arXiv subjects

Pierre-Nicolas Jolissaint

Publications and source records attributed to Pierre-Nicolas Jolissaint.

3 recordsLinked to original sources

arXiv2015

Functions conditionally of negative type on groups acting on regular trees

Let $\mathcal{T}_{q+1}$ be the $(q+1)$-regular tree and let $G$ be a group of automorphisms acting transitively on the vertices and on the boundary of $\mathcal{T}_{q+1}$. We give an upper bound for the growth of cocycles with values in any unitary representation of the group $G$. This bound is optimal by projecting the Haagerup cocycle onto an appropriate subspace of $\ell^{2}(E)$. We also obtain a description of functions conditionally of negative type which are unbounded.

math.GR↗

arXiv2013

$\ell^p$-distortion and $p$-spectral gap of finite regular graphs

We give a lower bound for the $\ell^p$-distortion $c_p(X)$ of finite graphs $X$, depending on the first eigenvalue $λ_1^{(p)}(X)$ of the $p$-Laplacian and the maximal displacement of permutations of vertices. For a $k$-regular vertex-transitive graph it takes the form $c_p(X)^{p}\geq diam(X)^{p}λ_{1}^{(p)}(X)/2^{p-1}k$. This bound is optimal for expander families and, for $p=2$, it gives the exact value for cycles and hypercubes. As a new application we give a non-trivial lower bound for the $\ell^2$-distortion of a family of Cayley graphs of $SL_n(q)$ ($q$ fixed, $n\geq 2$) with respect to a standard two-element generating set.

math.MG↗

arXiv2013

$L^p$ compression of some HNN extensions

Using recent developments on locally compact groups, we are able to obtain quantitative results on embeddings into Lebesgue spaces for a large class of HNN extensions.

math.GR↗