references for the quotient of a product of hyperbolic 2- and 3- spaces by $SL_2$ over a number ring

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Let $F$ be a number field of degree $r+2s$ with ring of integers $mathcalO_F$, $G=SL_2(Fotimes_mathbbQmathbbR)cong SL_2(mathbbR)^rtimes SL_2(mathbbC)^s$, $K=SO_2(mathbbR)^rtimes SU_2(mathbbC)^ssubseteq G$, and $Gamma=SL_2(mathcalO_F)$.

Is there a name for the locally symmetric space $X=Gammabackslash G/KcongGammabackslash(mathbbH^2)^rtimes(mathbbH^3)^s$? When $F$ is real quadratic (or more generally totally real) these go by the name of Hilbert or Hilbert-Blumenthal modular surfaces (varieties). When $F$ has complex places, these seem to be less well-studied (probably because they lack complex structure).

In any case, I'm looking for references on the geometry of $X$ (especially any discussion of compact totally geodesic subspaces).

[Edit: For instance, when $F$ has exactly one place, i.e. $F=mathbbQ$ or $F=mathbbQ(sqrt-d)$, $d>0$, we get the modular surface and the Bianchi orbifolds respectively. The closed geodesics on the modular surface and in the Bianchi orbifolds are associated to anisotropic indefinite binary quadratic forms, and the closed geodesic surfaces in the Bianchi orbifolds are associated to anisotropic indefinite binary Hermitian forms.]

number-theory reference-request hyperbolic-geometry symmetric-spaces

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edited Aug 2 at 19:05

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yoyo

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Let $F$ be a number field of degree $r+2s$ with ring of integers $mathcalO_F$, $G=SL_2(Fotimes_mathbbQmathbbR)cong SL_2(mathbbR)^rtimes SL_2(mathbbC)^s$, $K=SO_2(mathbbR)^rtimes SU_2(mathbbC)^ssubseteq G$, and $Gamma=SL_2(mathcalO_F)$.

Is there a name for the locally symmetric space $X=Gammabackslash G/KcongGammabackslash(mathbbH^2)^rtimes(mathbbH^3)^s$? When $F$ is real quadratic (or more generally totally real) these go by the name of Hilbert or Hilbert-Blumenthal modular surfaces (varieties). When $F$ has complex places, these seem to be less well-studied (probably because they lack complex structure).

In any case, I'm looking for references on the geometry of $X$ (especially any discussion of compact totally geodesic subspaces).

[Edit: For instance, when $F$ has exactly one place, i.e. $F=mathbbQ$ or $F=mathbbQ(sqrt-d)$, $d>0$, we get the modular surface and the Bianchi orbifolds respectively. The closed geodesics on the modular surface and in the Bianchi orbifolds are associated to anisotropic indefinite binary quadratic forms, and the closed geodesic surfaces in the Bianchi orbifolds are associated to anisotropic indefinite binary Hermitian forms.]

number-theory reference-request hyperbolic-geometry symmetric-spaces

share|cite|improve this question

edited Aug 2 at 19:05

asked Aug 2 at 18:12

yoyo

6,3291525

add a comment | 

up vote
2
down vote

favorite

up vote
2
down vote

favorite

Let $F$ be a number field of degree $r+2s$ with ring of integers $mathcalO_F$, $G=SL_2(Fotimes_mathbbQmathbbR)cong SL_2(mathbbR)^rtimes SL_2(mathbbC)^s$, $K=SO_2(mathbbR)^rtimes SU_2(mathbbC)^ssubseteq G$, and $Gamma=SL_2(mathcalO_F)$.

Is there a name for the locally symmetric space $X=Gammabackslash G/KcongGammabackslash(mathbbH^2)^rtimes(mathbbH^3)^s$? When $F$ is real quadratic (or more generally totally real) these go by the name of Hilbert or Hilbert-Blumenthal modular surfaces (varieties). When $F$ has complex places, these seem to be less well-studied (probably because they lack complex structure).

In any case, I'm looking for references on the geometry of $X$ (especially any discussion of compact totally geodesic subspaces).

[Edit: For instance, when $F$ has exactly one place, i.e. $F=mathbbQ$ or $F=mathbbQ(sqrt-d)$, $d>0$, we get the modular surface and the Bianchi orbifolds respectively. The closed geodesics on the modular surface and in the Bianchi orbifolds are associated to anisotropic indefinite binary quadratic forms, and the closed geodesic surfaces in the Bianchi orbifolds are associated to anisotropic indefinite binary Hermitian forms.]

number-theory reference-request hyperbolic-geometry symmetric-spaces

share|cite|improve this question

edited Aug 2 at 19:05

asked Aug 2 at 18:12

yoyo

6,3291525

Let $F$ be a number field of degree $r+2s$ with ring of integers $mathcalO_F$, $G=SL_2(Fotimes_mathbbQmathbbR)cong SL_2(mathbbR)^rtimes SL_2(mathbbC)^s$, $K=SO_2(mathbbR)^rtimes SU_2(mathbbC)^ssubseteq G$, and $Gamma=SL_2(mathcalO_F)$.

Is there a name for the locally symmetric space $X=Gammabackslash G/KcongGammabackslash(mathbbH^2)^rtimes(mathbbH^3)^s$? When $F$ is real quadratic (or more generally totally real) these go by the name of Hilbert or Hilbert-Blumenthal modular surfaces (varieties). When $F$ has complex places, these seem to be less well-studied (probably because they lack complex structure).

In any case, I'm looking for references on the geometry of $X$ (especially any discussion of compact totally geodesic subspaces).

[Edit: For instance, when $F$ has exactly one place, i.e. $F=mathbbQ$ or $F=mathbbQ(sqrt-d)$, $d>0$, we get the modular surface and the Bianchi orbifolds respectively. The closed geodesics on the modular surface and in the Bianchi orbifolds are associated to anisotropic indefinite binary quadratic forms, and the closed geodesic surfaces in the Bianchi orbifolds are associated to anisotropic indefinite binary Hermitian forms.]

number-theory reference-request hyperbolic-geometry symmetric-spaces

share|cite|improve this question

edited Aug 2 at 19:05

asked Aug 2 at 18:12

yoyo

6,3291525

share|cite|improve this question

share|cite|improve this question

edited Aug 2 at 19:05

edited Aug 2 at 19:05

edited Aug 2 at 19:05

asked Aug 2 at 18:12

yoyo

6,3291525

asked Aug 2 at 18:12

yoyo

6,3291525

asked Aug 2 at 18:12

yoyo

6,3291525

6,3291525

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When $r=0$ and $s=1$ the groups are known as Bianchi groups and the quotient space $X$ is a Bianchi manifold.

Perhaps as a class the spaces you are asking about are loosely called "arithmetic manifolds", but I'm not sure.

You might look up the references on that wikipedia page linked above, also the book by Dave Witte-Morris entitled "Introduction to arithmetic groups", although perhaps those references concentrate more on the groups than on the quotient spaces.

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edited Aug 2 at 18:52

answered Aug 2 at 18:37

Lee Mosher

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up vote
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When $r=0$ and $s=1$ the groups are known as Bianchi groups and the quotient space $X$ is a Bianchi manifold.

Perhaps as a class the spaces you are asking about are loosely called "arithmetic manifolds", but I'm not sure.

You might look up the references on that wikipedia page linked above, also the book by Dave Witte-Morris entitled "Introduction to arithmetic groups", although perhaps those references concentrate more on the groups than on the quotient spaces.

share|cite|improve this answer

edited Aug 2 at 18:52

answered Aug 2 at 18:37

Lee Mosher

45.3k33478

add a comment | 

up vote
0
down vote

When $r=0$ and $s=1$ the groups are known as Bianchi groups and the quotient space $X$ is a Bianchi manifold.

Perhaps as a class the spaces you are asking about are loosely called "arithmetic manifolds", but I'm not sure.

You might look up the references on that wikipedia page linked above, also the book by Dave Witte-Morris entitled "Introduction to arithmetic groups", although perhaps those references concentrate more on the groups than on the quotient spaces.

share|cite|improve this answer

edited Aug 2 at 18:52

answered Aug 2 at 18:37

Lee Mosher

45.3k33478

add a comment | 

up vote
0
down vote

up vote
0
down vote

When $r=0$ and $s=1$ the groups are known as Bianchi groups and the quotient space $X$ is a Bianchi manifold.

Perhaps as a class the spaces you are asking about are loosely called "arithmetic manifolds", but I'm not sure.

You might look up the references on that wikipedia page linked above, also the book by Dave Witte-Morris entitled "Introduction to arithmetic groups", although perhaps those references concentrate more on the groups than on the quotient spaces.

share|cite|improve this answer

edited Aug 2 at 18:52

answered Aug 2 at 18:37

Lee Mosher

45.3k33478

When $r=0$ and $s=1$ the groups are known as Bianchi groups and the quotient space $X$ is a Bianchi manifold.

Perhaps as a class the spaces you are asking about are loosely called "arithmetic manifolds", but I'm not sure.

You might look up the references on that wikipedia page linked above, also the book by Dave Witte-Morris entitled "Introduction to arithmetic groups", although perhaps those references concentrate more on the groups than on the quotient spaces.

share|cite|improve this answer

edited Aug 2 at 18:52

answered Aug 2 at 18:37

Lee Mosher

45.3k33478

share|cite|improve this answer

share|cite|improve this answer

edited Aug 2 at 18:52

edited Aug 2 at 18:52

edited Aug 2 at 18:52

answered Aug 2 at 18:37

Lee Mosher

45.3k33478

answered Aug 2 at 18:37

Lee Mosher

45.3k33478

answered Aug 2 at 18:37

Lee Mosher

45.3k33478

45.3k33478

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