TP-Groups and ℓp Rigidity

• TP-groups are second-countable, locally compact groups endowed with Property (Tℓp), ensuring no sequence of almost invariant vectors exists in the non-invariant ℓp subspace.
• They are characterized via a Kazhdan-pair formulation and quasi-regular representations, reducing Banach-space isometries to concrete Hilbert-space settings.
• TP-groups exhibit strong rigidity traits that impact lattice structures, compact generation, and connections to other properties like Property τ and fixed-point behaviors.

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 (Bekka et al., 2013).

1. Definition and basic formulations

Let $G$ be a second-countable locally compact group, let $1<p<\infty$ with $1<p<\infty$0, and let $1<p<\infty$1 denote the group of linear bijective isometries of $1<p<\infty$2. An orthogonal representation of $1<p<\infty$3 on $1<p<\infty$4 is a continuous homomorphism

$1<p<\infty$5

meaning that $1<p<\infty$6 is continuous from $1<p<\infty$7 to $1<p<\infty$8 for each $1<p<\infty$9. The closed subspace of invariant vectors is

$p\neq2$0

and $p\neq2$1 denotes its closed complement (Bekka et al., 2013).

A sequence $p\neq2$2 is a sequence of almost invariant vectors if $p\neq2$3 for all $p\neq2$4 and

$p\neq2$5

for every compact $p\neq2$6. The group $p\neq2$7 has Property $p\neq2$8 if for every orthogonal representation $p\neq2$9, the restriction of $\ell_p=\ell_p(\mathbb N)$0 to $\ell_p=\ell_p(\mathbb N)$1 does not admit any sequence of almost invariant vectors (Bekka et al., 2013).

The property has an equivalent “Kazhdan-pair” form: there exists a compact set $\ell_p=\ell_p(\mathbb N)$2 and $\ell_p=\ell_p(\mathbb N)$3 such that

$\ell_p=\ell_p(\mathbb N)$4

for every orthogonal representation $\ell_p=\ell_p(\mathbb N)$5 and every unit vector $\ell_p=\ell_p(\mathbb N)$6. This places TP-groups within the general spectral-gap framework associated with rigidity properties of locally compact groups (Bekka et al., 2013).

2. Detection by quasi-regular representations

For totally disconnected, second-countable, locally compact groups, Property $\ell_p=\ell_p(\mathbb N)$7 is characterized by a minimal family of unitary representations. If $\ell_p=\ell_p(\mathbb N)$8 is open, the associated quasi-regular representation is

$\ell_p=\ell_p(\mathbb N)$9

Theorem 4 states that the following are equivalent for such a group $(T_{\ell_p})$0: $(T_{\ell_p})$1 has Property $(T_{\ell_p})$2, and the trivial representation $(T_{\ell_p})$3 is isolated from all quasi-regular representations of $(T_{\ell_p})$4 on $(T_{\ell_p})$5 as $(T_{\ell_p})$6 ranges over open subgroups (Bekka et al., 2013).

Equivalently, there exist a compact $(T_{\ell_p})$7 and $(T_{\ell_p})$8 such that for every open subgroup $(T_{\ell_p})$9 and every unit vector $(T)$0 orthogonal to the invariants,

$(T)$1

This is a representation-theoretic reduction from Banach-space isometries to a concrete family of Hilbert-space representations (Bekka et al., 2013).

The proof idea given in the source uses the Banach–Lamperti description of $(T)$2, reducing orthogonal $(T)$3-representations to an $(T)$4-sum of quasi-regular representations. The converse is obtained by showing that any monomial representation $(T)$5 for an open $(T)$6 and character $(T)$7 weakly sits inside some $(T)$8 with $(T)$9 open. This establishes that open-subgroup quasi-regular representations are the decisive testing family for totally disconnected TP-groups (Bekka et al., 2013).

A connected second-countable locally compact group $\tau$0 has Property $\tau$1 if and only if its abelianization $\tau$2 is compact. The same source further states that, in the connected case, classical Property $\tau$3 is equivalent to $\tau$4 (Bekka et al., 2013).

3. Structural consequences and relation to Kazhdan rigidity

Property $\tau$5 has strong structural consequences. If $\tau$6 has Property $\tau$7 for some $\tau$8, then $\tau$9 is compactly generated, the abelianization $(F)$0 is compact, Property $(F)$1 is invariant under passage to finite-index subgroups and overgroups, and if $(F)$2 is amenable and totally disconnected then $(F)$3 is compact (Bekka et al., 2013).

The comparison with classical Property $(F)$4 is asymmetric. Property $(F)$5 implies Property $(F)$6, but the converse fails in general. Both properties share compact generation and compact abelianization, but unlike classical Property $(F)$7, $(F)$8 need not pass to arbitrary lattices (Bekka et al., 2013).

This relation to Kazhdan rigidity places TP-groups within a larger representation-theoretic landscape. For locally compact groups, classical Property $(F)$9 is equivalent to Property $G$0 for the full group $G$1-algebra $G$2, and $G$3 then has strong Property $G$4. For locally compact IN-groups, classical Property $G$5 is equivalent to strong Property $G$6 of the reduced group $G$7-algebra $G$8 (Bekka et al., 2018). These results concern classical Property $G$9, not $1<p<\infty$0, 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 $1<p<\infty$1. For any reduced irreducible classical root system $1<p<\infty$2 of rank at least $1<p<\infty$3 and a finitely generated commutative ring $1<p<\infty$4 with $1<p<\infty$5, the Steinberg group $1<p<\infty$6 and the elementary Chevalley group $1<p<\infty$7 have Property $1<p<\infty$8 (Ershov et al., 2011). Since Property $1<p<\infty$9, 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 $1<p<\infty$00 is a non-archimedean local field and $1<p<\infty$01 is a connected simple $1<p<\infty$02-group, then

$1<p<\infty$03

is totally disconnected, non-amenable, and has the Howe–Moore property. By Theorem 8, such a group has Property $1<p<\infty$04 for every $1<p<\infty$05, $1<p<\infty$06. If $1<p<\infty$07, then $1<p<\infty$08 also has classical Property $1<p<\infty$09; if $1<p<\infty$10, the source identifies this as a genuine $1<p<\infty$11-example without classical $1<p<\infty$12 (Bekka et al., 2013).

Second, let $1<p<\infty$13 be a regular or bi-regular tree of degree at least $1<p<\infty$14, and let $1<p<\infty$15 with the compact-open topology. Then $1<p<\infty$16 is locally compact, totally disconnected, non-amenable, and has Howe–Moore. It follows that $1<p<\infty$17 has Property $1<p<\infty$18 for $1<p<\infty$19, yet $1<p<\infty$20 fails classical $1<p<\infty$21 (Bekka et al., 2013).

These examples are subsumed by a general Howe–Moore criterion: if $1<p<\infty$22 is non-amenable, totally disconnected, and has the Howe–Moore mixing property, then $1<p<\infty$23 has Property $1<p<\infty$24 for all $1<p<\infty$25 (Bekka et al., 2013). The significance of this criterion is that it isolates a dynamical mechanism—mixing in the sense of Howe–Moore—that is sufficient for $1<p<\infty$26-rigidity even when classical Property $1<p<\infty$27 fails.

5. Discrete groups, Property $1<p<\infty$28, and irreducible lattices

For discrete groups, Property $1<p<\infty$29 sits between other finitary rigidity notions. A discrete group $1<p<\infty$30 has Property $1<p<\infty$31 if the trivial representation is isolated among the family $1<p<\infty$32 for normal subgroups $1<p<\infty$33 of finite index. Proposition 10 states that if $1<p<\infty$34 has Property $1<p<\infty$35, then $1<p<\infty$36 has Property $1<p<\infty$37 (Bekka et al., 2013).

In the opposite direction, Glasner–Monod’s Property $1<p<\infty$38—every amenable action on a countable set has a fixed point—implies $1<p<\infty$39 for discrete groups. Proposition 15 gives this implication explicitly (Bekka et al., 2013). 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 $1<p<\infty$40 be second-countable locally compact groups and let $1<p<\infty$41 be an irreducible lattice. If $1<p<\infty$42 has classical Property $1<p<\infty$43 and $1<p<\infty$44 is connected and minimally almost periodic, then $1<p<\infty$45 has Property $1<p<\infty$46 for every $1<p<\infty$47. If $1<p<\infty$48 lacks $1<p<\infty$49, then neither does $1<p<\infty$50 (Bekka et al., 2013).

The proof strategy proceeds by contradiction: failure of $1<p<\infty$51 produces a family of open, hence finite-index, subgroups $1<p<\infty$52 whose quasi-regular representations admit a net of almost invariant vectors; these are assembled into

$1<p<\infty$53

A classical resolution theorem for irreducible lattices in $1<p<\infty$54 then forces a nonzero subrepresentation factoring through $1<p<\infty$55, and a density argument in $1<p<\infty$56 yields a contradiction (Bekka et al., 2013). This result shows that TP-rigidity survives in lattice settings beyond the standard inheritance theory of classical Property $1<p<\infty$57.

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 $1<p<\infty$58. This should be distinguished from the triple product property, where one studies triples $1<p<\infty$59 of non-empty subsets or subgroups of a finite group $1<p<\infty$60 satisfying

$1<p<\infty$61

together with associated capacities $1<p<\infty$62, $1<p<\infty$63 and ratios $1<p<\infty$64, $1<p<\infty$65 (Murthy, 18 Dec 2025).

That finite-group theory is motivated in part by the Cohn–Umans framework for fast matrix multiplication and includes upper bounds such as

$1<p<\infty$66

for groups with an abelian normal subgroup of prime index $1<p<\infty$67, as well as bounds for finite nilpotent groups of class $1<p<\infty$68, including $1<p<\infty$69 and, in various $1<p<\infty$70-group cases, $1<p<\infty$71 or $1<p<\infty$72 (Murthy, 18 Dec 2025, Murthy, 17 Feb 2026). Search and test algorithms for such TPP triples form another separate line of work, with intersection-based algorithms identified as especially effective in practice (Hedtke et al., 2011).

A plausible implication is that “TP-groups” can be misconstrued unless the underlying property is specified. In the $1<p<\infty$73 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.