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A building (cf. also [[Tits building|Tits building]]) which is defined for a connected [[Reductive group|reductive group]] over a [[Field|field]] which is complete with respect to a non-trivial discrete valuation and has a perfect residue class field. It is the topological realization of the (poly-) simplicial complex of all parahoric subgroups. One way to define it more precisely is as follows.

Let  $  K $
be a [[Field|field]] which is complete with respect to the non-trivial valuation  $  \omega : {K  ^  \times  } \rightarrow \mathbf Z $
and has a perfect residue class field. Let  $  G $
be a connected, reductive  $  K $-
group. First, assume  $  G $
to be semi-simple (cf. [[Semi-simple group|Semi-simple group]]).

==Apartments.==
Let  $  S $
be a maximal  $  K $-
split [[Torus|torus]] in  $  G $
and denote by  $  Z $(
respectively,  $  N $) 
the [[Centralizer|centralizer]] (respectively, normalizer; cf. [[Normalizer of a subset|Normalizer of a subset]]) of  $  S $
in  $  G $. 
Let  $  X _ {*} ( S ) $(
respectively,  $  X  ^ {*} ( S ) $) 
denote the group of cocharacters (respectively, characters) of  $  S $
and let  $  {\langle  {\cdot, \cdot } \rangle } : {X  ^ {*} ( S ) \times X _ {*} ( S ) } \rightarrow \mathbf Z $
be the canonical perfect pairing. Then there is a unique group [[Homomorphism|homomorphism]]  $  \nu : {Z ( K ) } \rightarrow {V = X _ {*} ( S ) \otimes _ {\mathbf Z} \mathbf R } $
such that  $  \langle  {\chi, \nu ( z ) } \rangle = - \omega ( \chi ( z ) ) $
for all  $  \chi \in X  ^ {*} _ {K} ( Z ) $(
i.e., the group of  $  K $-
rational characters of  $  Z $). 
One can show that there is a unique affine  $  V $-
space  $  A $
together with a group homomorphism  $  \nu : {N ( K ) } \rightarrow { { \mathop{\rm Aff} } ( A ) } $(
i.e., the affine bijections  $  A \rightarrow A $) 
extending  $  \nu : {Z ( K ) } \rightarrow {V \subset  { \mathop{\rm Aff} } ( A ) } $, 
called the (empty) apartment associated with  $  S $.

==Filtrations of the root subgroups.==
Denote by  $  \Phi $
the [[Root system|root system]] of  $  G $
with respect to  $  S $
and, for  $  a \in \Phi $, 
by  $  U _ {a} $
the root subgroup of  $  G $
associated with  $  a $. 
Then, for  $  u \in U _ {a} ( K ) \backslash \{ 1 \} $, 
the set  $  U _ {- a }  ( K ) uU _ {- a }  ( K ) \cap N ( K ) $
contains exactly one element, denoted by  $  m ( u ) $. 
An affine mapping  $  \alpha : A \rightarrow \mathbf R $
is called an affine root if the vector part  $  a $
of  $  \alpha $
is contained in  $  \Phi $
and if there exists a  $  u \in U _ {a} ( K ) \backslash \{ 1 \} $
such that  $  \alpha ^ {-1 } ( 0 ) = \{ {x \in A } : {\nu ( m ( u ) ) ( x ) = x } \} $. 
In that case  $  \alpha $
is abbreviated as  $  \alpha ( a,u ) $. 
For  $  x \in A $
and  $  a \in \Phi $, 
let  $  U _ {a,x }  = \{ {u \in U _ {a} ( K ) } : {\alpha ( a,u ) ( x ) \geq  0 } \} \cup \{ 1 \} $
and let  $  U _ {x} $
be the subgroup of  $  G ( K ) $
generated by all  $  U _ {a,x }  $
for  $  a \in \Phi $.

==Simplicial structures.==
Two points  $  x,y \in A $
are called equivalent if  $  \alpha ( x ) $
and  $  \alpha ( y ) $
have the same sign or are both equal to  $  0 $
for all affine roots  $  \alpha $. 
One obtains a (poly-) simplicial complex (i.e., a direct product of simplicial complexes) in  $  A $
by defining the faces to be the equivalence classes.

==Building.==
Let  $  { \mathop{\rm BT} } ( G,K ) = G ( K ) \times A/ \sim $, 
where  $  ( g,x ) \sim ( h,y ) $
if there exists an  $  n \in N ( K ) $
such that  $  \nu ( n ) ( x ) = y $
and  $  g ^ {- 1 } hn \in U _ {x} $. 
There is a canonical  $  G ( K ) $-
action on  $  { \mathop{\rm BT} } ( G,K ) $
induced by left-multiplication on the first factor of  $  G ( K ) \times A $. 
One can identify  $  A $
with its canonical image in  $  { \mathop{\rm BT} } ( G,K ) $. 
The subsets of the form  $  gA $, 
for  $  g \in G ( K ) $, 
are called apartments and the subsets of the form  $  gF $, 
for  $  g \in G ( K ) $
and  $  F $
a face in  $  A $, 
are called faces. One can equip  $  { \mathop{\rm BT} } ( G,K ) $
with a [[Metric|metric]] which is  $  G ( K ) $-
invariant. This metric coincides on  $  A $
with the metric induced by the scalar product on  $  V $
which is invariant under the Weyl group of  $  \Phi $. 
The metric space  $  { \mathop{\rm BT} } ( G,K ) $
together with these structures is called the Bruhat–Tits building of  $  G $.

If  $  G $
is not semi-simple, the Bruhat–Tits building of  $  G $
is, by definition, the Bruhat–Tits building of the derived group (cf. [[Commutator subgroup|Commutator subgroup]]) of  $  G $.

===Example.===
Assume  $  G = { \mathop{\rm SL} } _ {2} $, 
and denote by  $  {\mathcal O} $
and  $  \pi $
the valuation ring of  $  K $
and a uniformizer of  $  {\mathcal O} $, 
respectively. An  $  {\mathcal O} $-
lattice is a free  $  {\mathcal O} $-
submodule of  $  K  ^ {2} $
of rank  $  2 $. 
Then the Bruhat–Tits building of  $  { \mathop{\rm SL} } _ {2} $
is the topological realization of the following simplicial complex: the  $  0 $-
simplices are the  $  {\mathcal O} $-
lattices in  $  K  ^ {2} $
up to homothety and the  $  1 $-
simplices are pairs  $  L \subset  L  ^  \prime  $
of  $  {\mathcal O} $-
lattices in  $  K  ^ {2} $
up to homothety with  $  \pi L  ^  \prime  \subset  L $.

General references for Bruhat–Tits buildings are [[#References|[a1]]] and [[#References|[a2]]]. (The situations considered there are more general.) A good overview can be found in [[#References|[a3]]].

Originally, the Bruhat–Tits building was the essential technical tool for the classification of reductive groups over local fields (cf. [[Reductive group|Reductive group]]). There are further applications, e.g. in the representation theory of reductive groups over local fields, in the theory of p-adic symmetric spaces and in the theory of Shimura varieties.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Bruhat,   J. Tits,   "Groupes réductifs sur un corps local I, II"  ''IHES Publ. Math.'' , '''41,50'''  (1972–1984)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Landvogt,   "A compactification of the Bruhat–Tits building" , ''Lecture Notes in Mathematics'' , '''1619''' , Springer  (1996)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Tits,   "Reductive groups over local fields" , ''Proc. Symp. Pure Math.'' , '''33''' , Amer. Math. Soc.  (1979)  pp. 29–69</TD></TR></table>