Random Triangular Groups

Żuk’s criterion for Banach spaces and random groups

“…The graph (𝑉, 𝐸) is called bipartite if V can be partitioned into two disjoint sets 𝑉 = 𝑆 1 ∪ 𝑆 2 called sides such that for each {𝑢, 𝑣} ∈ 𝐸, |{𝑢, 𝑣} ∩ 𝑆 1 | = |{𝑢, 𝑣} ∩ 𝑆 2 | = 1 (i.e., each edge has exactly one vertex in each side). We note that it follows that 𝑚(𝑆 1 ) = 𝑚(𝑆 2 ) = 1 2 𝑚(𝑉). We also define ℓ 2 (𝑉, 𝑚) as in section 2.2 above; that is, ℓ 2 (𝑉, 𝑚) is the space of functions 𝜙 : 𝑉 → C with an inner-product…”

Banach Zuk's criterion for partite complexes with application to random groups

“…For this, we will need to bound the second eigenvalue of the random walk on the link of the Cayley complex of a group in M + (m, ρ). Our approach for bounding the second eigenvalue is heavily based on [ALuS15] and [dLdlS21] and we claim very little originality here (a few adaptations were needed in order to provide a sharp result in the triangular positive model). In order to bound the second eigenvalue, we will first need some general lemmata regarding graphs and the bipartite Erdös-Rényi graph.…”

“…After this set-up, we argue as in [ALuS15]: Let Γ ∈ M + (m, ρ). The relations of Γ are of the form s i s j s k and thus the Cayley complex of Γ is a two-dimensional simplicial complex on which Γ acts transitively on the vertices.…”

“…Note that we allow double edges, i.e., if s i s j s k , s i s j s k ′ are two relations of Γ, then the edge {s j , s −1 i } will appear twice. The idea of [ALuS15] is that each of these graphs behave like G bipartite (m, ρ ′ ) where ρ ′ ≥ ρ and thus by Lemma 4.11 and Theorem 4.13, we can bound the second eigenvalue of the link. However, some analysis is needed to account for double edges and this will require Lemma 4.14.…”

“…Our proof of Theorem 1.5 follows the scheme of proof [KK13, Theorem B]: first, we reduce the problem to another model of random groups in which the Cayley complex of a random group is a partite simplicial complex and then we apply our version of Żuk's criterion (this is not completely straightforward and we heavily use the ideas of [ALuS15] in the analysis).…”