Tori_and_geometry Algebraic_torus

1 tori , geometry

1.1 flat subspaces , rank of symmetric spaces
1.2 q-rank of lattices
1.3 buildings

tori , geometry
flat subspaces , rank of symmetric spaces

if



g


{\displaystyle g}

semisimple lie group real rank




r



{\displaystyle \mathbb {r} }

-rank defined above (for




r



{\displaystyle \mathbb {r} }

-algebraic group group of real points isomorphic



g


{\displaystyle g}

), in other words maximal



r


{\displaystyle r}

such there exists embedding



(


r


×



)

r


→
g


{\displaystyle (\mathbb {r} ^{\times })^{r}\to g}

. example, real rank of





s
l


n


(

r

)


{\displaystyle \mathrm {sl} _{n}(\mathbb {r} )}

equal



n
−
1


{\displaystyle n-1}

, , real rank of




s
o

(
p
,
q
)


{\displaystyle \mathrm {so} (p,q)}

equal



min
(
p
,
q
)


{\displaystyle \min(p,q)}

.

if



x


{\displaystyle x}

symmetric space associated



g


{\displaystyle g}

,



t
⊂
g


{\displaystyle t\subset g}

maximal split torus there exists unique orbit of



t


{\displaystyle t}

in



x


{\displaystyle x}

totally geodesic flat subspace in



x


{\displaystyle x}

. in fact maximal flat subspace , maximal such obtained orbits of split tori in way. there geometric definition of real rank, maximal dimension of flat subspace in



x


{\displaystyle x}

.

q-rank of lattices

if lie group



g


{\displaystyle g}

obtained real points of algebraic group




g



{\displaystyle \mathbf {g} }

on rational field




q



{\displaystyle \mathbb {q} }






q



{\displaystyle \mathbb {q} }

-rank of




g



{\displaystyle \mathbf {g} }

has geometric significance. 1 has introduce arithmetic group



Γ


{\displaystyle \gamma }

associated




g



{\displaystyle \mathbf {g} }

, group of integer points of




g



{\displaystyle \mathbf {g} }

, , quotient space



m
=
Γ
∖
x


{\displaystyle m=\gamma \backslash x}

, riemannian orbifold , hence metric space. asymptotic cone of



m


{\displaystyle m}

homeomorphic finite simplicial complex top-dimensional simplices of dimension equal




q



{\displaystyle \mathbb {q} }

-rank of




g



{\displaystyle \mathbf {g} }

. in particular,



m


{\displaystyle m}

compact if , if




g



{\displaystyle \mathbf {g} }

anisotropic.

note allows define




q



{\displaystyle \mathbf {q} }

-rank of lattice in semisimple lie group, dimension of asymptotic cone.

buildings

if




g



{\displaystyle \mathbf {g} }

semisimple group on





q


p




{\displaystyle \mathbb {q} _{p}}

maximal split tori in




g



{\displaystyle \mathbf {g} }

correspond apartments of bruhat-tits building



x


{\displaystyle x}

associated




g



{\displaystyle \mathbf {g} }

. in particular dimension of



x


{\displaystyle x}

equal





q


p




{\displaystyle \mathbb {q} _{p}}

-rank of




g



{\displaystyle \mathbf {g} }

.