A tuple (or subgroup) in a group is said to degenerate to another if the latter is an endomorphic image of the former. In a countable reduced abelian group, it is shown that if tuples (or finite subgroups) degenerate to each other, then they lie in the same automorphism orbit. The proof is based on techniques that were developed by Kaplansky and Mackey in order to give an elegant proof of Ulm's theorem. Similar results hold for reduced countably generated torsion modules over principal ideal domains. It is shown that the depth and the description of atoms of the resulting poset of orbits of tuples depend only on the Ulm invariants of the module in question (and not on the underlying ring). A complete description of the poset of orbits of elements in terms of the Ulm invariants of the module is given. The relationship between this description of orbits and a very different-looking one obtained by Dutta and Prasad for torsion modules of bounded order is explained.
For a subgroup $L$ of the symmetric group $S_\ell$, we determine the minimal base size of $GL_d(q)\wr L$ acting on $V_d(q)^\ell$ as an imprimitive linear group. This is achieved by computing the number of orbits of $GL_d(q)$ on spanning $m$-tuples, which turns out to be the number of $d$-dimensional subspaces of $V_m(q)$. We then use these results to prove that for certain families of subgroups $L$, the affine groups whose stabilisers are large subgroups of $GL_d(q)\wr L$ satisfy a conjecture of Pyber concerning bases.
We prove that for a connected, semisimple linear Lie group $G$ the spaces of generating pairs of elements or subgroups are well-behaved in a number of ways: the set of pairs of elements generating a dense subgroup is Zariski-open in the compact case, Euclidean-open in general, and always dense. Similarly, for sufficiently generic circle subgroups $H_i$, $i=1,2$ of $G$, the space of conjugates of $H_i$ that generate a dense subgroup is always Zariski-open and dense. Similar statements hold for pairs of Lie subalgebras of the Lie algebra $Lie(G)$.
We study Morse representations of discrete subgroups in higher rank semi-simple Lie groups defined by M. Kapovich, B. Leeb and J. Porti. We show that, if a sequence of Morse representations $ρ_n : Γ\rightarrow G$ is (strongly) unbounded in the character variety, the group must have a very particular structure.
Given a GGS-group $G$ with non-constant defining tuple over a prime-regular rooted tree, we calculate the indices $|G:G^{(n)}|$ and describe the structure of the higher derived subgroups $G^{(n)}$ for all $n \in \mathbb{N}$. We find that the values $|G:G^{(n)}|$ depend only mildly on the structure of the defining tuple.
Generalising Solomon's theorem, C. Gordon and F. Rodriguez-Villegas have proven recently that, in any group, the number of solutions to a system of coefficient-free equations is divisible by the order of this group whenever the rank of the matrix composed of the exponent sums of i-th unknown in j-th equation is less than the number of unknowns. We generalise this result in two directions: first, we consider equations with coefficients, and secondly, we consider not only systems of equations but also any first-order formulae in the group language (with constants). Our theorem implies some amusi…
An action of a group $G$ is highly transitive if $G$ acts transitively on $k$-tuples of distinct points for all $k \geq 1$. Many examples of groups with a rich geometric or dynamical action admit highly transitive actions. We prove that if a group $G$ admits a highly transitive action such that $G$ does not contain the subgroup of finitary alternating permutations, and if $H$ is a confined subgroup of $G$, then the action of $H$ remains highly transitive, possibly after discarding finitely many points. This result provides a tool to rule out the existence of highly transitive actions, and to c…
Let $G \simeq M \rtimes C$ be an $n$-generator group with $M$ Abelian and $C$ cyclic. We study the Nielsen equivalence classes and T-systems of generating $n$-tuples of $G$. The subgroup $M$ can be turned into a finitely generated faithful module over a suitable quotient $R$ of the integral group ring of $C$. When $C$ is infinite, we show that the Nielsen equivalence classes of the generating $n$-tuples of $G$ correspond bijectively to the orbits of unimodular rows in $M^{n -1}$ under the action of a subgroup of $GL_{n - 1}(R)$. Making no assumption on the cardinality of $C$, we exhibit a comp…
This paper studies the generic behavior of $k$-tuple elements for $k\ge 2$ in a proper group action with contracting elements, with applications towards relatively hyperbolic groups, CAT(0) groups and mapping class groups. For a class of statistically convex-cocompact action, we show that an exponential generic set of $k$ elements for any fixed $k\ge 2$ generates a quasi-isometrically embedded free subgroup of rank $k$. For $k=2$, we study the sprawl property of group actions and establish that the class of statistically convex-cocompact actions is statistically hyperbolic in a sense of M. Duc…
Let $G$ be a finite group. We consider the problem of counting simultaneous conjugacy classes of $n$-tuples and simultaneous conjugacy classes of commuting $n$-tuples in $G$. Let $α_{G,n}$ denote the number of simultaneous conjugacy classes of $n$-tuples, and $β_{G,n}$ the number of simultaneous conjugacy classes of commuting $n$-tuples in $G$. The generating functions $A_G(t) = \sum_{n\geq 0} α_{G,n}t^n,$ and $B_G(t) = \sum_{n\geq 0} β_{G,n}t^n$ are rational functions of $t$. We show that $A_G(t)$ determines and is completely determined by the class equation of $G$. We show that $α_{G,n}$ gro…
It is known that the notion of a transitive subgroup of a permutation group $G$ extends naturally to subsets of $G$. We consider subsets of the general linear group $\operatorname{GL}(n,q)$ acting transitively on flag-like structures, which are common generalisations of $t$-dimensional subspaces of $\mathbb{F}_q^n$ and bases of $t$-dimensional subspaces of $\mathbb{F}_q^n$. We give structural characterisations of transitive subsets of $\operatorname{GL}(n,q)$ using the character theory of $\operatorname{GL}(n,q)$ and interprete such subsets as designs in the conjugacy class association scheme …
Sharp bounds for intervals in finite subgroup lattices
Beijing Institute of Mathematical Sciences and Applications（北京数学与应用高等研究院）；Dipartimento di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca（米兰比可卡大学数学与应用系）
Let $G$ be a reductive algebraic group---possibly non-connected---over a field $k$ and let $H$ be a subgroup of $G$. If $G= GL_n$ then there is a degeneration process for obtaining from $H$ a completely reducible subgroup $H'$ of $G$; one takes a limit of $H$ along a cocharacter of $G$ in an appropriate sense. We generalise this idea to arbitrary reductive $G$ using the notion of $G$-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a $G$-completely reducible subgroup $H'$ of $G$, uniq…
Let $S_\infty$ denote the topological group of permutations of the natural numbers. We study the complexity of the isomorphism relation on classes of closed subgroups $S_\infty$ in the setting of Borel reducibility between equivalence relations on Polish spaces. Given a closed subgroup $G$ of $S_\infty$, the coarse group $\mathcal M(G)$ is the structure with domain the cosets of open subgroups of $G$, and a ternary relation $AB \sqsubseteq C$. If $G$ has only countably many open subgroups, then $\mathcal M(G)$ is a countable structure. Coarse groups form our main tool in studying such closed s…
