Regular subgroup of a Lie group

Regular subsets of a Lie group are a class of discrete subsets of the Lie group that share a number of properties with discrete subsets in rank 1 Lie groups. (In particular, all discrete subsets of rank 1 Lie groups are regular.)

They are important in representation theory and differential geometry , among other things the term is used in the study of Anosov representations and Morse representations . In particular, the definition of the regularity part RCA-groups , which in the theory of group effects on symmetric spaces higher rank to from the theory of small shear groups known term convexo-kokompakter groups generalize.

definition

Let it be a symmetrical space of a non-compact type . Let it be a firmly chosen Weyl chamber in a maximum flat . There is a clear point for each ${\displaystyle X=G/K}$ ${\displaystyle \Delta \subset F}$ ${\displaystyle F\subset X}$ ${\displaystyle x\in X}$

${\displaystyle {\overline {x}}\in \Delta }$

in - orbit of . ${\displaystyle G}$ ${\displaystyle x}$

A sequence is called regular if ${\displaystyle x_{n}\in X}$

${\displaystyle \lim _{n\to \infty }d({\overline {x_{n}}},\partial \Delta )=\infty }$

applies.

A discrete subgroup is called regular if for each sequence and a the sequence ${\displaystyle \Gamma \subset G}$ ${\displaystyle (\gamma _{n}\in \Gamma )_{n\in \mathbb {N} }}$ ${\displaystyle x\in X}$

${\displaystyle x_{n}:=\gamma _{n}x\in X}$

is regular. (This definition is independent of the choice of the base point .) ${\displaystyle x\in X}$

It's called evenly regular if there's one with ${\displaystyle \epsilon >0}$

${\displaystyle d({\overline {\gamma x}},\partial \Delta )>\epsilon \parallel \gamma x\parallel }$

for everyone there. (This definition is also independent of the choice of the base point .) ${\displaystyle \gamma \in \Gamma }$ ${\displaystyle x\in X}$

Lie Theoretical Formulation

Algebraically, regularity can be defined using the Cartan decomposition and the exponential mapping as follows. ${\displaystyle {\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}}}$ ${\displaystyle exp_{x}\colon {\mathfrak {p}}\to G/K=X}$

Choose a base of simple roots . A sequence is regular if and only if ${\displaystyle \left\{\alpha _{1},\ldots ,\alpha _{r}\right\}\subset {\mathfrak {g}}}$ ${\displaystyle x_{n}\in X}$

${\displaystyle \lim _{n\to \infty }\alpha _{i}(x_{n})=\infty }$ For ${\displaystyle i=1,\ldots ,r}$

applies.

Examples

If a symmetric space is of rank , then every discrete subgroup is regular; this follows tautologically . ${\displaystyle X=G/K}$ ${\displaystyle rank(G/K)=1}$ ${\displaystyle \Gamma \subset G}$ ${\displaystyle \partial \Delta =\emptyset }$

Simple examples of regular subgroups for symmetric spaces of rank are obtained by means of Lie group homomorphisms of a Lie group with . For each discrete subgroup its picture is a regular subgroup of . ${\displaystyle X^{\prime }=G^{\prime }/K^{\prime }}$ ${\displaystyle \geq 2}$ ${\displaystyle \phi \colon G\to G^{\prime }}$ ${\displaystyle G}$ ${\displaystyle rank(G/K)=1}$ ${\displaystyle \Gamma \subset G}$ ${\displaystyle \Gamma ^{\prime }:=\phi (\Gamma )}$ ${\displaystyle G^{\prime }}$

There are numerous other examples of regular subgroups. Potrie-Sambarino have shown that the images of all Anosov representations (especially all hyperconvex representations ) are regular subsets of . ${\displaystyle PGL(d,\mathbb {R} )}$

literature

M. Kapovich, B. Leeb, J. Porti: Morse actions of discrete groups on symmetric spaces pdf

Individual evidence

1. ↑ R. Potrie, A. sambarino: Eigenvalues and entropy of a Hitchin representation pdf