Thin Beauville p-groups and Beauville surfaces
Şükran Gül
University of the Basque Country, Bilbao, Spain
(joint work with Gustavo Fernández-Alcober, Norberto Gavioli and
Carlo M. Scoppola)
Groups and Topological Groups
Trento, June 16-17, 2017
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Contents
1
Background material
2
Thin p-groups
Metabelian thin p-groups
p-Groups of maximal class
3
Beauville surfaces
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Beauville surfaces and groups
Definition
A Beauville surface of unmixed type is a compact complex surface which is
the quotient of a product C1 × C2 of two algebraic curves C1 and C2 of
genera at least 2 by the action of a finite group G acting freely by
holomorphic transformations, in such a way that:
(i) Ci /G ∼
= P1 (C) for i = 1, 2.
(ii) The covering map Ci → Ci /G is ramified over three points for i = 1, 2.
The group G is then called a Beauville group of unmixed type.
Question: Which finite groups are Beauville groups?
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Thin Beauville p-groups and Beauville surfaces
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Background material
Beauville surfaces and groups
Definition
A Beauville surface of unmixed type is a compact complex surface which is
the quotient of a product C1 × C2 of two algebraic curves C1 and C2 of
genera at least 2 by the action of a finite group G acting freely by
holomorphic transformations, in such a way that:
(i) Ci /G ∼
= P1 (C) for i = 1, 2.
(ii) The covering map Ci → Ci /G is ramified over three points for i = 1, 2.
The group G is then called a Beauville group of unmixed type.
Question: Which finite groups are Beauville groups?
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Beauville surfaces and groups
Definition
A Beauville surface of unmixed type is a compact complex surface which is
the quotient of a product C1 × C2 of two algebraic curves C1 and C2 of
genera at least 2 by the action of a finite group G acting freely by
holomorphic transformations, in such a way that:
(i) Ci /G ∼
= P1 (C) for i = 1, 2.
(ii) The covering map Ci → Ci /G is ramified over three points for i = 1, 2.
The group G is then called a Beauville group of unmixed type.
Question: Which finite groups are Beauville groups?
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Beauville surfaces and groups
Definition
A Beauville surface of unmixed type is a compact complex surface which is
the quotient of a product C1 × C2 of two algebraic curves C1 and C2 of
genera at least 2 by the action of a finite group G acting freely by
holomorphic transformations, in such a way that:
(i) Ci /G ∼
= P1 (C) for i = 1, 2.
(ii) The covering map Ci → Ci /G is ramified over three points for i = 1, 2.
The group G is then called a Beauville group of unmixed type.
Question: Which finite groups are Beauville groups?
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Group-theoretical reformulation
Definition
Let G be a group. For every x, y ∈ G , we define
[
Σ(x, y ) =
(hxig ∪ hy ig ∪ hxy ig ).
g ∈G
Theorem (Bauer, Catanese, Grunewald)
A finite group G is a Beauville group if and only if
(i) G is a 2-generator group.
(ii) G has two sets of generators, {x1 , y1 } and {x2 , y2 }, such that
Σ(x1 , y1 ) ∩ Σ(x2 , y2 ) = 1.
We say that {x1 , y1 } and {x2 , y2 } is a Beauville structure for G . We call
{xi , yi , xi yi } the triple associated to {xi , yi } for i = 1, 2.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Group-theoretical reformulation
Definition
Let G be a group. For every x, y ∈ G , we define
[
Σ(x, y ) =
(hxig ∪ hy ig ∪ hxy ig ).
g ∈G
Theorem (Bauer, Catanese, Grunewald)
A finite group G is a Beauville group if and only if
(i) G is a 2-generator group.
(ii) G has two sets of generators, {x1 , y1 } and {x2 , y2 }, such that
Σ(x1 , y1 ) ∩ Σ(x2 , y2 ) = 1.
We say that {x1 , y1 } and {x2 , y2 } is a Beauville structure for G . We call
{xi , yi , xi yi } the triple associated to {xi , yi } for i = 1, 2.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Group-theoretical reformulation
Definition
Let G be a group. For every x, y ∈ G , we define
[
Σ(x, y ) =
(hxig ∪ hy ig ∪ hxy ig ).
g ∈G
Theorem (Bauer, Catanese, Grunewald)
A finite group G is a Beauville group if and only if
(i) G is a 2-generator group.
(ii) G has two sets of generators, {x1 , y1 } and {x2 , y2 }, such that
Σ(x1 , y1 ) ∩ Σ(x2 , y2 ) = 1.
We say that {x1 , y1 } and {x2 , y2 } is a Beauville structure for G . We call
{xi , yi , xi yi } the triple associated to {xi , yi } for i = 1, 2.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
4 / 26
Background material
Group-theoretical reformulation
Definition
Let G be a group. For every x, y ∈ G , we define
[
Σ(x, y ) =
(hxig ∪ hy ig ∪ hxy ig ).
g ∈G
Theorem (Bauer, Catanese, Grunewald)
A finite group G is a Beauville group if and only if
(i) G is a 2-generator group.
(ii) G has two sets of generators, {x1 , y1 } and {x2 , y2 }, such that
Σ(x1 , y1 ) ∩ Σ(x2 , y2 ) = 1.
We say that {x1 , y1 } and {x2 , y2 } is a Beauville structure for G . We call
{xi , yi , xi yi } the triple associated to {xi , yi } for i = 1, 2.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Known results
In the sequel, all groups will be finite and p will always be a prime.
Theorem (Catanese, 2000)
An abelian group is a Beauville group if and only if it is isomorphic
to Cn × Cn , where n > 1 and gcd(n, 6) = 1.
Corollary
There are no abelian Beauville 2-groups or 3-groups.
Corollary
Let p ≥ 5, and let G be an abelian 2-generator p-group. If the exponent
of G is p e , then:
G is a Beauville group
Şükran Gül
⇐⇒
|G p
e−1
| = p2.
Thin Beauville p-groups and Beauville surfaces
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Background material
Known results
In the sequel, all groups will be finite and p will always be a prime.
Theorem (Catanese, 2000)
An abelian group is a Beauville group if and only if it is isomorphic
to Cn × Cn , where n > 1 and gcd(n, 6) = 1.
Corollary
There are no abelian Beauville 2-groups or 3-groups.
Corollary
Let p ≥ 5, and let G be an abelian 2-generator p-group. If the exponent
of G is p e , then:
G is a Beauville group
Şükran Gül
⇐⇒
|G p
e−1
| = p2.
Thin Beauville p-groups and Beauville surfaces
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Background material
Known results
In the sequel, all groups will be finite and p will always be a prime.
Theorem (Catanese, 2000)
An abelian group is a Beauville group if and only if it is isomorphic
to Cn × Cn , where n > 1 and gcd(n, 6) = 1.
Corollary
There are no abelian Beauville 2-groups or 3-groups.
Corollary
Let p ≥ 5, and let G be an abelian 2-generator p-group. If the exponent
of G is p e , then:
G is a Beauville group
Şükran Gül
⇐⇒
|G p
e−1
| = p2.
Thin Beauville p-groups and Beauville surfaces
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Background material
Known results
In the sequel, all groups will be finite and p will always be a prime.
Theorem (Catanese, 2000)
An abelian group is a Beauville group if and only if it is isomorphic
to Cn × Cn , where n > 1 and gcd(n, 6) = 1.
Corollary
There are no abelian Beauville 2-groups or 3-groups.
Corollary
Let p ≥ 5, and let G be an abelian 2-generator p-group. If the exponent
of G is p e , then:
G is a Beauville group
Şükran Gül
⇐⇒
|G p
e−1
| = p2.
Thin Beauville p-groups and Beauville surfaces
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Background material
Known results
Theorem (Bauer, Catanese, Grunewald, 2005; Fuertes, González-Diez,
2009)
The alternating groups An , for n ≥ 6, and the symmetric groups Sn ,
for n ≥ 5, are Beauville groups.
Theorem (Bauer, Catanese, Grunewald, 2005)
The groups SL(2, p), PSL(2, p) for p 6= 2, 3, 5, are Beauville groups.
Theorem (Guralnick, Malle, 2012)
Any non-abelian finite simple group other than A5 is a Beauville group.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Known results
Theorem (Bauer, Catanese, Grunewald, 2005; Fuertes, González-Diez,
2009)
The alternating groups An , for n ≥ 6, and the symmetric groups Sn ,
for n ≥ 5, are Beauville groups.
Theorem (Bauer, Catanese, Grunewald, 2005)
The groups SL(2, p), PSL(2, p) for p 6= 2, 3, 5, are Beauville groups.
Theorem (Guralnick, Malle, 2012)
Any non-abelian finite simple group other than A5 is a Beauville group.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Known results
Theorem (Bauer, Catanese, Grunewald, 2005; Fuertes, González-Diez,
2009)
The alternating groups An , for n ≥ 6, and the symmetric groups Sn ,
for n ≥ 5, are Beauville groups.
Theorem (Bauer, Catanese, Grunewald, 2005)
The groups SL(2, p), PSL(2, p) for p 6= 2, 3, 5, are Beauville groups.
Theorem (Guralnick, Malle, 2012)
Any non-abelian finite simple group other than A5 is a Beauville group.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Known results: p-groups
Barker, Boston, and Fairbairn (2012)
The smallest non-abelian Beauville p-groups for p = 2 and p = 3 are
of order 27 and 35 .
For p ≥ 5, the smallest Beauville p-group is of order p 3 .
All Beauville p-groups of order at most p 4 are determined.
The number of Beauville groups of orders p 5 and p 6 are estimated.
However, not all the cases are settled.
Fernández-Alcober, Gül (2016)
The exact number of Beauville groups of order p 5 , for p ≥ 5, and of
order p 6 , for p ≥ 7 are determined.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Known results: p-groups
Barker, Boston, and Fairbairn (2012)
The smallest non-abelian Beauville p-groups for p = 2 and p = 3 are
of order 27 and 35 .
For p ≥ 5, the smallest Beauville p-group is of order p 3 .
All Beauville p-groups of order at most p 4 are determined.
The number of Beauville groups of orders p 5 and p 6 are estimated.
However, not all the cases are settled.
Fernández-Alcober, Gül (2016)
The exact number of Beauville groups of order p 5 , for p ≥ 5, and of
order p 6 , for p ≥ 7 are determined.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Known results: p-groups
Barker, Boston, and Fairbairn (2012)
The smallest non-abelian Beauville p-groups for p = 2 and p = 3 are
of order 27 and 35 .
For p ≥ 5, the smallest Beauville p-group is of order p 3 .
All Beauville p-groups of order at most p 4 are determined.
The number of Beauville groups of orders p 5 and p 6 are estimated.
However, not all the cases are settled.
Fernández-Alcober, Gül (2016)
The exact number of Beauville groups of order p 5 , for p ≥ 5, and of
order p 6 , for p ≥ 7 are determined.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Known results: p-groups
Barker, Boston, and Fairbairn (2012)
The smallest non-abelian Beauville p-groups for p = 2 and p = 3 are
of order 27 and 35 .
For p ≥ 5, the smallest Beauville p-group is of order p 3 .
All Beauville p-groups of order at most p 4 are determined.
The number of Beauville groups of orders p 5 and p 6 are estimated.
However, not all the cases are settled.
Fernández-Alcober, Gül (2016)
The exact number of Beauville groups of order p 5 , for p ≥ 5, and of
order p 6 , for p ≥ 7 are determined.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
7 / 26
Background material
Known results: p-groups
Barker, Boston, and Fairbairn (2012)
The smallest non-abelian Beauville p-groups for p = 2 and p = 3 are
of order 27 and 35 .
For p ≥ 5, the smallest Beauville p-group is of order p 3 .
All Beauville p-groups of order at most p 4 are determined.
The number of Beauville groups of orders p 5 and p 6 are estimated.
However, not all the cases are settled.
Fernández-Alcober, Gül (2016)
The exact number of Beauville groups of order p 5 , for p ≥ 5, and of
order p 6 , for p ≥ 7 are determined.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
7 / 26
Background material
Known results: p-groups
Barker, Boston, and Fairbairn (2012)
The smallest non-abelian Beauville p-groups for p = 2 and p = 3 are
of order 27 and 35 .
For p ≥ 5, the smallest Beauville p-group is of order p 3 .
All Beauville p-groups of order at most p 4 are determined.
The number of Beauville groups of orders p 5 and p 6 are estimated.
However, not all the cases are settled.
Fernández-Alcober, Gül (2016)
The exact number of Beauville groups of order p 5 , for p ≥ 5, and of
order p 6 , for p ≥ 7 are determined.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Known results: p-groups
Theorem (Barker, Boston, and Fairbairn, 2012)
There are non-abelian Beauville p-groups of order p n for every p ≥ 5 and
n ≥ 3.
Theorem (Barker, Boston, Peyerimhoff, and Vdovina, 2015)
There are infinitely many Beauville 2-groups.
Theorem (González-Diez, Jaikin-Zapirain, 2015; Stix, Vdovina, 2015)
There are infinitely many Beauville 3-groups.
(But no explicit example is provided.)
Theorem (Fernández-Alcober, Gül, 2016)
There are quotients of the Nottingham group over F3 which are Beauville
groups of order 3n , for every n ≥ 5.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Known results: p-groups
Theorem (Barker, Boston, and Fairbairn, 2012)
There are non-abelian Beauville p-groups of order p n for every p ≥ 5 and
n ≥ 3.
Theorem (Barker, Boston, Peyerimhoff, and Vdovina, 2015)
There are infinitely many Beauville 2-groups.
Theorem (González-Diez, Jaikin-Zapirain, 2015; Stix, Vdovina, 2015)
There are infinitely many Beauville 3-groups.
(But no explicit example is provided.)
Theorem (Fernández-Alcober, Gül, 2016)
There are quotients of the Nottingham group over F3 which are Beauville
groups of order 3n , for every n ≥ 5.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Known results: p-groups
Theorem (Barker, Boston, and Fairbairn, 2012)
There are non-abelian Beauville p-groups of order p n for every p ≥ 5 and
n ≥ 3.
Theorem (Barker, Boston, Peyerimhoff, and Vdovina, 2015)
There are infinitely many Beauville 2-groups.
Theorem (González-Diez, Jaikin-Zapirain, 2015; Stix, Vdovina, 2015)
There are infinitely many Beauville 3-groups.
(But no explicit example is provided.)
Theorem (Fernández-Alcober, Gül, 2016)
There are quotients of the Nottingham group over F3 which are Beauville
groups of order 3n , for every n ≥ 5.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
8 / 26
Background material
Known results: p-groups
Theorem (Barker, Boston, and Fairbairn, 2012)
There are non-abelian Beauville p-groups of order p n for every p ≥ 5 and
n ≥ 3.
Theorem (Barker, Boston, Peyerimhoff, and Vdovina, 2015)
There are infinitely many Beauville 2-groups.
Theorem (González-Diez, Jaikin-Zapirain, 2015; Stix, Vdovina, 2015)
There are infinitely many Beauville 3-groups.
(But no explicit example is provided.)
Theorem (Fernández-Alcober, Gül, 2016)
There are quotients of the Nottingham group over F3 which are Beauville
groups of order 3n , for every n ≥ 5.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Generalization of Catanese’s criterion
Definition
Le G be a Beauville p-group such that every lift of a Beauville structure of
G /Φ(G ) ∼
= Cp × Cp is a Beauville structure of G . Then we say that G is of
tame type. Otherwise, we say that G is of wild type.
Theorem (Fernández-Alcober, Gül, 2016)
Let G be a p-group of exponent p e satisfying the following condition:
xp
e−1
= yp
e−1
(xy −1 )p
⇐⇒
e−1
= 1.
(SA)
Then G is a Beauville group if and only if
p ≥ 5 and |G p
e−1
| ≥ p2.
Furthermore, G is of tame type.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Generalization of Catanese’s criterion
Definition
Le G be a Beauville p-group such that every lift of a Beauville structure of
G /Φ(G ) ∼
= Cp × Cp is a Beauville structure of G . Then we say that G is of
tame type. Otherwise, we say that G is of wild type.
Theorem (Fernández-Alcober, Gül, 2016)
Let G be a p-group of exponent p e satisfying the following condition:
xp
e−1
= yp
e−1
(xy −1 )p
⇐⇒
e−1
= 1.
(SA)
Then G is a Beauville group if and only if
p ≥ 5 and |G p
e−1
| ≥ p2.
Furthermore, G is of tame type.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
9 / 26
Background material
Generalization of Catanese’s criterion
Definition
Le G be a Beauville p-group such that every lift of a Beauville structure of
G /Φ(G ) ∼
= Cp × Cp is a Beauville structure of G . Then we say that G is of
tame type. Otherwise, we say that G is of wild type.
Theorem (Fernández-Alcober, Gül, 2016)
Let G be a p-group of exponent p e satisfying the following condition:
xp
e−1
= yp
e−1
(xy −1 )p
⇐⇒
e−1
= 1.
(SA)
Then G is a Beauville group if and only if
p ≥ 5 and |G p
e−1
| ≥ p2.
Furthermore, G is of tame type.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
9 / 26
Background material
Generalization of Catanese’s criterion
Definition
Le G be a Beauville p-group such that every lift of a Beauville structure of
G /Φ(G ) ∼
= Cp × Cp is a Beauville structure of G . Then we say that G is of
tame type. Otherwise, we say that G is of wild type.
Theorem (Fernández-Alcober, Gül, 2016)
Let G be a p-group of exponent p e satisfying the following condition:
xp
e−1
= yp
e−1
(xy −1 )p
⇐⇒
e−1
= 1.
(SA)
Then G is a Beauville group if and only if
p ≥ 5 and |G p
e−1
| ≥ p2.
Furthermore, G is of tame type.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Groups satisfying condition SA
Almost all families of p-groups which are known to have a good behaviour
with respect to powers satisfy condition SA.
Obviously, abelian groups and groups of exponent p.
Regular p-groups.
Thus in particular groups of order ≤ p p and groups with γp−1 (G )
cyclic.
Powerful p-groups, i.e. groups in which G 0 ≤ G p if p is odd, or
G 0 ≤ G 4 if p = 2.
The so-called p-central p-groups, i.e. groups in which Ω1 (G ) ≤ Z (G )
if p is odd, or Ω2 (G ) ≤ Z (G ) if p = 2.
i
If G is a p-group, recall that Ωi (G ) = hg ∈ G | g p = 1i.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
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Background material
Groups satisfying condition SA
Almost all families of p-groups which are known to have a good behaviour
with respect to powers satisfy condition SA.
Obviously, abelian groups and groups of exponent p.
Regular p-groups.
Thus in particular groups of order ≤ p p and groups with γp−1 (G )
cyclic.
Powerful p-groups, i.e. groups in which G 0 ≤ G p if p is odd, or
G 0 ≤ G 4 if p = 2.
The so-called p-central p-groups, i.e. groups in which Ω1 (G ) ≤ Z (G )
if p is odd, or Ω2 (G ) ≤ Z (G ) if p = 2.
i
If G is a p-group, recall that Ωi (G ) = hg ∈ G | g p = 1i.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
10 / 26
Background material
Groups satisfying condition SA
Almost all families of p-groups which are known to have a good behaviour
with respect to powers satisfy condition SA.
Obviously, abelian groups and groups of exponent p.
Regular p-groups.
Thus in particular groups of order ≤ p p and groups with γp−1 (G )
cyclic.
Powerful p-groups, i.e. groups in which G 0 ≤ G p if p is odd, or
G 0 ≤ G 4 if p = 2.
The so-called p-central p-groups, i.e. groups in which Ω1 (G ) ≤ Z (G )
if p is odd, or Ω2 (G ) ≤ Z (G ) if p = 2.
i
If G is a p-group, recall that Ωi (G ) = hg ∈ G | g p = 1i.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
10 / 26
Background material
Groups satisfying condition SA
Almost all families of p-groups which are known to have a good behaviour
with respect to powers satisfy condition SA.
Obviously, abelian groups and groups of exponent p.
Regular p-groups.
Thus in particular groups of order ≤ p p and groups with γp−1 (G )
cyclic.
Powerful p-groups, i.e. groups in which G 0 ≤ G p if p is odd, or
G 0 ≤ G 4 if p = 2.
The so-called p-central p-groups, i.e. groups in which Ω1 (G ) ≤ Z (G )
if p is odd, or Ω2 (G ) ≤ Z (G ) if p = 2.
i
If G is a p-group, recall that Ωi (G ) = hg ∈ G | g p = 1i.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
10 / 26
Background material
Groups satisfying condition SA
Almost all families of p-groups which are known to have a good behaviour
with respect to powers satisfy condition SA.
Obviously, abelian groups and groups of exponent p.
Regular p-groups.
Thus in particular groups of order ≤ p p and groups with γp−1 (G )
cyclic.
Powerful p-groups, i.e. groups in which G 0 ≤ G p if p is odd, or
G 0 ≤ G 4 if p = 2.
The so-called p-central p-groups, i.e. groups in which Ω1 (G ) ≤ Z (G )
if p is odd, or Ω2 (G ) ≤ Z (G ) if p = 2.
i
If G is a p-group, recall that Ωi (G ) = hg ∈ G | g p = 1i.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
10 / 26
Background material
Groups satisfying condition SA
Almost all families of p-groups which are known to have a good behaviour
with respect to powers satisfy condition SA.
Obviously, abelian groups and groups of exponent p.
Regular p-groups.
Thus in particular groups of order ≤ p p and groups with γp−1 (G )
cyclic.
Powerful p-groups, i.e. groups in which G 0 ≤ G p if p is odd, or
G 0 ≤ G 4 if p = 2.
The so-called p-central p-groups, i.e. groups in which Ω1 (G ) ≤ Z (G )
if p is odd, or Ω2 (G ) ≤ Z (G ) if p = 2.
i
If G is a p-group, recall that Ωi (G ) = hg ∈ G | g p = 1i.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
10 / 26
Thin p-groups
3
Thin p-groups
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
11 / 26
Thin p-groups
Thin p-groups
Definition
A p-group is thin if the following two conditions hold
(i) N C G =⇒ γi+1 (G ) ≤ N ≤ γi (G ) for some i.
(ii) |γi (G ) : γi+1 (G )| ≤ p 2 for all i.
Examples of thin p-groups
(i) p-groups of maximal class.
(ii) All non-trivial quotients of the Nottingham group over Fp , with p odd.
(iii) All non-trivial quotients of the binary p-adic group, a Sylow pro-p
subgroup of GL2 (Zp ).
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
12 / 26
Thin p-groups
Thin p-groups
Definition
A p-group is thin if the following two conditions hold
(i) N C G =⇒ γi+1 (G ) ≤ N ≤ γi (G ) for some i.
(ii) |γi (G ) : γi+1 (G )| ≤ p 2 for all i.
Examples of thin p-groups
(i) p-groups of maximal class.
(ii) All non-trivial quotients of the Nottingham group over Fp , with p odd.
(iii) All non-trivial quotients of the binary p-adic group, a Sylow pro-p
subgroup of GL2 (Zp ).
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
12 / 26
Thin p-groups
Metabelian thin p-groups
Beauville structures in metabelian thin p-groups
We study Beauville structures in metabelian thin p-groups which are not
groups of maximal class.
Theorem (Brandl, Caranti, Scoppola, 1992)
Let G be a metabelian thin p-group which is not of maximal class. Then
|G | ≤ p 2p and G is of class at most p + 1.
Since we have a criterion for groups of class < p, we can focus on
metabelian thin p-groups of class p or p + 1.
If p = 3 then |G | = 35 or 36 , and a computer search shows that there are
exactly 3 isomorphism classes of Beauville groups. So we assume p ≥ 5 in
the sequel.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
13 / 26
Thin p-groups
Metabelian thin p-groups
Beauville structures in metabelian thin p-groups
We study Beauville structures in metabelian thin p-groups which are not
groups of maximal class.
Theorem (Brandl, Caranti, Scoppola, 1992)
Let G be a metabelian thin p-group which is not of maximal class. Then
|G | ≤ p 2p and G is of class at most p + 1.
Since we have a criterion for groups of class < p, we can focus on
metabelian thin p-groups of class p or p + 1.
If p = 3 then |G | = 35 or 36 , and a computer search shows that there are
exactly 3 isomorphism classes of Beauville groups. So we assume p ≥ 5 in
the sequel.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
13 / 26
Thin p-groups
Metabelian thin p-groups
Beauville structures in metabelian thin p-groups
We study Beauville structures in metabelian thin p-groups which are not
groups of maximal class.
Theorem (Brandl, Caranti, Scoppola, 1992)
Let G be a metabelian thin p-group which is not of maximal class. Then
|G | ≤ p 2p and G is of class at most p + 1.
Since we have a criterion for groups of class < p, we can focus on
metabelian thin p-groups of class p or p + 1.
If p = 3 then |G | = 35 or 36 , and a computer search shows that there are
exactly 3 isomorphism classes of Beauville groups. So we assume p ≥ 5 in
the sequel.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
13 / 26
Thin p-groups
Metabelian thin p-groups
Beauville structures in metabelian thin p-groups
We study Beauville structures in metabelian thin p-groups which are not
groups of maximal class.
Theorem (Brandl, Caranti, Scoppola, 1992)
Let G be a metabelian thin p-group which is not of maximal class. Then
|G | ≤ p 2p and G is of class at most p + 1.
Since we have a criterion for groups of class < p, we can focus on
metabelian thin p-groups of class p or p + 1.
If p = 3 then |G | = 35 or 36 , and a computer search shows that there are
exactly 3 isomorphism classes of Beauville groups. So we assume p ≥ 5 in
the sequel.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
13 / 26
Thin p-groups
Metabelian thin p-groups
The analysis of the general case
relies on the following special
case:
(i) G is of class p.
(ii) γp (G ) is a diamond,
that is case II.
Lemma
If G is as above, the following holds:
(i) G p ≤ γp (G ).
(ii) |M p | ≤ p for every maximal
subgroup M of G .
(iii) No more than 3 maximal
subgroups of G can have
the same pth power.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
14 / 26
Thin p-groups
Metabelian thin p-groups
The analysis of the general case
relies on the following special
case:
(i) G is of class p.
(ii) γp (G ) is a diamond,
that is case II.
Lemma
If G is as above, the following holds:
(i) G p ≤ γp (G ).
(ii) |M p | ≤ p for every maximal
subgroup M of G .
(iii) No more than 3 maximal
subgroups of G can have
the same pth power.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
14 / 26
Thin p-groups
Metabelian thin p-groups
The analysis of the general case
relies on the following special
case:
(i) G is of class p.
(ii) γp (G ) is a diamond,
that is case II.
Lemma
If G is as above, the following holds:
(i) G p ≤ γp (G ).
(ii) |M p | ≤ p for every maximal
subgroup M of G .
(iii) No more than 3 maximal
subgroups of G can have
the same pth power.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
14 / 26
Thin p-groups
Metabelian thin p-groups
The analysis of the general case
relies on the following special
case:
(i) G is of class p.
(ii) γp (G ) is a diamond,
that is case II.
Lemma
If G is as above, the following holds:
(i) G p ≤ γp (G ).
(ii) |M p | ≤ p for every maximal
subgroup M of G .
(iii) No more than 3 maximal
subgroups of G can have
the same pth power.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
14 / 26
Thin p-groups
Metabelian thin p-groups
The analysis of the general case
relies on the following special
case:
(i) G is of class p.
(ii) γp (G ) is a diamond,
that is case II.
Lemma
If G is as above, the following holds:
(i) G p ≤ γp (G ).
(ii) |M p | ≤ p for every maximal
subgroup M of G .
(iii) No more than 3 maximal
subgroups of G can have
the same pth power.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
14 / 26
Thin p-groups
Metabelian thin p-groups
The analysis of the general case
relies on the following special
case:
(i) G is of class p.
(ii) γp (G ) is a diamond,
that is case II.
Lemma
If G is as above, the following holds:
(i) G p ≤ γp (G ).
(ii) |M p | ≤ p for every maximal
subgroup M of G .
(iii) No more than 3 maximal
subgroups of G can have
the same pth power.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
14 / 26
Thin p-groups
Metabelian thin p-groups
Theorem
Let G be as in case II. Then G has a Beauville structure in which
one of the two triples has all elements of order p 2 .
Theorem
Let G be a metabelian thin p-group, where p ≥ 5. Assume that
G is of class ≥ p, and not of maximal class. Then G is a Beauville group
if and only if one of the following cases holds:
(i) G is of class p + 1.
(ii) G is of class p and |γp (G )| = p 2 .
(iii) G is of class p, |γp (G )| = p and G p = γp−1 (G ).
(iv) G is of class p, |γp (G )| = p, G p = γp (G ), and G has at least three
maximal subgroups of exponent p.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
15 / 26
Thin p-groups
Metabelian thin p-groups
Theorem
Let G be as in case II. Then G has a Beauville structure in which
one of the two triples has all elements of order p 2 .
Theorem
Let G be a metabelian thin p-group, where p ≥ 5. Assume that
G is of class ≥ p, and not of maximal class. Then G is a Beauville group
if and only if one of the following cases holds:
(i) G is of class p + 1.
(ii) G is of class p and |γp (G )| = p 2 .
(iii) G is of class p, |γp (G )| = p and G p = γp−1 (G ).
(iv) G is of class p, |γp (G )| = p, G p = γp (G ), and G has at least three
maximal subgroups of exponent p.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
15 / 26
Thin p-groups
p-Groups of maximal class
Beauville structures in p-groups of maximal class
Theorem (Blackburn, 1958)
Let G be a p-group of maximal class of order ≤ p p+1 . Then
exp G /Z (G ) = p.
As a consequence, exp G = p or p 2 and G p ≤ Z (G ).
Theorem
Let G be a p-group of maximal class of order ≤ p p . Then
G is a Beauville group
⇐⇒
p ≥ 5 and exp G = p.
Theorem
There are no Beauville p-groups of maximal class for p = 2 or 3.
Thus, in the rest of the analysis we assume that p ≥ 5 and ≥ p p+1 .
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
16 / 26
Thin p-groups
p-Groups of maximal class
Beauville structures in p-groups of maximal class
Theorem (Blackburn, 1958)
Let G be a p-group of maximal class of order ≤ p p+1 . Then
exp G /Z (G ) = p.
As a consequence, exp G = p or p 2 and G p ≤ Z (G ).
Theorem
Let G be a p-group of maximal class of order ≤ p p . Then
G is a Beauville group
⇐⇒
p ≥ 5 and exp G = p.
Theorem
There are no Beauville p-groups of maximal class for p = 2 or 3.
Thus, in the rest of the analysis we assume that p ≥ 5 and ≥ p p+1 .
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
16 / 26
Thin p-groups
p-Groups of maximal class
Beauville structures in p-groups of maximal class
Theorem (Blackburn, 1958)
Let G be a p-group of maximal class of order ≤ p p+1 . Then
exp G /Z (G ) = p.
As a consequence, exp G = p or p 2 and G p ≤ Z (G ).
Theorem
Let G be a p-group of maximal class of order ≤ p p . Then
G is a Beauville group
⇐⇒
p ≥ 5 and exp G = p.
Theorem
There are no Beauville p-groups of maximal class for p = 2 or 3.
Thus, in the rest of the analysis we assume that p ≥ 5 and ≥ p p+1 .
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
16 / 26
Thin p-groups
p-Groups of maximal class
Beauville structures in p-groups of maximal class
Theorem (Blackburn, 1958)
Let G be a p-group of maximal class of order ≤ p p+1 . Then
exp G /Z (G ) = p.
As a consequence, exp G = p or p 2 and G p ≤ Z (G ).
Theorem
Let G be a p-group of maximal class of order ≤ p p . Then
G is a Beauville group
⇐⇒
p ≥ 5 and exp G = p.
Theorem
There are no Beauville p-groups of maximal class for p = 2 or 3.
Thus, in the rest of the analysis we assume that p ≥ 5 and ≥ p p+1 .
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
16 / 26
Thin p-groups
p-Groups of maximal class
Beauville structures in p-groups of maximal class
Theorem (Blackburn, 1958)
Let G be a p-group of maximal class of order ≤ p p+1 . Then
exp G /Z (G ) = p.
As a consequence, exp G = p or p 2 and G p ≤ Z (G ).
Theorem
Let G be a p-group of maximal class of order ≤ p p . Then
G is a Beauville group
⇐⇒
p ≥ 5 and exp G = p.
Theorem
There are no Beauville p-groups of maximal class for p = 2 or 3.
Thus, in the rest of the analysis we assume that p ≥ 5 and ≥ p p+1 .
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
16 / 26
Thin p-groups
p-Groups of maximal class
Beauville structures in p-groups of maximal class
Theorem (Blackburn, 1958)
Let G be a p-group of maximal class of order ≤ p p+1 . Then
exp G /Z (G ) = p.
As a consequence, exp G = p or p 2 and G p ≤ Z (G ).
Theorem
Let G be a p-group of maximal class of order ≤ p p . Then
G is a Beauville group
⇐⇒
p ≥ 5 and exp G = p.
Theorem
There are no Beauville p-groups of maximal class for p = 2 or 3.
Thus, in the rest of the analysis we assume that p ≥ 5 and ≥ p p+1 .
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
16 / 26
Thin p-groups
p-Groups of maximal class
Beauville structures in p-groups of maximal class
If G is a p-group of maximal class, we set G1 = CG (G 0 /γ4 (G )).
It is a maximal subgroup of G .
The existence of Beauville structures in groups of maximal class will
depend on the orders of the elements outside G1 .
Lemma (Blackburn, 1958)
Let G be a p-group of maximal class, and let M be a maximal subgroup
of G . Then
(i) All elements in M r Φ(G ) have the same order.
(ii) If M 6= G1 then that order is either p or p 2 .
Definition
If M 6= G1 is a maximal subgroup of G , we call a difference M r Φ(G )
a maximal branch.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
17 / 26
Thin p-groups
p-Groups of maximal class
Beauville structures in p-groups of maximal class
If G is a p-group of maximal class, we set G1 = CG (G 0 /γ4 (G )).
It is a maximal subgroup of G .
The existence of Beauville structures in groups of maximal class will
depend on the orders of the elements outside G1 .
Lemma (Blackburn, 1958)
Let G be a p-group of maximal class, and let M be a maximal subgroup
of G . Then
(i) All elements in M r Φ(G ) have the same order.
(ii) If M 6= G1 then that order is either p or p 2 .
Definition
If M 6= G1 is a maximal subgroup of G , we call a difference M r Φ(G )
a maximal branch.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
17 / 26
Thin p-groups
p-Groups of maximal class
Beauville structures in p-groups of maximal class
If G is a p-group of maximal class, we set G1 = CG (G 0 /γ4 (G )).
It is a maximal subgroup of G .
The existence of Beauville structures in groups of maximal class will
depend on the orders of the elements outside G1 .
Lemma (Blackburn, 1958)
Let G be a p-group of maximal class, and let M be a maximal subgroup
of G . Then
(i) All elements in M r Φ(G ) have the same order.
(ii) If M 6= G1 then that order is either p or p 2 .
Definition
If M 6= G1 is a maximal subgroup of G , we call a difference M r Φ(G )
a maximal branch.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
17 / 26
Thin p-groups
p-Groups of maximal class
Beauville structures in p-groups of maximal class
If G is a p-group of maximal class, we set G1 = CG (G 0 /γ4 (G )).
It is a maximal subgroup of G .
The existence of Beauville structures in groups of maximal class will
depend on the orders of the elements outside G1 .
Lemma (Blackburn, 1958)
Let G be a p-group of maximal class, and let M be a maximal subgroup
of G . Then
(i) All elements in M r Φ(G ) have the same order.
(ii) If M 6= G1 then that order is either p or p 2 .
Definition
If M 6= G1 is a maximal subgroup of G , we call a difference M r Φ(G )
a maximal branch.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
17 / 26
Thin p-groups
p-Groups of maximal class
Beauville structures in p-groups of maximal class
Theorem
Let G be a p-group of maximal class of order p n , with n ≥ p + 1, and
assume that either G is metabelian or G1 is of class at most 2. Then G is a
Beauville group if and only if p ≥ 5 and one of the following two cases
holds:
(i) All maximal branches have elements of order p.
(ii) There are exactly 2 maximal branches with elements of order p, and
either n 6≡ 2 (mod p − 1), or if n = p + 1 and exp G1 = p.
In the first case, all Beauville groups are of tame type, and in the second
case, all Beauville groups are of wild type.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
18 / 26
Beauville surfaces
3
Beauville surfaces
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
19 / 26
Beauville surfaces
Non-isomorphic Beauville surfaces
One Beauville group can have many different Beauville structures.
Different Beauville structures in the same group can give rise to
non-isomorphic Beauville surfaces.
Problem: Determine the number of non-isomorphic Beauville surfaces
coming from Beauville group G .
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
20 / 26
Beauville surfaces
Non-isomorphic Beauville surfaces
One Beauville group can have many different Beauville structures.
Different Beauville structures in the same group can give rise to
non-isomorphic Beauville surfaces.
Problem: Determine the number of non-isomorphic Beauville surfaces
coming from Beauville group G .
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
20 / 26
Beauville surfaces
Non-isomorphic Beauville surfaces
One Beauville group can have many different Beauville structures.
Different Beauville structures in the same group can give rise to
non-isomorphic Beauville surfaces.
Problem: Determine the number of non-isomorphic Beauville surfaces
coming from Beauville group G .
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
20 / 26
Beauville surfaces
Non-isomorphic Beauville surfaces
We define a specific group A(G ) of transformations of the Beauville
structures of G .
Remark: The group A(G ) contains Aut(G ), with the natural action on
Beauville structures.
Theorem (Bauer, Catanese, Grunewald, 2004)
Let G be a Beauville group. Then the Beauville structures S1 and S2 define
isomorphic Beauville surfaces if and only if S1 is in the A(G )-orbit of S2 .
Thus, the number of non-isomorphic Beauville surfaces with G matches
with the number of A(G )-orbits on the set of Beauville structures.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
21 / 26
Beauville surfaces
Non-isomorphic Beauville surfaces
We define a specific group A(G ) of transformations of the Beauville
structures of G .
Remark: The group A(G ) contains Aut(G ), with the natural action on
Beauville structures.
Theorem (Bauer, Catanese, Grunewald, 2004)
Let G be a Beauville group. Then the Beauville structures S1 and S2 define
isomorphic Beauville surfaces if and only if S1 is in the A(G )-orbit of S2 .
Thus, the number of non-isomorphic Beauville surfaces with G matches
with the number of A(G )-orbits on the set of Beauville structures.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
21 / 26
Beauville surfaces
Non-isomorphic Beauville surfaces
We define a specific group A(G ) of transformations of the Beauville
structures of G .
Remark: The group A(G ) contains Aut(G ), with the natural action on
Beauville structures.
Theorem (Bauer, Catanese, Grunewald, 2004)
Let G be a Beauville group. Then the Beauville structures S1 and S2 define
isomorphic Beauville surfaces if and only if S1 is in the A(G )-orbit of S2 .
Thus, the number of non-isomorphic Beauville surfaces with G matches
with the number of A(G )-orbits on the set of Beauville structures.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
21 / 26
Beauville surfaces
Non-isomorphic Beauville surfaces
We define a specific group A(G ) of transformations of the Beauville
structures of G .
Remark: The group A(G ) contains Aut(G ), with the natural action on
Beauville structures.
Theorem (Bauer, Catanese, Grunewald, 2004)
Let G be a Beauville group. Then the Beauville structures S1 and S2 define
isomorphic Beauville surfaces if and only if S1 is in the A(G )-orbit of S2 .
Thus, the number of non-isomorphic Beauville surfaces with G matches
with the number of A(G )-orbits on the set of Beauville structures.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
21 / 26
Beauville surfaces
Known results
Garion, Penegini, (2014)
For Sn , An or PSL(2, p) for odd prime p, the estimates for the number
of non-isomorphic Beauville surfaces coming from these groups are
given.
González-Diez, Jones, Torres-Teigell, (2014)
For abelian Beauville groups, an exact formula for this number is given.
For example, the number of non-isomorphic Beauville surfaces coming from
Cp × Cp for p ≥ 5 is:

 1 p 4 − 10p 3 + 45p 2 − 110p + 122 if p ≡ −1 (mod 3),
72
1
p 4 − 10p 3 + 45p 2 − 118p + 154 if p ≡ 1 (mod 3).
 72
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
22 / 26
Beauville surfaces
Known results
Garion, Penegini, (2014)
For Sn , An or PSL(2, p) for odd prime p, the estimates for the number
of non-isomorphic Beauville surfaces coming from these groups are
given.
González-Diez, Jones, Torres-Teigell, (2014)
For abelian Beauville groups, an exact formula for this number is given.
For example, the number of non-isomorphic Beauville surfaces coming from
Cp × Cp for p ≥ 5 is:

 1 p 4 − 10p 3 + 45p 2 − 110p + 122 if p ≡ −1 (mod 3),
72
1
p 4 − 10p 3 + 45p 2 − 118p + 154 if p ≡ 1 (mod 3).
 72
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
22 / 26
Beauville surfaces
Known results
Garion, Penegini, (2014)
For Sn , An or PSL(2, p) for odd prime p, the estimates for the number
of non-isomorphic Beauville surfaces coming from these groups are
given.
González-Diez, Jones, Torres-Teigell, (2014)
For abelian Beauville groups, an exact formula for this number is given.
For example, the number of non-isomorphic Beauville surfaces coming from
Cp × Cp for p ≥ 5 is:

 1 p 4 − 10p 3 + 45p 2 − 110p + 122 if p ≡ −1 (mod 3),
72
1
p 4 − 10p 3 + 45p 2 − 118p + 154 if p ≡ 1 (mod 3).
 72
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
22 / 26
Beauville surfaces
Metabelian p-groups of maximal class
Let G be a metabelian Beauville p-group of maximal class, and let
|G | = p n where n = m(p − 1) + r for 3 ≤ r ≤ p.
Recall: We have two cases:
(i) All maximal branches have elements of order p −→ tame type.
(ii) There are exactly 2 maximal branches with elements of order p −→
wild type.
For both cases, we determine the number of all Beauville structures.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
23 / 26
Beauville surfaces
Metabelian p-groups of maximal class
Let G be a metabelian Beauville p-group of maximal class, and let
|G | = p n where n = m(p − 1) + r for 3 ≤ r ≤ p.
Recall: We have two cases:
(i) All maximal branches have elements of order p −→ tame type.
(ii) There are exactly 2 maximal branches with elements of order p −→
wild type.
For both cases, we determine the number of all Beauville structures.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
23 / 26
Beauville surfaces
Metabelian p-groups of maximal class
Let G be a metabelian Beauville p-group of maximal class, and let
|G | = p n where n = m(p − 1) + r for 3 ≤ r ≤ p.
Recall: We have two cases:
(i) All maximal branches have elements of order p −→ tame type.
(ii) There are exactly 2 maximal branches with elements of order p −→
wild type.
For both cases, we determine the number of all Beauville structures.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
23 / 26
Beauville surfaces
Metabelian p-groups of maximal class
Let G be a metabelian Beauville p-group of maximal class, and let
|G | = p n where n = m(p − 1) + r for 3 ≤ r ≤ p.
Recall: We have two cases:
(i) All maximal branches have elements of order p −→ tame type.
(ii) There are exactly 2 maximal branches with elements of order p −→
wild type.
For both cases, we determine the number of all Beauville structures.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
23 / 26
Beauville surfaces
Tame type
Theorem
Let G be of tame type of order p n . Then the number of non-isomorphic
Beauville surfaces with G is at least
1 p+1 p−2
p
p−2
2 n−3
(p −1) p
+
(p −1)2 (p n−3 −p n−r ) ,
2d
3
3
2
2
where | Aut(G )| = dp 2n−4 and
(
p(p − 1)2 if G1 is abelian,
d=
pd1
if [G1 , G2 ] = Gn−k and 1 ≤ k ≤ p − 3
for some d1 |p − 1.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
24 / 26
Beauville surfaces
Wild type
Theorem
Let G be of wild type of order p n . Then, if n is even, the number of
non-isomorphic Beauville surfaces with G is exactly
1 p − 2
p−2
2 n−3
(p − 1) p
+
(p − 1)2 (p n−3 − p n−r ) .
d
3
2
If n is odd, then this number is at most
1 p − 2
p−2
2 n−3
(p − 1) p
+2
(p − 1)2 (p n−3 − p n−r ) ,
d
3
2
where | Aut(G )| = dp 2n−4 for some d|p − 1.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
25 / 26
Beauville surfaces
Wild type
Theorem
Let G be of wild type of order p n . Then, if n is even, the number of
non-isomorphic Beauville surfaces with G is exactly
1 p − 2
p−2
2 n−3
(p − 1) p
+
(p − 1)2 (p n−3 − p n−r ) .
d
3
2
If n is odd, then this number is at most
1 p − 2
p−2
2 n−3
(p − 1) p
+2
(p − 1)2 (p n−3 − p n−r ) ,
d
3
2
where | Aut(G )| = dp 2n−4 for some d|p − 1.
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
25 / 26
Beauville surfaces
THANK YOU
Şükran Gül
Thin Beauville p-groups and Beauville surfaces
26 / 26