
∑k
j=1 Zj ∈ g with νj = ∥Zj∥j, let
({
X
(1)
ni
}
, . . . ,
{
X
(k)
ni
})
be a k-tuple
of sets of horizontal vectors adjusted to Z. For j = 1, . . . , k − 1 and l = j + 1, . . . , k, there are
polynomials Qlj(β1, β2, . . . , βk) such that
Qlj(0, . . . , 0) = 0 and
∥∥B(j)
l
∥∥
l
≤ Qlj
(
ν1,
√
ν2, . . . , k
√
νk
)
.
Moreover, their coefficients depend only on the dimensions d1 and k.
Proof. We prove the assertion by inductions on j. When j = 1, then B
(1)
l = 0 for all l =
2, . . . , k, so the lemma trivially holds.
Assume that the lemma holds for j > 1. From the definition of the l-error vector Al =
Al
({
X
(j+1)
ni
})
and the (l, j)-error vector B
(j)
l = B
(j)
l
({
X
(1)
ni
}
, . . . ,
{
X
(j)
ni
})
, we have
y
({
X
(j+1)
ni
})
= Zj+1 +
k∑
l=j+2
Al,
and
j∏
l=1
y
({
X
(l)
ni
})
=
j∑
l=1
Zl +
k∑
l=j+1
B
(j)
l .
In particular, we can see that the (l, j+1)-error vector B
(j+1)
l can be written by using the l-error
vector Al and (l, j)-error vector B
(j)
l ,
j+1∑
l=1
Zl +
k∑
l=j+2
B
(j+1)
l =
j+1∏
l=1
y
({
X
(l)
ni
})
=
(
j∑
l=1
Zl +
k∑
l=j+1
B
(j)
l
)(
Zj+1 +
k∑
l=j+2
Al
)
.
More precisely, let {Sp}p=1,...,2k be the finite set of vectors defined by
Sp =
Zp, p = 1, . . . , j,
B
(j)
p , p = j + 1, . . . , k,
0, p = k + 1, . . . , k + j,
Zj+1, p = k + j + 1,
Ap−k, p = k + j + 2, . . . , 2k.
14 K. Tashiro
Denote by deg(Sp) the number of the layer in which the vector Sp ∈ g =
⊕
Vj is. For p =
1, . . . , 2k, let Q̃p
(
ν1, . . . , k
√
νk
)
be the polynomial which controls ∥Sp∥p (∥Sp∥p−k for k + 1 ≤ p
≤ 2k) so that it attains zero at (0, . . . , 0) and the coefficients depend on d1 and k. We can
choose such polynomials by the induction hypothesis and Lemma 3.6.
By applying the BCH formula (2.2), we have(
j∑
l=1
Zl +
k∑
l=j+1
B
(j)
l
)(
Zj+1 +
k∑
l=j+2
Al
)
=
(
k∑
i=1
Sp
)(
k∑
i=1
Sp+i
)
=
2k∑
i=1
Si +
∑
q≥2
∑
(p1,...,pq)∈Iq2k
α(p1,...,pq)[Sp1 , . . . , Spq ].
Thus the (l, j + 1)-error vector B
(j+1)
l can be written by
B
(j+1)
l =
k∑
q=2
∑
(p1,...,pq)∈Iq2k, deg(Sp1 )+···+deg(Spq )=l
α(p1,...,pq)[Sp1 , . . . , Spq ].
By the triangle inequality and Lemma 3.5,
∥∥B(j+1)
l
∥∥
l
≤
k∑
q=2
∑
(p1,...,pq)∈Iq2k, deg(Sp1 )+···+deg(Spq )=l
|αp1,...,pq |2p1∧···∧pq
q∏
i=1
Q̃pi
(
ν1, . . . , k
√
νk
)
,
where p1 ∧ · · · ∧ pq is the minimum of p1, . . . , pq. ■
We introduce a function d
(j)
com : X j → R which can be regarded as the combinatorial distance
function for a j-tuple of sets of horizontal vectors.
Definition 3.11. For a j-tuple of sets of horizontal vectors
({
X
(1)
ni
}
, . . . ,
{
X
(j)
ni
})
, define
d(j)com
({
X
(1)
ni
}
, . . . ,
{
X
(j)
ni
})
=
j∑
l=1
dcom
({
X
(l)
ni
})
.
Lemma 2.7 and the triangle inequality imply the following lemma.
Lemma 3.12. For any j-tuple of sets of horizontal vectors
({
X
(1)
ni
}
, . . . ,
{
X
(j)
ni
})
, we have
dcc
(
j∏
l=1
y
({
X
(l)
ni
}))
≤ 2k−1d(j)com
({
X
(1)
ni
}
, . . . ,
{
X
(j)
ni
})
.
Now we are ready to prove Theorem 3.2. We show the following technical proposition.
Proposition 3.13. There are positive constants ϵ1, . . . , ϵk, which depend only on d1 and k, such
that if a vector Z =
∑k
j=1 Zj satisfies ∥Zj∥j ≤ ϵj, then there is a k-tuple of sets of horizontal
vectors
({
X
(1)
ni
}
, . . . ,
{
X
(k)
ni
})
adjusted to Z such that
d(k)com
({
X
(1)
ni
}
, . . . ,
{
X
(k)
ni
})
≤ 1
2k−1
.
This proposition implies Theorem 3.2. Indeed, we can check that Z =
∏k
j=1 y
({
X
(j)
ni
})
since G is k-step. Combined with Lemma 3.12,
dcc(Z) = dcc
(
k∏
j=1
y
({
X
(j)
ni
}))
≤ 2k−1d(k)com
({
X
(1)
ni
}
, . . . ,
{
X
(k)
ni
})
≤ 1.
Systolic Inequalities for Compact Quotients of Carnot Groups with Popp’s Volume 15
Proof. We will prove by induction on k. We have already shown the assertion for 2-step Carnot
group with ϵ1 =
1
2 and ϵ2 =
1
64d31
in Theorem 3.1.
Assume that the assertion is true for (k−1)-step Carnot groups with the grading dimVi = di
with positive numbers ϵ̃1, . . . , ϵ̃k−1. Let Gk be the subgroup of G generated by {[x1, . . . , xk]c |
x1, . . . , xk ∈ G}. Notice that the quotient group G/Gk is (k − 1)-step Carnot group which has
the grading
⊕k−1
j=1 Vj . From the induction hypothesis, if a vector
∑k−1
j=1 Zj satisfies ∥Zj∥j ≤ ϵ̃j ,
then there are (k − 1)-tuple of sets of horizontal vectors
({
X
(1)
ni
}
, . . . ,
{
X
(k−1)
ni
})
adjusted to∑k−1
j=1 Zj such that
d(k−1)
com
({
X
(1)
ni
}
, . . . ,
{
X
(k−1)
ni
})
≤ 1
2k−2
.
Moreover, for a positive number t > 0, the (k − 1)-tuple of sets of horizontal vectors({
tX
(1)
ni
}
, . . . ,
{
tX
(k−1)
ni
})
is adjusted to the vector
∑k−1
j=1 t
jZj , and satisfies
d(k−1)
com
({
tX
(1)
ni
}
, . . . ,
{
tX
(k−1)
ni
})
≤ t
2k−2
. (3.10)
Next we consider the product
∏k−1
j=1 y
({
tX
(j)
ni
})
in the original group G. By using the
(k, k − 1)-error vector B
(k−1)
k , we can write
k−1∏
j=1
y
({
tX
(j)
ni
})
=
k−1∑
j=1
tjZj +B
(k−1)
k .
By Lemma 3.10, there is a polynomial Qkk−1 such that∥∥B(k−1)
k
∥∥
k
≤ Qkk−1
(
tϵ̃1, . . . , t
k
√
ϵ̃k
)
.
Now let Zk be a vector in Vk, νk = ∥Zk∥k, and
{
X
(k)
ni
}
a set of horizontal vectors adjusted
to Zk −B
(k−1)
k . By the definition of d
(k)
com and (3.10), we have
d(k)com
({
tX
(1)
ni
}
, . . . ,
{
tX
(k−1)
ni
}
,
{
X
(k)
ni
})
= d(k−1)
com
({
tX
(1)
ni
}
, . . . ,
{
tX
(k−1)
ni
})
+ dcom
({
X
(k)
ni
})
≤ t
2k−2
+ dcom
({
X
(k)
ni
})
.
Since the set of horizontal vectors
{
X
(k)
ni
}
is adjusted to Zk −B
(k−1)
k , Lemma 3.7 yields
dcom
({
X
(k)
ni
})
≤ kd
2k−1
2
1
∥∥Zk −B
(k−1)
k
∥∥ 1
k
k
≤ kd
2k−1
2
1
(
νk +Qkk−1
(
tϵ̃1, . . . , t
k
√
ϵ̃k
)) 1
k .
Since the polynomial Qkk−1 attains zero at (0, . . . , 0) ∈ Rk and the coefficients depend only
on d1 and k, there are positive numbers T , ϵ̂k, which depend only on the dimension d1 and k,
such that if t ≤ T and νk ≤ ϵ̂k, then
t
2k−2
+ dcom
({
X
(k)
ni
})
≤ 1
2k−1
.
We conclude the proposition by letting ϵj = T j ϵ̃j for j = 1, . . . , k−1 and ϵk = min
{
T k ϵ̃k, ϵ̂k
}
. ■
Acknowledgements
The author would appreciate to thank Professor Takumi Yokota for many insightful suggestions.
The author thanks the referees for many helpful comments on earlier drafts of the manuscript.
This research is supported by JSPS KAKENHI grant number 18K03298 and 20J13261.
16 K. Tashiro
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1 Introduction
2 Carnot groups
3 Proof of the main theorem
3.1 2-step case
3.2 Higher step cases
References
|
| id | nasplib_isofts_kiev_ua-123456789-211729 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-16T09:46:39Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Tashiro, Kenshiro 2026-01-09T12:53:47Z 2022 Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume. Kenshiro Tashiro. SIGMA 18 (2022), 058, 16 pages 1815-0659 2020 Mathematics Subject Classification: 53C17; 26B15; 22E25 arXiv:2201.00128 https://nasplib.isofts.kiev.ua/handle/123456789/211729 https://doi.org/10.3842/SIGMA.2022.058 In this paper, we give a systolic inequality for a quotient space of a Carnot group Γ∖ with Popp's volume. Namely, we show the existence of a positive constant C such that the systole of Γ∖ is less than Cvol(Γ∖ )¹ᐟQ, where Q is the Hausdorff dimension. Moreover, the constant depends only on the dimension of the grading of the Lie algebra = ⨁ ᵢ. To prove this fact, the scalar product on introduced in the definition of Popp's volume plays a key role. The author would appreciate to thank Professor Takumi Yokota for many insightful suggestions. The author thanks the referees for many helpful comments on earlier drafts of the manuscript. This research is supported by JSPS KAKENHI grant numbers 18K03298 and 20J13261. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume Article published earlier |
| spellingShingle | Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume Tashiro, Kenshiro |
| title | Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume |
| title_full | Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume |
| title_fullStr | Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume |
| title_full_unstemmed | Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume |
| title_short | Systolic Inequalities for Compact Quotients of Carnot Groups with Popp's Volume |
| title_sort | systolic inequalities for compact quotients of carnot groups with popp's volume |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211729 |
| work_keys_str_mv | AT tashirokenshiro systolicinequalitiesforcompactquotientsofcarnotgroupswithpoppsvolume |