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Ulla Karhumäki

Publications and source records attributed to Ulla Karhumäki.

10 recordsLinked to original sources
arXiv2026

The Borovik-Cherlin conjecture holds in ACF

We show that every faithful, transitive, and generically $(n+2)$-transitive action of a connected group $G$ on an irreducible variety $X$ of dimension $n > 0$, all defined over an algebraically closed field $F$, is isomorphic to the natural action of the projective linear group $PGL_{n+1}(F)$ on the projective space $\mathbb{P}^n(F)$. More precisely, we establish the Borovik-Cherlin conjecture for permutation groups $(G,X)$ definable in models of $ACF$.

math.GR↗
arXiv2025

Primitive pseudo-finite permutation groups of finite SU-rank

We study definably primitive pseudo-finite permutation groups of finite $SU$-rank. We show that if $(G,X)$ is such a permutation group, then the rank of $G$ can be bounded in terms of the rank of $X$, providing an analogue of a theorem of Borovik and Cherlin in the setting of definably primitive permutation groups of finite Morley rank.

math.LO↗
arXiv2024

Finite-dimensional pseudofinite groups of small dimension, without CFSG

Any simple pseudofinite group G is known to be isomorphic to a (twisted) Chevalley group over a pseudofinite field. This celebrated result mostly follows from the work of Wilson in 1995 and heavily relies on the classification of finite simple groups (CFSG). It easily follows that G is finite-dimensional with additive and fine dimension and, in particular, that if dim(G)=3 then G is isomorphic to PSL(2,F) for some pseudofinite field F. We describe pseudofinite finite-dimensional groups when the dimension is fine, additive and \<4 and, in particular, show that the classification G isomorphic to PSL(2,F) is independent from CFSG.

math.GR↗
arXiv2024

Fixed point subgroups of a supertight automorphism

Let $G$ be an infinite simple group of finite Morley rank and $α$ a supertight automorphism of $G$ so that the fixed point subgroup $P_n:=C_G(α^n)$ is pseudofinite for all $n\in \mathbb{N}\setminus\{0\}$. It is know (using CFSG) that the socle $S_n:={\rm Soc}(P_n)$ is a (twisted) Chevalley group over a pseudofinite field. We prove that there is $r\in \mathbb{N}\setminus\{0\}$ so that for each $n$ we have $[P_n:S_n] < r$ and that there is no $m \in \mathbb{N}\setminus \{0\}$ so that for each $n$ the sizes of the Sylow $2$-subgroups of $S_n$ are bounded by $m$. We also note that in the recent identification result of $G$ under the assumption ${\rm pr}_2(G)=1$, the use of CFSG is not needed.

math.GR↗
arXiv2023

Small groups of finite Morley rank with a supertight automorphism

Let $G$ be an infinite simple group of finite Morley rank and of Prüfer $2$-rank $1$ which admits a supertight automorphism $α$ such that the fixed-point subgroup $C_G(α^n)$ is pseudofinite for all integers $n > 0$. We prove that $G \cong {\rm PGL}_2(K)$ for some algebraically closed field $K$ of characteristic $\neq 2$.

math.GR↗
arXiv2019

Definably simple stable groups with finitary groups of automorphisms

We prove that infinite definably simple locally finite groups of finite centraliser dimension are simple groups of Lie type over locally finite fields. Then, we identify conditions on automorphisms of a stable group that make it resemble the Frobenius maps, and allow us to classify definably simple stable groups in the specific case when they admit such automorphisms.

math.LO↗
arXiv2016

Online Exploration of Rectangular Grids

In this paper, we consider the problem of exploring unknown environments with autonomous agents. We model the environment as a graph with edge weights and analyze the task of visiting all vertices of the graph at least once. The hardness of this task heavily depends on the knowledge and the capabilities of the agent. In our model, the agent sees the whole graph in advance, but does not know the weights of the edges. As soon as it arrives in a vertex, it can see the weights of all the outgoing edges. We consider the special case of two different edge weights $1$ and $k$ and prove that the problem remains hard even in this case. We prove a lower bound of $11/9$ on the competitive ratio of any deterministic strategy for exploring a ladder graph and complement this result by a $4$-competitive algorithm. All of these results hold for undirected graphs. Exploring directed graphs, where the direction of the edges is not known beforehand, seems to be much harder. Here, we prove that a natural greedy strategy has a linear lower bound on the competitive ratio both in ladders and square grids.

cs.CC↗