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Piotr Mizerka

Publications and source records attributed to Piotr Mizerka.

10 recordsLinked to original sources
arXiv2025

Sampling elements of a finite group: efficiency of the product replacement algorithm with an accumulator

Let $G$ be a finite group generated by $k$ elements. The well-known product replacement algorithm provides an effective method for sampling generating sets of $G$. We study a refinement of this algorithm that is designed to output individual elements of $G$. We show that after $O(k^2\log|G|)$ steps, the distribution of the output is close to uniform on $G$, which improves upon the best results known to date. The proof proceeds via spectral gap estimates and uses computer assisted calculations.

math.GR↗
arXiv2025

Inducing spectral gaps for the cohomological Laplacians of $\operatorname{Sp}_{2n}(\mathbb{Z})$

We show that the spectral gap of the first cohomological Laplacian $\Delta_1$ for $\operatorname{Sp}_{2n}(\mathbb{Z})$ follows once a slightly stronger assumption holds for some $\operatorname{Sp}_{2m}(\mathbb{Z})$, where $n\geq m$. As an application of this result, we provide explicit lower bounds for some quotients of $\operatorname{Sp}_{2n}(\mathbb{Z})$ for any $n\geq 2$.

math.GR↗
arXiv2024

Non-vanishing unitary cohomology of low-rank integral special linear groups

We construct explicit finite-dimensional orthogonal representations $\pi_N$ of $\operatorname{SL}_{N}(\mathbb{Z})$ for $N \in \{3,4\}$ all of whose invariant vectors are trivial, and such that $H^{N - 1}(\operatorname{SL}_{N}(\mathbb{Z}),\pi_N)$ is non-trivial. This implies that for $N$ as above, the group $\operatorname{SL}_{N}(\mathbb{Z})$ does not have property $(T_{N-1})$ of Bader-Sauer and therefore is not $(N-1)$-Kazhdan in the sense of De Chiffre-Glebsky-Lubotzky-Thom, both being higher versions of Kazhdan's property $T$.

math.GR↗
arXiv2024

Inducing spectral gaps for the cohomological Laplacians of $\operatorname{SL}_n(\mathbb{Z})$ and $\operatorname{SAut}(F_n)$

The technique of inducing spectral gaps for cohomological Laplacians in degree zero was used by Kaluba, Kielak and Nowak to prove property (T) for $\operatorname{SAut}(F_n)$ and $\operatorname{SL}_n(\mathbb{Z})$. In this paper, we adapt this technique to Laplacians in degree one. This allows to provide a lower bound for the cohomological Laplacian in degree one for $\operatorname{SL}_n(\mathbb{Z})$ for every unitary representation. In particular, one gets in that way an alternative proof of property (T) for $\operatorname{SL}_n(\mathbb{Z})$ whenever $n\geq 3$.

math.GR↗
arXiv2023

On order units in the augmentation ideal

We study order units in the real group ring and the augmentation ideal, as well as in matrix algebras. We identify an infinite family of order units in the powers of the augmentation ideal, that includes the Laplacian, and show that these order units are naturally obtained via cohomological operations from more simpler diagonal order units in matrix algebras.

math.GR↗
arXiv2021

Exclusions of smooth actions on spheres of the non-split extension of $C_2$ by $SL(2,5)$

There are four groups $G$ fitting into a short exact sequence $ 1\rightarrow SL(2,5)\rightarrow G\rightarrow C_2\rightarrow 1, $ where $SL(2,5)$ is the special linear group of $(2\times 2)$-matrices with entries in the field of five elements. Except for the direct product of $SL(2,5)$ and $C_2$, there are two other semidirect products of these two groups and just one non-semidirect product $SL(2,5).C_2$, considered in this paper. It is known that each finite nonsolvable group can act on spheres with arbitrary positive number of fixed points. Clearly, $SL(2,5).C_2$ is a nonsolvable group. Moreover, it turns out that $SL(2,5).C_2$ possesses a free representation and as such, can potentially act pseudofreely with nonempty fixed point set on manifolds of arbitrarily large dimension. We prove that $SL(2,5).C_2$ cannot act effectively with odd number of fixed points on low-dimensional spheres. In the special case of effective one fixed point actions, we are able to exclude a wider class of spheres. Moreover, we prove that specific pseudofree one fixed point actions of $SL(2,5).C_2$ on spheres do not exist.

math.GT↗
arXiv2020

A new family of finite Oliver groups satisfying the Laitinen Conjecture

This paper is concerned with the Laitinen Conjecture. The conjecture predicts an answer to the Smith question which reads as follows. Is it true that for a finite group acting smoothly on a sphere with exactly two fixed points, the tangent spaces at the fixed points have always isomorphic group module structures defined by differentiation of the action? Using the technique of induction of group representations, we indicate a new infinite family of finite Oliver groups for which the Laitinen Conjecture holds.

math.GR↗
arXiv2020

Inducing of exotic smooth two fixed point actions on spheres

This paper is concerned with the Smith question which reads as follows. Is it true that for a finite group acting smoothly on a sphere with exactly two fixed points, the tangent spaces at the fixed points have always isomorphic group module structures defined by differentiation of the action? We show that one can answer this question negatively by using the technique of induction of group representations. We apply our results to indicate new dimensions of spheres admitting actions of specific Oliver groups, which give the negative answer to the Smith question. In particular, for the first time, we indicate some solvable non-nilpotent Oliver groups which yield negative answers to the Smith question.

math.GT↗