---
title: TP-Groups and ℓp Rigidity
url: https://www.emergentmind.com/topics/tp-groups
type: topic
---

# TP-Groups and ℓp Rigidity

In the terminology summarized here, TP-groups are second-countable locally compact groups with Property \((T_{\ell_p})\), where \(1<p<\infty\) and \(p\neq2\). This property is formulated in terms of orthogonal representations on the Banach space \(\ell_p=\ell_p(\mathbb N)\) and requires that, after removing the invariant vectors, no sequence of almost invariant unit vectors remains. Property \((T_{\ell_p})\) is presented as a weak version of Kazhdan’s Property \((T)\), but it retains a substantial rigidity content: it admits a Kazhdan-pair formulation, has a sharp characterization for totally disconnected groups through quasi-regular representations associated to open subgroups, and interacts closely with Property \(\tau\), Property \((F)\), and the theory of irreducible lattices [1303.5183].

## 1. Definition and basic formulations

Let \(G\) be a second-countable locally compact group, let \(1<p<\infty\) with \(p\neq2\), and let \(O(\ell_p)\) denote the group of linear bijective isometries of \(\ell_p\). An orthogonal representation of \(G\) on \(\ell_p\) is a continuous homomorphism
\[
\pi:G\to O(\ell_p),
\]
meaning that \(g\mapsto \pi(g)f\) is continuous from \(G\) to \(\ell_p\) for each \(f\in\ell_p\). The closed subspace of invariant vectors is
\[
\ell_p^{\pi(G)}=\{f\in\ell_p:\pi(g)f=f\ \forall\,g\in G\},
\]
and \(\ell_p'(\pi)\) denotes its closed complement [1303.5183].

A sequence \((f_n)\subset \ell_p\) is a sequence of almost invariant vectors if \(\|f_n\|=1\) for all \(n\) and
\[
\lim_{n\to\infty}\sup_{g\in K}\|\pi(g)f_n-f_n\|=0
\]
for every compact \(K\subset G\). The group \(G\) has Property \((T_{\ell_p})\) if for every orthogonal representation \(\pi:G\to O(\ell_p)\), the restriction of \(\pi\) to \(\ell_p'(\pi)\) does not admit any sequence of almost invariant vectors [1303.5183].

The property has an equivalent “Kazhdan-pair” form: there exists a compact set \(Q\subset G\) and \(\varepsilon>0\) such that
\[
\sup_{g\in Q}\|\pi(g)f-f\|\ge \varepsilon
\]
for every orthogonal representation \(\pi\) and every unit vector \(f\in\ell_p'(\pi)\). This places TP-groups within the general spectral-gap framework associated with rigidity properties of locally compact groups [1303.5183].

## 2. Detection by quasi-regular representations

For totally disconnected, second-countable, locally compact groups, Property \((T_{\ell_p})\) is characterized by a minimal family of unitary representations. If \(H\le G\) is open, the associated quasi-regular representation is
\[
\lambda_{G/H}:G\to U(\ell_2(G/H)),\quad \lambda_{G/H}(g)(f)(x)=f(g^{-1}x).
\]
Theorem 4 states that the following are equivalent for such a group \(G\): \(G\) has Property \((T_{\ell_p})\), and the trivial representation \(1_G\) is isolated from all quasi-regular representations of \(G\) on \(\ell_2(G/H)\) as \(H\) ranges over open subgroups [1303.5183].

Equivalently, there exist a compact \(Q\subset G\) and \(\varepsilon>0\) such that for every open subgroup \(H\) and every unit vector \(f\in\ell_2(G/H)\) orthogonal to the invariants,
\[
\sup_{g\in Q}\|\lambda_{G/H}(g)f-f\|\ge\varepsilon.
\]
This is a representation-theoretic reduction from Banach-space isometries to a concrete family of Hilbert-space representations [1303.5183].

The proof idea given in the source uses the Banach–Lamperti description of \(O(\ell_p)\), reducing orthogonal \(\ell_p\)-representations to an \(\ell_2\)-sum of quasi-regular representations. The converse is obtained by showing that any monomial representation \(\mathrm{Ind}_H^G\chi\) for an open \(H\) and character \(\chi\) weakly sits inside some \(\ell_2(G/L)\) with \(L\le H\) open. This establishes that open-subgroup quasi-regular representations are the decisive testing family for totally disconnected TP-groups [1303.5183].

A connected second-countable locally compact group \(G\) has Property \((T_{\ell_p})\) if and only if its abelianization \(G/[G,G]\) is compact. The same source further states that, in the connected case, classical Property \((T)\) is equivalent to \((T_{\ell_p})\) [1303.5183].

## 3. Structural consequences and relation to Kazhdan rigidity

Property \((T_{\ell_p})\) has strong structural consequences. If \(G\) has Property \((T_{\ell_p})\) for some \(1<p\neq2\), then \(G\) is compactly generated, the abelianization \(G/[G,G]\) is compact, Property \((T_{\ell_p})\) is invariant under passage to finite-index subgroups and overgroups, and if \(G\) is amenable and totally disconnected then \(G\) is compact [1303.5183].

The comparison with classical Property \((T)\) is asymmetric. Property \((T)\) implies Property \((T_{\ell_p})\), but the converse fails in general. Both properties share compact generation and compact abelianization, but unlike classical Property \((T)\), \((T_{\ell_p})\) need not pass to arbitrary lattices [1303.5183].

This relation to Kazhdan rigidity places TP-groups within a larger representation-theoretic landscape. For locally compact groups, classical Property \((T)\) is equivalent to Property \((T)\) for the full group \(C^*\)-algebra \(C^*(G)\), and \(C^*(G)\) then has strong Property \((T)\). For locally compact IN-groups, classical Property \((T)\) is equivalent to strong Property \((T)\) of the reduced group \(C^*\)-algebra \(C_r^*(G)\) [1803.10933]. These results concern classical Property \((T)\), not \((T_{\ell_p})\), but they clarify the rigidity background against which TP-groups are studied.

A substantial source of examples comes from groups already known to have Kazhdan’s Property \((T)\). For any reduced irreducible classical root system \(\Phi\) of rank at least \(2\) and a finitely generated commutative ring \(R\) with \(1\), the Steinberg group \(\mathrm{St}_\Phi(R)\) and the elementary Chevalley group \(\mathrm{E}_\Phi(R)\) have Property \((T)\) [1102.0031]. Since Property \((T)\Rightarrow(T_{\ell_p})\), this suggests a broad supply of TP-groups inside the theory of root-graded and Chevalley-type groups.

## 4. Principal examples

Two primary classes of examples are singled out. First, if \(k\) is a non-archimedean local field and \(\mathbf G\) is a connected simple \(k\)-group, then
\[
G=\mathbf G(k)
\]
is totally disconnected, non-amenable, and has the Howe–Moore property. By Theorem 8, such a group has Property \((T_{\ell_p})\) for every \(1\le p<\infty\), \(p\neq2\). If \(\mathrm{rank}_k\mathbf G\ge2\), then \(G\) also has classical Property \((T)\); if \(\mathrm{rank}_k\mathbf G=1\), the source identifies this as a genuine \((T_{\ell_p})\)-example without classical \((T)\) [1303.5183].

Second, let \(T\) be a regular or bi-regular tree of degree at least \(3\), and let \(G=\mathrm{Aut}(T)\) with the compact-open topology. Then \(G\) is locally compact, totally disconnected, non-amenable, and has Howe–Moore. It follows that \(G\) has Property \((T_{\ell_p})\) for \(p\neq2\), yet \(G\) fails classical \((T)\) [1303.5183].

These examples are subsumed by a general Howe–Moore criterion: if \(G\) is non-amenable, totally disconnected, and has the Howe–Moore mixing property, then \(G\) has Property \((T_{\ell_p})\) for all \(p\neq2\) [1303.5183]. The significance of this criterion is that it isolates a dynamical mechanism—mixing in the sense of Howe–Moore—that is sufficient for \(\ell_p\)-rigidity even when classical Property \((T)\) fails.

## 5. Discrete groups, Property \(\tau\), and irreducible lattices

For discrete groups, Property \((T_{\ell_p})\) sits between other finitary rigidity notions. A discrete group \(\Gamma\) has Property \(\tau\) if the trivial representation is isolated among the family \(\{\ell_2(\Gamma/H)\}\) for normal subgroups \(H\triangleleft\Gamma\) of finite index. Proposition 10 states that if \(\Gamma\) has Property \((T_{\ell_p})\), then \(\Gamma\) has Property \(\tau\) [1303.5183].

In the opposite direction, Glasner–Monod’s Property \((F)\)—every amenable action on a countable set has a fixed point—implies \((T_{\ell_p})\) for discrete groups. Proposition 15 gives this implication explicitly [1303.5183]. Accordingly, TP-groups in the discrete category inherit a notable part of the expansion and fixed-point behavior traditionally associated with stronger rigidity properties.

The lattice construction in Theorem 12 is especially significant. Let \(G_1,G_2\) be second-countable locally compact groups and let \(\Gamma\subset G_1\times G_2\) be an irreducible lattice. If \(G_1\) has classical Property \((T)\) and \(G_2\) is connected and minimally almost periodic, then \(\Gamma\) has Property \((T_{\ell_p})\) for every \(1<p\neq2\). If \(G_2\) lacks \((T)\), then neither does \(\Gamma\) [1303.5183].

The proof strategy proceeds by contradiction: failure of \((T_{\ell_p})\) produces a family of open, hence finite-index, subgroups \(H_i\le\Gamma\) whose quasi-regular representations admit a net of almost invariant vectors; these are assembled into
\[
\Pi=\bigoplus_i \ell_2(\Gamma/H_i).
\]
A classical resolution theorem for irreducible lattices in \(G_1\times G_2\) then forces a nonzero subrepresentation factoring through \(G_2\), and a density argument in \(G_2\) yields a contradiction [1303.5183]. This result shows that TP-rigidity survives in lattice settings beyond the standard inheritance theory of classical Property \((T)\).

## 6. Terminological distinctions and adjacent uses of “TP”

The abbreviation “TP” is not unique across group theory. In the present usage, TP-groups are groups with Property \((T_{\ell_p})\). This should be distinguished from the triple product property, where one studies triples \((S,T,U)\) of non-empty subsets or subgroups of a finite group \(G\) satisfying
\[
s's^{-1}t't^{-1}u'u^{-1}=1 \iff s=s',\ t=t',\ u=u',
\]
together with associated capacities \(\beta(G)\), \(\beta_0(G)\) and ratios \(\rho(G)\), \(\rho_0(G)\) [2512.16730].

That finite-group theory is motivated in part by the Cohn–Umans framework for fast matrix multiplication and includes upper bounds such as
\[
\rho_0(G)\le \frac{p^2}{2p-1}
\]
for groups with an abelian normal subgroup of prime index \(p\), as well as bounds for finite nilpotent groups of class \(2\), including \(\rho_0(G)<\sqrt{|G:Z(G)|}\) and, in various \(p\)-group cases, \(\rho_0(G)\le p\) or \(\rho_0(G)=1\) [2512.16730; 2602.15796]. Search and test algorithms for such TPP triples form another separate line of work, with intersection-based algorithms identified as especially effective in practice [1104.5097].

A plausible implication is that “TP-groups” can be misconstrued unless the underlying property is specified. In the \((T_{\ell_p})\) literature, the subject is Banach-space rigidity for locally compact groups; in the TPP literature, the subject is a combinatorial condition on triples of subsets of finite groups. The two theories use similar initials but address different mathematical problems.

Source: https://www.emergentmind.com/topics/tp-groups