arXiv subjects

Mikolaj Fraczyk

Publications and source records attributed to Mikolaj Fraczyk.

21 records · Page 2Linked to original sources

arXiv2018

Growth of mod$-2$ homology in higher rank locally symmetric spaces

Let $X$ be a higher rank symmetric space or a Bruhat-Tits building of dimension at least $2$ such that the isometry group of $X$ has property $(T)$. We prove that for every torsion free lattice $Γ\subset {\rm Isom} X$ any homology class in $H_1(Γ\backslash X,\mathbb F_2)$ has a representative cycle of total length $o_X({\rm Vol}(Γ\backslash X))$. As an application we show that $\dim_{\mathbb F_2} H_1(Γ\backslash X,\mathbb F_2)=o_X({\rm Vol}(Γ\backslash X)).$

math.AT↗

arXiv2018

Kesten's theorem for uniformly recurrent subgroups

We prove an inequality on the difference between the spectral radius of the Cayley graph of a group $G$ and the spectral radius of the Schreier graph $H\backslash G$ for any subgroup $H$. As an application we extend Kesten's theorem on spectral radii to uniformly recurrent subgroups and give a short proof that the result of Lyons and Peres on cycle density in Ramanujan graphs holds on average. More precisely, we show that if $\mathcal G$ is an infinite deterministic Ramanujan graph, then the time spent in short cycles by a random walk of length $n$ is $o(n)$.

math.GR↗

arXiv2014

The covering number of the difference sets in partitions of $G$-spaces and groups

We prove that for every finite partition $G=A_1\cup\dots\cup A_n$ of a group $G$ either $cov(A_iA_i^{-1})\le n$ for all cells $A_i$ or else $cov(A_iA_i^{-1}A_i)<n$ for some cell $A_i$ of the partition. Here $cov(A)=\min\{|F|:F\subset G,\;G=FA\}$ is the covering number of $A$ in $G$. A similar result is proved also of partitions of $G$-spaces. This gives two partial answers to a problem of Protasov posed in 1995.

math.CO↗