- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- super abelian group
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

**Classical groups**

**Finite groups**

**Group schemes**

**Topological groups**

**Lie groups**

**Super-Lie groups**

**Higher groups**

**Cohomology and Extensions**

**Related concepts**

**∞-Lie theory** (higher geometry)

**Background**

*Smooth structure*

*Higher groupoids*

*Lie theory*

**∞-Lie groupoids**

**∞-Lie algebroids**

**Formal Lie groupoids**

**Cohomology**

**Homotopy**

**Related topics**

**Examples**

*$\infty$-Lie groupoids*

*$\infty$-Lie groups*

*$\infty$-Lie algebroids*

*$\infty$-Lie algebras*

In group theory, but particularly in Lie group-theory, the term “biquotient” tends to mean the quotient space of a topological group or Lie group $G$ by the action of *two* subgroups $H_1, H_2 \subset G$, hence by the action of their direct product group $H_1 \times H_2$, one factor regarded as acting by group multiplication from the left, the other (more precisely: its opposite) acting by multiplication from the right.

This is typically and suggestively denoted as

$H_1 \backslash G / H_2
\;\coloneqq\;
G/( H_1 \times H_2^{op} )
\;\coloneqq\;
G/( g \sim h_1 \cdot g \cdot h_2 \vert h_i \in H_i )
\,.$

Another way to think of a biquotient is as a double coset space, see there for more.

Typically extra conditions are imposed on $H_1, H_2 \subset G$, such as that $H_i \subset G$ are closed subgroups and notably that the induced action of $H_1$ on the single quotient space/coset space $G/H_2$ of the other, is still free.

More generally, one can consider biquotients of $G$ by subgroups of the direct product group (e.g. Kapovitch).

The Gromoll-Meyer sphere – an exotic 7-sphere – arises as the biquotient of Sp(2) by two copies of Sp(1).

For instance:

- Burt Totaro,
*Cheeger manifolds and the classification of biquotients*(arXiv:math/0210247)

- Vitali Kapovitch,
*A note on rational homotopy of biquotients*(pdf)

Created on April 27, 2019 at 15:37:53. See the history of this page for a list of all contributions to it.