Searcharxiv⌕ Search

arXiv subjects

Paweł Piwek

Publications and source records attributed to Paweł Piwek.

7 recordsLinked to original sources
arXiv2024

Profinite rigidity for free-by-cyclic groups with centre

A free-by-cyclic group $F_N\rtimes_ϕ\mathbb{Z}$ has non-trivial centre if and only if $[ϕ]$ has finite order in ${\rm{Out}}(F_N)$. We establish a profinite ridigity result for such groups: if $Γ_1$ is a free-by-cyclic group with non-trivial centre and $Γ_2$ is a finitely generated free-by-cyclic group with the same finite quotients as $Γ_1$, then $Γ_2$ is isomorphic to $Γ_1$. One-relator groups with centre are similarly rigid. We prove that finitely generated free-by-(finite cyclic) groups are profinitely rigid in the same sense; the proof revolves around a finite poset $\mathbf{fsc}(G)$ that carries information about the centralisers of finite subgroups of $G$ -- it is a complete invariant for these groups. These results provide contrasts with the lack of profinite rigidity among surface-by-cyclic groups and (free abelian)-by-cyclic groups, as well as general virtually-free groups.

math.GR↗
arXiv2024

Solvable and non-solvable finite groups of the same order type

We construct two finite groups of size $2^{365}\cdot 3^{105}\cdot 7^{104}$: a solvable group $G$ and a non-solvable group $H$, such that for every integer $n$ the groups have the same number of elements of order $n$. This answers a question posed in 1987 by John G. Thompson.

math.GR↗
arXiv2023

Lack of profinite rigidity among extensions with free quotient

We present a construction that yields infinite families of non-isomorphic semidirect products $N \rtimes F_m$ sharing a specified profinite completion. Within each family, $m \ge 2$ is constant and $N$ is a fixed group. For $m=2$ we can take $N$ to be free of rank $\ge 10$, free abelian of rank $\ge 12$, or a surface group of genus $\ge 5$.

math.GR↗
arXiv2023

Profinite rigidity properties of central extensions of 2-orbifold groups

We extend Wilkes' results on the profinite rigidity of SFSs to the setting of central extensions of 2-orbifold groups with higher-rank centre. We prove that both rigid and non-rigid phenomena arise in this setting and that the non-rigid phenomena are transient in the sense that if $\widehat{G}_1 \cong \widehat{G}_2$, then $G_1 \times \mathbb{Z} \cong G_2 \times \mathbb{Z}$.

math.GR↗
arXiv2023

Exact Count of Boundary Pieces of ReLU Classifiers: Towards the Proper Complexity Measure for Classification

Classic learning theory suggests that proper regularization is the key to good generalization and robustness. In classification, current training schemes only target the complexity of the classifier itself, which can be misleading and ineffective. Instead, we advocate directly measuring the complexity of the decision boundary. Existing literature is limited in this area with few well-established definitions of boundary complexity. As a proof of concept, we start by analyzing ReLU neural networks, whose boundary complexity can be conveniently characterized by the number of affine pieces. With the help of tropical geometry, we develop a novel method that can explicitly count the exact number of boundary pieces, and as a by-product, the exact number of total affine pieces. Numerical experiments are conducted and distinctive properties of our boundary complexity are uncovered. First, the boundary piece count appears largely independent of other measures, e.g., total piece count, and $l_2$ norm of weights, during the training process. Second, the boundary piece count is negatively correlated with robustness, where popular robust training techniques, e.g., adversarial training or random noise injection, are found to reduce the number of boundary pieces.

stat.ML↗
arXiv2019

Investigating transversals as generating sets for groups

In [3] is was shown that for any group $G$ whose rank (i.e., minimal number of generators) is at most 3, and any finite index subgroup $H\leq G$ with index $[G:H]\geq rank(G)$, one can always find a left-right transversal of $H$ which generates $G$. In this paper we extend this result to groups of rank at most 4. We also extend this to groups $G$ of arbitrary (finite) rank $r$ provided all the non-trivial divisors of $[G:core_G(H)]$ are at least $2r-1$. Finally, we extend this to groups $G$ of arbitrary (finite) rank provided $H$ is malnormal in $G$.

math.GR↗