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Constructing Bieberbach Groups froma quotient group of the orbit braid group
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Toric Topology 2022 in Osaka
March, 24, 2022

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1

Crystallographic Groups and Bieberbach Groups

2

Conclusions related to a quotient of the Artin braid groups

3

Constructing Bieberbach Groups from Bnorb /[Pn , Pn ]4

Further Research

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Constructing Bieberbach Groups from Bn/[Pn , Pn ]Further ResearchCharlap L. S., Bieberbach Groups and Flat Manifolds;
Dekimpe K., Almost-Bieberbach groups: Affine and Polynomial
Structures.

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Constructing Bieberbach Groups from Bn/[Pn , Pn ]Further ResearchCharlap L. S., Bieberbach Groups and Flat Manifolds;
Dekimpe K., Almost-Bieberbach groups: Affine and Polynomial
Structures.
Definition
Let G be a Hausdorff topological group. A subgroup H of G is
said to be uniform if G/H is compact.

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Constructing Bieberbach Groups from Bn/[Pn , Pn ]Further ResearchCharlap L. S., Bieberbach Groups and Flat Manifolds;
Dekimpe K., Almost-Bieberbach groups: Affine and Polynomial
Structures.
Definition
Let G be a Hausdorff topological group. A subgroup H of G is
said to be uniform if G/H is compact.
Definition
A discrete and uniform subgroup Π of Rn ⋊ O(n, R) ⊂ Aff(Rn ) is
said to be a crystallographic Group of dimension n. If in addition Π
is torsion free then Π is called a Bieberbach group of dimension n.

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If Φ is a group, an integral representation of rank m of Φ isdefined to be a homomorphism Θ : Φ → Aut(Zm ). Two such
representations are said to be equivalent if their images areconjugate in Aut(Zm ). We say that Θ is a faithful representation
if it is injective.

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Lemma
Let Π be a group. Then Π is a crystallographic group if and only ifthere exists an integer n ∈ N and a short exact sequenceζ

1 → Zn → Π →− Φ→1such that:(a)(b)Φ is finite, andthe integral representation Θ : Φ → Aut(Zn ), induced by
conjugation on Z and defined by Θ(φ)(x) = πxπ −1 , for allx ∈ Zn , φ ∈ Φ, where π ∈ Π is such that ζ(π) = φ, is faithful.

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Lemma
Let Π be a group. Then Π is a crystallographic group if and only ifthere exists an integer n ∈ N and a short exact sequenceζ

1 → Zn → Π →− Φ→1such that:(a)(b)Φ is finite, andthe integral representation Θ : Φ → Aut(Zn ), induced by
conjugation on Z and defined by Θ(φ)(x) = πxπ −1 , for allx ∈ Zn , φ ∈ Φ, where π ∈ Π is such that ζ(π) = φ, is faithful.

The integer n is called the dimension of Π. The finite group Φ iscalled the holonomy group of Π. And the integral representationΘ : Φ → Aut(Zn ) is called the holonomy representation of Π.
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1 → Zn → Π →− Φ→1CorollaryLet Π be a crystallographic group of dimension n and holonomy
group Φ, and let H be a subgroup of Φ. ζ −1 (H) is a
crystallographic subgroup of Π of dimension n with holonomy
group H.

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Definition
A Riemannian manifold M is called flat if it has zero curvature at
every point.

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Definition
A Riemannian manifold M is called flat if it has zero curvature at
every point.
Theorem (the first Bieberbach Theorem)
There is a correspondence between Bieberbach groups andfundamental groups of closed flat Riemannian manifolds.

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Theorem (Wolf J.A.)
The holonomy group of the corresponding flat manifold M is
isomorphic to the group Φ.
Theorem (Auslander and Kuranishi)
Any finite group is the holonomy group of some flat manifold.

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Lemma
Let M be the flat manifold whose fundamental group is the
Bieberbach group Π. Then M is orientable if and only if theintegral representation Θ : Φ → Aut(Zn ) satisfies
Im(Θ) ⊂ SO(n, Z).

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Lemma
Let M be the flat manifold whose fundamental group is the
Bieberbach group Π. Then M is orientable if and only if theintegral representation Θ : Φ → Aut(Zn ) satisfies
Im(Θ) ⊂ SO(n, Z).
Lemma
Let M be the flat manifold whose fundamental group is the
Bieberbach group Π. The holonomy group Φ is generated by
U1 , · · · , Us . Then the first Betti number of M is:π[π, π]= n − rank(Θ(U1 ) − I, · · · , Θ(Us ) − I).

b1 (M ) = rankH1 (M, Z) = rank

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Lemma
Let M be the flat manifold whose fundamental group is the
Bieberbach group Π. The holonomy group Φ is cyclic. Supposethat A = Θ(1). Then M supports an Anosv diffeomorphism if andonly if A has none of the following numbers as simple eigenvalues:±1, ±i, ±ω, ±ω 2 , where

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Lemma
Let M be the flat manifold whose fundamental group is the
Bieberbach group Π. The holonomy group Φ is cyclic. Supposethat A = Θ(1). Then M supports an Anosv diffeomorphism if andonly if A has none of the following numbers as simple eigenvalues:±1, ±i, ±ω, ±ω 2 , whereω 3 = 1.

Lemma
Let M be the flat manifold whose fundamental group is the
Bieberbach group Π. Then M is Kähler if and only if(a)the dimension of M is even, and(b)each R-irreducible summand of φ which is also C-irreducible
occurs with an even multiplicity.
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Definition
The Artin braid group Bn on n strands is defined by the presentation:
U generators: σ1 , · · · , σn−1 ;U relations:

σi σj = σj σi|i − j| ≥ 2,σi σi+1 σi = σi+1 σi σi+1

i = 1, · · · , n − 2.

Definition
The Artin pure braid group Pn is defined by the presentation:
U generators: Sij , 1 ≤ i < j ≤ n;
U relations:−1
Srs
Sij Srs =


Sij ,

 Sij ,−1
Srj Sij Srj,

r < s < i < j;i < r < s < j;r < i = s < j;−1,ij (Sij Ssj ) (Sij Ssj )S−1
−1

−1 −1 −1
(Srj Ssj Srj Ssj )Sij (Srj Ssj Srj
Ssj ),r = i < s < j;r < i < s < j.

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Proposition (Gonçalves, Guaschi, Ocampo)
Let n ⩾ 2. There is a short exact sequence:1→Zn(n−1)
2s→ Bn /[Pn , Pn ] →− Σn → 1,

and the middle group Bn /[Pn , Pn ] is a crystallographic group.

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Proposition (Gonçalves, Guaschi, Ocampo)
Let n ⩾ 2. There is a short exact sequence:1→Zn(n−1)
2s→ Bn /[Pn , Pn ] →− Σn → 1,

and the middle group Bn /[Pn , Pn ] is a crystallographic group.
Corollaryen
Let n ⩾ 3, and let H be a subgroup of Σn . Then the group H
defined by
e n = s−1 (H)/[Pn , Pn ]H
(1)
is a crystallographic group of dimension n(n−1)
with holonomy
2group H.
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Theorem (Gonçalves, Guaschi, Ocampo)
If n ⩾ 3 then the quotient group Bn /[Pn , Pn ] has no finite-orderelement of even order.

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Theorem (Gonçalves, Guaschi, Ocampo)
If n ⩾ 3 then the quotient group Bn /[Pn , Pn ] has no finite-orderelement of even order.
Corollaryen
Let n ⩾ 3, and let H be a 2-subgroup of Σn . Then the group Hn(n−1)
given by equation (1) is a Bieberbach group of dimension.
2

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Constructing Bieberbach Groups from Bn/[Pn , Pn ]Further ResearchConsider the cyclic subgroup H2d ,k ⊂ Σn :H2d ,k = ⟨µ⟩,µ = (2d , · · · , 1)(2 · 2d , · · · , 1 + 2d ) · · · (k, · · · , k − (2d − 1))with k = 2d m such that 2d ⩽ k ⩽ n and m is a positive integer.
e 2d ,k = s−1 (H2d ,k )/[Pn , Pn ] is a Bieberbach group with
Then Hholonomy group H2d ,k and holonomy representation
ψH2d ,k : H2d ,k → Aut(Pn /[Pn , Pn ]).

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Theorem (Ocampo, Rodriguez-Nieto)Let k = 2d m. Let χH2d ,k be the flat manifold of dimension n(n−1)
with2efundamental group H2d ,k and holonomy group H2d ,k = Z2d . Then(a)

χH2d ,k is orientable if and only if one of n or m is even.

(b)

The first homology group of the flat manifold χH2d ,k is
H1 (χH2d ,k , Z) = Z|T| ⊕ Z2d−1 , where T is the transversal of theaction by conjugation of H2d ,k on the basis {Ai,j |1 ⩽ i < j ⩽ n} of
Pn /[Pn , Pn ] satisying|T| =k(2n − k − 2) (n − k)(n − k − 1)
k
+
+
.
d
2
2d+1
2

So, the first Betti number of the flat manifold χH2d ,k is
b1 (χH2d ,k ) =(2d − 1)k 2 − 2kn(2d − 1) + 2d n2 − 2d n.
2d+1

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Theorem (Ocampo, Rodriguez-Nieto)(c)e 2d ,kThe flat manifold χH2d ,k with fundamental group H
admits Anosov diffeomorphism if and only if
(i)
(ii)
(iii)

(c)

n ⩾ 4 in the cased = 1,n ⩾ 5 in the case d = 2,n ⩾ 2d in the case d ⩾ 3.

Suppose that n(n−1)
is even, so n = 4q or n = 4q + 1, for2some q. Let d = 1, then the flat manifold χH2d ,k is Kähler if
and only if n = 4q and m is even. Let d2 then the flat
manifold χH2d ,k is Kähler if and only if one of the following
conditions holds
(i)
(ii)

n = 4qn = 4q + 1 and m is even.

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Theorem (Ocampo)
Let G be a finite abelian group.
(a)

There exists n and a Bieberbach subgroup ΓG of Bn /[Pn , Pn ]of dimension n(n−1)
with holonomy group G.
2(b)

The finite abelian group G is the holonomy group of a flat
manifold χΓG of dimension n(n−1), where n is an integer for2
which G embeds in the symmetric group Σn , and thefundamental group of χΓG is isomorphic to a subgroup ΓG of
Bn /[Pn , Pn ].

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Theorem (Ocampo)
Let q = pr11 pr22 · · · prt t be an odd number, where pi are distinct odd
primes and ri ⩾ 1 for all 1 ⩽ i ⩽ t. Let χΓq be the flat manifold ofwith fundamental group Γq ⊂ Bn /[Pn , Pn ] anddimension n(n−1)
2holonomy group Zq . Then(a)χΓq is orientable.
trPpi i −1(b)
The first Betti number of χΓq is b1 (χΓq ) =+ t(t−1)
2
2 .
i=1

(c)

The flat manifold χΓq with fundamental group Γq admits Anosov
diffeomorphism if and only if q ̸= 3.

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Theorem (Ocampo)
Let q = pr11 pr22 · · · prt t be an odd number, where pi are distinct odd
primes and ri ⩾ 1 for all 1 ⩽ i ⩽ t. Let χΓq be the flat manifold ofwith fundamental group Γq ⊂ Bn /[Pn , Pn ] anddimension n(n−1)
2holonomy group Zq . Then(a)χΓq is orientable.
trPpi i −1(b)
The first Betti number of χΓq is b1 (χΓq ) =+ t(t−1)
2
2 .
i=1

(c)

The flat manifold χΓq with fundamental group Γq admits Anosov
diffeomorphism if and only if q ̸= 3.

Theorem (Ocampo)
Let p be an odd prime and let r ⩾ 1. Let χΓpr be the flat manifold ofrr
dimension p (p2 −1) with fundamental group Γpr ⊂ Bn /[Pn , Pn ] andholonomy group Zpr . Then the flat manifold χΓpr is Kähler if and only ifthere is an integer u ⩾ 1 such that pr = 4u + 1.
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Surface braid group
Proposition (Goncalves, Guaschi, Ocampo, Pereiro)
Let M be an orientable, compact connected surface of genus g ⩾ 1
without boundary, and let n ⩾ 2. Then there exists a split
extension of the form:s

1 → Z2ng → Bn (M )/[Pn (M ), Pn (M )] →− Σn → 1.
The quotient Bn (M )/[Pn (M ), Pn (M )] is a crystallographic groupof dimension 2ng, whose holonomy group is Σn .

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Surface braid group
Proposition (Goncalves, Guaschi, Ocampo, Pereiro)
Let M be an orientable, compact connected surface of genus g ⩾ 1
without boundary, and let n ⩾ 2. Then there exists a split
extension of the form:s

1 → Z2ng → Bn (M )/[Pn (M ), Pn (M )] →− Σn → 1.
The quotient Bn (M )/[Pn (M ), Pn (M )] is a crystallographic groupof dimension 2ng, whose holonomy group is Σn .
Proposition (Goncalves, Guaschi, Ocampo, Pereiro)
Let M = S2 or Ng , where g ⩾ 1. Then for all n ⩾ 1, the quotient
Bn (M )/[Pn (M ), Pn (M )] is not a crystallographic group.
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Surface braid group
Proposition (Goncalves, Guaschi, Ocampo, Pereiro)
Let M be an orientable, compact connected surface of genus g ⩾ 1
without boundary, and let n ⩾ 1. The quotient
Bn (M )/[Pn (M ), Pn (M )] has the following presentation:
Generators: σ1 , · · · , σn−1 , ai,r , 1 ⩽ i ⩽ n, 1 ⩽ r ⩽ 2g.
Relations:1

the Artin relations.

2

σi2 = 1, for all i = 1, · · · , n − 1

3[ai,r , aj,s ] = 1, for all i, j = 1, · · · , n and r, s = 1, · · · , 2g.

4

σi aj,r σi−1 = aτi (j),r for all 1 ⩽ i ⩽ n − 1, 1 ⩽ j and1 ⩽ r ⩽ 2g.

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Surface braid group
Theorem (Goncalves, Guaschi, Ocampo, Pereiro)
Let n ⩾ 2, and let M be an orientable surface of genus g ⩾ 1
without boundary. Let Gn be the cyclic subgroup⟨(n, n − 1, · · · , 2, 1)⟩ of Σn . Then there exists a subgroup G̃n,g of
σ −1 (Gn )/[Pn (M ), Pn (M )] ⊂ Bn (M )/[Pn (M ), Pn (M )] that is a
Bieberbach group of dimension 2ng whose holonomy group is Gn .
Further, the centre Z(G̃n,g ) of G̃n,g is a free Abelian group of rank
2g.

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Surface braid group
Theorem (Goncalves, Guaschi, Ocampo, Pereiro)
Let n ⩾ 2, and let M be an orientable surface of genus g ⩾ 1
without boundary. Let Gn be the cyclic subgroup⟨(n, n − 1, · · · , 2, 1)⟩ of Σn . Then there exists a subgroup G̃n,g of
σ −1 (Gn )/[Pn (M ), Pn (M )] ⊂ Bn (M )/[Pn (M ), Pn (M )] that is a
Bieberbach group of dimension 2ng whose holonomy group is Gn .
Further, the centre Z(G̃n,g ) of G̃n,g is a free Abelian group of rank
2g.
Theorem (Goncalves, Guaschi, Ocampo, Pereiro)
Let n ⩾ 2, and let χn,g be a 2ng-dimension flat manifold whose
fundamental group is the Bieberbach group G̃n,g . Then χn,g is anorientable Kahler manifold with first Betti number 2g that admits
Anosov diffeomorphisms.
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Theorem (Marin I.)
For every complex reflection group W , the group B/[P, P ] is
crystallographic with holonomy group W/Z(W ) of dimension
N = |A|. The kernel of the projection map B/[P, P ] → W/Z(W )is the subgroup P0 generated by P ab and Z0 (B). We haveP0 ∼= ZN .

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Theorem (Marin I.)
For every complex reflection group W , the group B/[P, P ] is
crystallographic with holonomy group W/Z(W ) of dimension
N = |A|. The kernel of the projection map B/[P, P ] → W/Z(W )is the subgroup P0 generated by P ab and Z0 (B). We haveP0 ∼= ZN .
Theorem (Marin I.)
For every complex reflection group W , the group B/[P, P ] has noelement of order 2.

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Orbit Braid Group

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Orbit Braid GroupG↷M
Definition (orbit configuration space, Xicontencatle)
FG (M, n) = {(x1 , · · · , xn ) ∈ M n |G(xi )Shuya Cai @ Fudan University\

G(xj ) = ∅Constructing Bieberbach Groups

if

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Orbit Braid GroupG↷M
Definition (orbit configuration space, Xicontencatle)
FG (M, n) = {(x1 , · · · , xn ) ∈ M n |G(xi )\G(xj ) = ∅if

Definition (Hao Li, Zhi Lü, Fengling Li) (orbit braid group)
Bnorb (M, G) ∼= π1E (FG (M, n), x, xorb )P orb (M, G) ∼= π E (FG (M, n), x, Gn (x))
n1

Bn (M, G) ∼= π1E (FG (M, n), x, Σn x)
Pn (M, G) ∼= π1 (FG (M, n), x)
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Theorem (Hao Li, Zhi Lü, Fengling Li)
Bnorb (C, Z2 ) admits the following presentation:
U generators: σ; σi , i = 1, · · · , n − 1.
U relations:
1
2
3
4
5

σ 2 = 1;σσ1 σσ1 = σ1 σσ1 σ;
σσi = σi σ, i = 2, · · · , n − 1;σi σj = σj σi , |i − j| > 1;σi σi+1 σi = σi+1 σi σi+1 .

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TheoremPnorb (C, Z2 ) admits the following presentation:
U generators: zk , k = 1, · · · , n; Sij , 1 ≤ i < j ≤ n.
U relations:
1
2

2
zk
= 1, k = 1, · · · , n;
 Si,j ,zj Sij zj ,zk Sij zk = zj Skj zj S −1 Sij Skj zj S −1 zj ,
kj
kj
3
4

k < i < j or
k = i < j;i < k < j;

i < j < k;−1 −1
zk zs zk = zs Sks zs Sks
zs , k < s;


Sij ,




Sij ,


−1
−1
Srj Sij Srj,
Srs Sij Srs = (Sij Ssj )Sij (Sij Ssj )−1 ,



−1 −1
−1 −1 −1
(Srj Ssj Srj Ssj )Sij (Srj Ssj Srj
Ssj ),Shuya Cai @ Fudan University

Constructing Bieberbach Groupsr < s < i < j;i < r < s < j;r < i = s < j;r = i < s < j;r < i < s < j.

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DenotePnorb = Pnorb (C, Z2 );Pn = Pn (C, Z2 );Bnorb = Bnorb (C, Z2 );Bn = Bn (C, Z2 ).

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DenotePnorb = Pnorb (C, Z2 );Pn = Pn (C, Z2 );Bnorb = Bnorb (C, Z2 );Bn = Bn (C, Z2 ).

Proposition
Bnorb /[Pnorb , Pnorb ] is not a crystallographic group.

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Bnorb /[Pn , Pn ] is a crystallographic group?

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Bnorb /[Pn , Pn ] is a crystallographic group?

s

1 → Pn → Bnorb →− (Z2 )n ⋊ Σn → 1.

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Bnorb /[Pn , Pn ] is a crystallographic group?

s

1 → Pn → Bnorb →− (Z2 )n ⋊ Σn → 1.
We obtains

1 → Pn /[Pn , Pn ] → Bnorb /[Pn , Pn ] →− (Z2 )n ⋊ Σn → 1.

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Bnorb /[Pn , Pn ] is a crystallographic group?

s

1 → Pn → Bnorb →− (Z2 )n ⋊ Σn → 1.
We obtains

1 → Pn /[Pn , Pn ] → Bnorb /[Pn , Pn ] →− (Z2 )n ⋊ Σn → 1.

RemarkΘ : Zn2 ⋊ Σn → Aut(Zn(n−1) ) induced by the conjugation is notfaithful.

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Lemma
LetS
E = s−1 ((−1, · · · , −1), 1). Then
(E Pn )/[Pn , Pn ] ∼= Zn(n−1) is a normal subgroup oforb
Bn /[Pn , Pn ] and it is generated by:ω = σ(σ1 σσ1 ) · · · (σn−1 · · · σ1 σσ1 · · · σn−1 ),Sn−1,n , Si,j , 1 ⩽ |i| < n − 1, 1 < j ⩽ n, |i| < j.

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Lemma
LetS
E = s−1 ((−1, · · · , −1), 1). Then
(E Pn )/[Pn , Pn ] ∼= Zn(n−1) is a normal subgroup oforb
Bn /[Pn , Pn ] and it is generated by:ω = σ(σ1 σσ1 ) · · · (σn−1 · · · σ1 σσ1 · · · σn−1 ),Sn−1,n , Si,j , 1 ⩽ |i| < n − 1, 1 < j ⩽ n, |i| < j.
Proposition
Let n ⩾ 3 and F = Z((Z2 )n ⋊ Σn ) = ⟨((−1, · · · , −1), 1)⟩. Then
there is a short exact sequence:s

1 → Zn(n−1) → Bnorb /[Pn , Pn ] →− (Z2 )n ⋊ Σn /F → 1,and Bnorb /[Pn , Pn ] is a crystallographic group.
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Theorem
Let n ⩾ 3.And let ϑ ∈ Zn2 ⋊ Σn /F satisfy the following conditions:U ϑ is of order 2
U ϑ is not conjugate to ((−1, 1, · · · , 1), 1).
e = σ −1 (H)/[Pn , Pn ] is a Bieberbach group
Then for H =< ϑ >, H
of dimension n(n − 1) with the holonomy group H.

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Theorem
Let n ⩾ 3.And let ϑ ∈ Zn2 ⋊ Σn /F satisfy the following conditions:U ϑ is of order 2
U ϑ is not conjugate to ((−1, 1, · · · , 1), 1).
e = σ −1 (H)/[Pn , Pn ] is a Bieberbach group
Then for H =< ϑ >, H
of dimension n(n − 1) with the holonomy group H.
Corollary
Bnorb /[Pn , Pn ] has no elements of order 2k, k ⩾ 2.

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Theorem
Let n ⩾ 3.And let ϑ ∈ Zn2 ⋊ Σn /F satisfy the following conditions:U ϑ is of order 2
U ϑ is not conjugate to ((−1, 1, · · · , 1), 1).
e = σ −1 (H)/[Pn , Pn ] is a Bieberbach group
Then for H =< ϑ >, H
of dimension n(n − 1) with the holonomy group H.
Corollary
Bnorb /[Pn , Pn ] has no elements of order 2k, k ⩾ 2.
Theorem
Let n ⩾ 3 and let m be odd integers greater than or equal to 3.
Then Bnorb /[Pn , Pn ] possesses infinitely many elements of order m.
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Constructing Bieberbach Groups from Bn/[Pn , Pn ]Further ResearchConsider the cyclic subgroup G ⊂ (Z2 )n ⋊ Σn /F :G = ⟨µ = ((ε1 , · · · , εn ), θ)⟩,θ = (2d · · · 1) · · · (m2d · · · (m − 1)2d + 1),
where d ⩾ 1, 0 ⩽ m2d ⩽ n.

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Constructing Bieberbach Groups from Bn/[Pn , Pn ]Further ResearchConsider the cyclic subgroup G ⊂ (Z2 )n ⋊ Σn /F :G = ⟨µ = ((ε1 , · · · , εn ), θ)⟩,θ = (2d · · · 1) · · · (m2d · · · (m − 1)2d + 1),
where d ⩾ 1, 0 ⩽ m2d ⩽ n.
order µ = 2u , u ⩾ 1,
satisfying u = d or = 2d.

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Constructing Bieberbach Groups from Bn/[Pn , Pn ]Further ResearchConsider the cyclic subgroup G ⊂ (Z2 )n ⋊ Σn /F :G = ⟨µ = ((ε1 , · · · , εn ), θ)⟩,θ = (2d · · · 1) · · · (m2d · · · (m − 1)2d + 1),
where d ⩾ 1, 0 ⩽ m2d ⩽ n.
order µ = 2u , u ⩾ 1,
satisfying u = d or = 2d.

M is orientable?
M has an Ansov structure?

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Constructing Bieberbach groups with non Abelianholonomy groups

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Constructing Bieberbach groups with non Abelianholonomy groupsr

For the semi-direct product (Z2 )2 +1 ⋊ Σ2r +1 , r ⩾ 2, we have
r

r

s

r

1 → Z(2 +1)(2 +2) → B2orb− (Z2 )2 +1 ⋊Σ2r +1 → 1.
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Constructing Bieberbach groups with non Abelianholonomy groupsr

For the semi-direct product (Z2 )2 +1 ⋊ Σ2r +1 , r ⩾ 2, we have
r

s

r

r

1 → Z(2 +1)(2 +2) → B2orb− (Z2 )2 +1 ⋊Σ2r +1 → 1.
r +1 /[P2r +1 , P2r +1 ] →rr

Consider (Z2 )2 ⋊ Z2r ⊂ (Z2 )2 +1 ⋊ Σ2r +1 generated byω i = ((1, 1, · · · , 1, −1, 1, · · · , 1), 1),

1 ⩽ i ⩽ 2r ,α = ((1, · · · , 1), (2, · · · , 2r + 1)).

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Constructing Bieberbach groups with non Abelianholonomy groupsr

For the semi-direct product (Z2 )2 +1 ⋊ Σ2r +1 , r ⩾ 2, we have
r

s

r

r

1 → Z(2 +1)(2 +2) → B2orb− (Z2 )2 +1 ⋊Σ2r +1 → 1.
r +1 /[P2r +1 , P2r +1 ] →rr

Consider (Z2 )2 ⋊ Z2r ⊂ (Z2 )2 +1 ⋊ Σ2r +1 generated byω i = ((1, 1, · · · , 1, −1, 1, · · · , 1), 1),

1 ⩽ i ⩽ 2r ,α = ((1, · · · , 1), (2, · · · , 2r + 1)).
r

r

Let F = ⟨((−1, · · · , −1), 1)⟩ ⊂ (Z2 )2 ⋊ Z2r . (Z2 )2 ⋊ Z2r /F is
non Abelian.

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The subgroup Γ ⊂ B2orb
r +1 /[P2r +1 , P2r +1 ] is generated by X1
where

S

X2 ,

X1 = {ωi , 1 ⩽ i ⩽ 2r , α = σ2r · · · σ2 },X2 = {S1,2 S−1,2 , · · · , S1,2r +1 S−1,2r +1 , Si,j , 2 ⩽ |i| < j ⩽ 2r + 1}.

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The subgroup Γ ⊂ B2orb
r +1 /[P2r +1 , P2r +1 ] is generated by X1
where

S

X2 ,

X1 = {ωi , 1 ⩽ i ⩽ 2r , α = σ2r · · · σ2 },X2 = {S1,2 S−1,2 , · · · , S1,2r +1 S−1,2r +1 , Si,j , 2 ⩽ |i| < j ⩽ 2r + 1}.
Γ is a Bieberbach group. The holonomy group is the non Abelianrgroup (Z2 )2 ⋊ Z2r /F .

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Definition
An almost-crystallographic group is a discrete subgroup Π of thesemi-direct product N ⋊ C that acts properly and discontinuously
on N such that N/Π is compact. If in addition Π is torsion freethen Π is called an almost-Bieberbach group.

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Definition
An almost-crystallographic group is a discrete subgroup Π of thesemi-direct product N ⋊ C that acts properly and discontinuously
on N such that N/Π is compact. If in addition Π is torsion freethen Π is called an almost-Bieberbach group.
Infra-nilmanifolds are determined completely by their fundamentalgroups that are almost-Bieberbach groups.

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Theorem (Gasior, Petrosyan, Szczepa)
Let M be an almost-flat manifold with holonomy group F . Then
M is orientable if and only if det = 1. Suppose M is orientabletand a 2-Sylow subgroup of F is cyclic, i.e. C2t = ⟨q | q 2 = 1⟩ forsome t ⩾ 0. Let πab denote the abelianisation of the fundamentalgroup π of M .
(a)

(b)

If 21 (n − T race[θ(q)2
structure.

t−1]) ̸≡ 2(mod 4), then M has a Spint−1

If 21 (n − T race[θ(q)2 ]) ≡ 2(mod 4), then M has a Spinstructure if and only if the epimorphism q∗ : πab → C2t
resulting from projection q : π → C2t factors through a cyclic
group of order 2t+1 .

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Theorem (Gonalves, Guaschi, Ocampo)
Let n, k ⩾ 3. Bn /Γk (Pn ) is an almost-crystallographic group.

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Theorem (Gonalves, Guaschi, Ocampo)
Let n, k ⩾ 3. Bn /Γk (Pn ) is an almost-crystallographic group.
Theorem (Gonalves, Guaschi, Ocampo)
For n, k ⩾ 3, Bn /Γk (Pn ) has no elements order 2 or 3.

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Theorem (Gonalves, Guaschi, Ocampo)
Let n, k ⩾ 3. Bn /Γk (Pn ) is an almost-crystallographic group.
Theorem (Gonalves, Guaschi, Ocampo)
For n, k ⩾ 3, Bn /Γk (Pn ) has no elements order 2 or 3.
TheoremBnorb /Γk (Pn ) is an almost-crystallographic group.

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Thank you!

Shuya Cai @ Fudan University

Constructing Bieberbach Groups