Polynomial Sequence

Filtration

Let $G$ be a group, a prefiltration on $G$, $G_{\cdot }=(G_{(i)}{)}_{i=0}^{\infty }$ is a rested sequence $G=G_{(0)}\supseteq G_{(1)}\supseteq G_{(2)}\dots$ of subgroups of $G$. It’s a filtration if $[G_{(i)},G_{(j)}]\subseteq G_{(i+j)},\forall i,j\ge 0$ and $G_{(0)}=G_{(1)}$. It’s a class (at most) $S$ if $G_{(s+1)}=G_{(s+2)}\cdots =\{e_G\}$. $[H,K]:=\text{group genertaed by}[h,k],h\in H,k\in K,[x,y]=x^{-1}{y}^{-1}xy$

Example: An example for this filtration would be lower central series $G_0=G_1=G$, $G_{i+1}=[G,G_i]$. (It’s the minimal filtration on G).

Remark

If we take ${\mathfrak{g}}_i=G_{(i)}/G_{(i+1)}$, then each ${\mathfrak{g}}_i$ is an abelian group and $\mathfrak{g}={⨁}_{i=1}^{\infty }{\mathfrak{g}}_i$ is a graded Lie algebra. Recall that a group $G$ is nilpotent of class $\le s$ if $G_{s+1}=\{e_G\}$, with $G_i$ defined as the lower central series, so from definition we get a group $G$ with above filtration of class $s$ is nilpotent of class $\le s$.

Polynomial Sequence

We’ll only interested in proper filtrations in which $G$ is a simply-connected Lie group and each $G_{(i)}$ is closed subgroup.

Polynomial Sequences

Let $G_{\cdot }=(G_{(i)}{)}_{i=0}^{\infty }$ be a filtration of finite class, let $p_i(n)$ to be an integer valued polynomial, that is $p_i(n):\mathbb{Z}\to \mathbb{Z}$ The group $\text{poly}(\mathbb{Z},G_{\cdot })$ is the group generated by sequences of the form $g_j^{p_i(n)}$ when $g_i\in G_{\text{deg}(p_i)}$, we define $p:\mathbb{Z}\to G$ to be a polynomial sequence if $p\in \text{poly}(\mathbb{Z},G_{\cdot })$.

Leibman's Theorem (Host, Kra)

Let $G_{\cdot }$ be a prefiltration and suppose $p:\mathbb{Z}\to G$. Then the following are equivalent: (i) $p$ is a polynomial sequence as defined above i.e. $p(n)\in \text{poly}(\mathbb{Z},G_{\cdot })$. (ii) ${\partial }_{h_1}\dots {\partial }_{h_k}p(n)$ takes values in $G_{(k)}$ all $k\ge 1$ and all $h_1,\dots ,h_k,n\in \mathbb{Z}$. where ${\partial }_{h}p(n)=p(n)p(n+h{)}^{-1}$

Lattice and automorphic functions

• Let $G$ be a locally compact topological group, $\Gamma \le G$ is a Lattice if it’s discrete and cocompact.
• We say that the filtration $G_{\cdot }$ is rational if $\Gamma \cap G_{(i)}$ is a lattice in $G_{(i)}$ for all $i$.
• A smooth $\Gamma$-automorphism function is some $\phi \in C^{\infty }(G)$ satisfying $\phi (\gamma g)=\phi (g)$