Joerg Winkelmann
Let G be a connected real Lie group of dimension n. Then there exists a relatively compact open neighbourhood W of e in G such that for n+1 randomly chosen elements g_0,..,g_n the generated subgroup will be dense in G with probability one.
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It is known that in any free group the isolator of finitely generated subgroup is finitely generated subgroup. A very simple proof of this statement is proposed.
We prove that the free product of two finitely presented locally tame groups is locally tame and describe many examples of tame subgroups of finitely presented groups. We also include some open problems related to tame subgroups.
It is proved that all finitely generated subgroups of generalized free product of two groups are finitely separable provided that free factors have this property and amalgamated subgroups are normal in corresponding factors and satisfy the maximum condition for subgroups.
In this note we prove that every finitely presented subgroup of a systolic group is itself systolic.
We prove the pro-supersolvable closure of a finitely generated subgroup of the free group is finitely generated. It extends similar results for pro-$p$ closures proved by Ribes-Zalesskii and pro-Nilpotent closures proved by Margolis-Sapir-Weil.
In this article we generalize the theory of subgroup graphs of subgroups of free groups to finite index subgroups $H$ of finitely generated groups $G$. We study and prove various properties of $H$ in relation to its subgroup graph $Γ(H)$. For a finitely generated group $G$ we consider the poset $P_{\text{fi}}(G)$ of all right cosets of all proper finite index subgroups of $G$. We use the theory of subgroup graphs to prove that for many finitely generated infinite groups the order complex $ΔP_{\text{fi}} (G)$ and the corresponding nerve complex are contractible.
We prove that every finitely generated soluble group which is not virtually abelian has a subgroup of one of a small number of types.
A subgroup $Δ\leq Γ$ is commensurated if $|Δ:Δ\cap γΔγ^{-1}|<\infty$ for all $γ\in Γ$. We show a finitely generated branch group is just infinite if and only if every commensurated subgroup is either finite or of finite index. As a consequence, every commensurated subgroup of the Grigorchuk group is either finite or finite index.
Given a reflection $r$ in a Coxeter group $W$ (possibly of infinite rank), we consider the subgroup of $W$ generated by the reflections in $W$ having (-1)-eigenvectors orthogonal to the (-1)-eigenvector of $r$. In this paper, we determine completely when this subgroup is finitely generated, by using preceding results on centralizers of reflections. This result provides a new and naturally constructed example of a family of non-finitely generated subgroups of finitely generated groups.
We prove that for arbitrary two finitely generated subgroups A and B having infinite index in a free group F, there is a subgroup H of finite index in B such that the subgroup generated by A and H has infinite index in F. The main corollary of this theorem says that a noncyclic free group of finite rank admits a faithful highly transitive action on an infinite set, whereas the restriction of this action to any finitely generated subgroup of infinite index in F has no infinite orbits.
Brady proved that there are hyperbolic groups with finitely presented subgroups that are not of type $FP_3$ (and hence not hyperbolic). We reprove Brady's theorem by presenting a new construction. Our construction uses Bestvina-Brady Morse theory, but does not involve branched coverings.
We construct new examples of CAT(0) groups containing non finitely presented subgroups that are of type $FP_2$, these CAT(0) groups do not contain copies of $\mathbb{Z}^3$. We also give a construction of groups which are of type $F_n$ but not $F_{n+1}$ with no free abelian subgroups of rank greater than $\lceil\frac{n}{3}\rceil$.
We describe a general technique for embedding certain amalgamated products into direct products. This technique provides us with a way of constructing a host of finitely presented subgroups of automatic groups which are not even asynchronously automatic. We can also arrange that such subgroups satisfy, at best, an exponential isoperimetric inequality.
We study the relative growth of finitely generated subgroups in finitely generated groups, and the corresponding distortion function of the embeddings. We explore which functions are equivalent to the relative growth functions and distortion functions of finitely generated subgroups. We also study the connections between these two asymptotic invariants of group embeddings. We give conditions under which a length function on a finitely generated group can be extended to a length function on a larger group.
For all sufficiently large odd integers $n$, the following version of Higman's embedding theorem is proved in the variety ${\cal B}_n$ of all groups satisfying the identity $x^n=1$. A finitely generated group $G$ from ${\cal B}_n$ has a presentation $G=\langle A\mid R\rangle$ with a finite set of generators $A$ and a recursively enumerable set $R$ of defining relations if and only if it is a subgroup of a group $H$ finitely presented in the variety ${\cal B}_n$. It follows that there is a 'universal' $2$-generated finitely presented in ${\cal B}_n$ group containing isomorphic copies of all fin…
This paper studies effective separability for subgroups of finitely generated nilpotent groups and more broadly effective subgroup separability of finitely generated nilpotent groups. We provide upper and lower bounds that are polynomial with respect to the logarithm of the word length for infinite index subgroups of nilpotent groups. In the case of normal subgroups, we provide an exact computation generalizing work of the second author. We introduce a function that quantifies subgroup separability, and we provide polynomial upper and lower bounds. We finish by demonstrating that our results e…
On the structure of finitely generated subgroups of branch groups
本文描述了具有子群归纳性质的分支群（如第一Grigorchuk群和挠GGS群）有限生成子群的块结构。
For a family of group words $w$ we show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely many subgroups with a prescribed property, then $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite exponent $e$ whose union contains all $γ_k$-values in $G$, it is shown that $γ_k(G)$ has finite $(e,k,s)$-bounded exponent. If $G$ contains finitely many subgroups $G_1,G_2,...,G_s$ of finite rank $r$ whose union contain…
Let G be a right-angled Artin group. We use geometric methods to compute a presentation of the subgroup H of Aut(G) consisting of the automorphisms that send each generator to a conjugate of itself. This generalizes a result of McCool on basis-conjugating automorphisms of free groups.
We construct the first examples of finitely presented groups with cubic Dehn function containing a finitely generated infinite torsion subgroup. Moreover, we show that any infinite free Burnside group with sufficiently large odd exponent can be embedded as a subgroup of a finitely presented group with cubic Dehn function.
We give new and improved results on the freeness of subgroups of free profinite groups: A subgroup containing the normal closure of a finite word in the elements of a basis is free; Every infinite index subgroup of a finitely generated nonabelian free profinite group is contained in an infinitely generated free profinite subgroup. These results are combined with the twisted wreath product approach of Haran, an observation on the action of compact groups, and a rank counting argument to prove a conjecture of Bary-Soroker, Fehm, and Wiese, thus providing a quite general sufficient condition for …
A classical result of Schreier states that nontrivial finitely generated normal subgroups of free groups are of finite index, that is, free groups can only quotient to finite groups with finitely generated kernel. In this note we extend this result to the class of right-angled Artin groups (RAAGs). More precisely, we prove that the quotient of a RAAG by a finitely generated (full) normal subgroup is abelian-by-finite and finite-by-abelian. As Schreier's result extends to nontrivial free products of groups, we further show that our result extends to graph products of groups. As a corollary, we …
A subgroup Q is commensurated in a group G if each G conjugate of Q intersects Q in a group that has finite index in both Q and the conjugate. So commensurated subgroups are similar to normal subgroups. Semistability and simple connectivity at infinity are geometric asymptotic properties of finitely presented groups. In this paper we generalize several of the classic semistability and simple connectivity at infinity results for finitely presented groups. In particular, we show that if a finitely generated group G contains an infinite finitely generated commensurated subgroup Q of infinite inde…
We generalise the constructions of Brady and Lodha to give infinite families of hyperbolic groups, each having a finitely presented subgroup that is not of type $F_3$. By calculating the Euler characteristic of the hyperbolic groups constructed, we prove that infinitely many of them are pairwise non isomorphic. We further show that the first of these constructions cannot be generalised to dimensions higher than $3$.
We prove a variant of the well-known Reidemeister-Schreier theorem for finitely $L$-presented groups. More precisely, we prove that each finite index subgroup of a finitely $L$-presented group is itself finitely $L$-presented. Our proof is constructive and it yields a finite $L$-presentation for the subgroup. We further study conditions on a finite index subgroup of an invariantly finitely $L$-presented group to be invariantly $L$-presented itself.
In this note we introduce the class of $\mathcal H$-groups (or Hall groups) related to the class of $\mathcal B$-groups defined by Ph. Hall in 1950's. Establishing some basic properties of Hall groups we use them to obtain results concerning embeddings of abelian groups. In particular, we give an explicit classification of all abelian groups that can occur as subgroups in finitely-generated metabelian groups. Hall groups allow to give a negative answer to the Baumslag's conjecture of 1990 on the cardinality of the set of isomorphism classes for abelian subgroups in finitely generated metabelia…
We show that every countable group embeds in a group of type $FP_2$.
We construct families of $CAT(0)$ groups containing finitely presented groups whose divergence functions are of the form $r^α$ for a dense set of exponents $α\in [2,\infty)$ and $r^q\log(r)$ for integers $q \geq 2$. The same construction also yields examples of $CAT(0)$ groups containing contracting elements which are not contracting in certain finitely presented subgroups.
We will give an example of a branch group $G$ that has exponential growth but does not contain any non-abelian free subgroups. This answers question 16 from \cite{Bartholdi} positively. The proof demonstrates how to construct a non-trivial word $w_{a,b}(x,y)$ for any $a,b \in G$ such that $w_{a,b}(a,b) = 1$. The group $G$ is not just-infinite. We prove that every normal subgroup of $G$ is finitely generated as an abstract group and every proper quotient soluble. Further, $G$ has infinite virtual first Betti number but is not large.
Maximal subgroups of finitely presented special inverse monoids
本文研究了有限表示特殊逆幺半群的最大子群，证明其可能的最大子群恰好是递归表示群，并指出单位群恰好是有限生成递归表示群，首次揭示了超越单位群的子群结构复杂性。
Stallings folding theory is modified, using double coset representatives, and to applied to the study of subgroups of amalgamated products of finite rank free groups. As a first application the subgroup membership problem for such groups is shown to be decidable. An algorithm for this problem is constructed and its complexity is analysed. Groups in this class are also shown to possess the Howson property.
Just infinite quotients of finitely generated subgroups of $PL^+[0,1]$
本文证明 Richard Thompson 群 F 及两类分段同胚群的有限生成子群的恰无限商实际上交换，从而回答 Grigorchuk 的问题，并给出众多非顺从群例子。
A group G is (finitely) co-Hopfian if it does not contain any proper (finite-index) subgroups isomorphic to itself. We study finitely generated groups G that admit a descending chain of proper normal finite-index subgroups, each of which is isomorphic to G. We prove that up to finite index, these are always obtained by pulling back a chain of subgroups from a free abelian quotient. We give two applications: First, we show any characteristic proper finite-index subgroup isomorphic to G arises by pulling back a finite-index subgroup of the abelianization, and secondly, we prove special cases (fo…
Finitely generated subgroups of branch groups and subdirect products of just infinite groups
本文通过块子群概念描述了包含第一Grigorchuk群等分支群的有限生成子群结构，并基于恰好无穷群子直积的一般性结构陈述，证明了该族所有群均具有子群可分性（LERF）。
A subgroup $H$ of a group $G$ is $commensurated$ in $G$ if for each $g\in G$, $gHg^{-1}\cap H$ has finite index in both $H$ and $gHg^{-1}$. If there is a sequence of subgroups $H=Q_0\prec Q_1\prec ...\prec Q_{k}\prec Q_{k+1}=G$ where $Q_i$ is commensurated in $Q_{i+1}$ for all $i$, then $Q_0$ is $subcommensurated$ in $G$. In this paper we introduce the notion of the simple connectivity at infinity of a finitely generated group (in analogy with that for finitely presented groups). Our main result is: If a finitely generated group $G$ contains an infinite, finitely generated, subcommensurated su…
We construct the first examples of finitely presented groups with quadratic Dehn function containing a finitely generated infinite torsion subgroup. These examples are "optimal" in the sense that the Dehn function of any such finitely presented group must be at least quadratic. Moreover, we show that for any $n\geq2^{48}$ such that $n$ is either odd or divisible by $2^9$, any infinite free Burnside group with exponent $n$ is a quasi-isometrically embedded subgroup of a finitely presented group with quadratic Dehn function satisfying the Congruence Extension Property.
We prove that Thompson's group $F$ has a subgroup $H$ such that the conjugacy problem in $H$ is undecidable and the membership problem in $H$ is easily decidable. The subgroup $H$ of $F$ is a closed subgroup of $F$. That is, every function in $F$ which is a piecewise-$H$ function belongs to $H$. Other interesting examples of closed subgroups of $F$ include Jones' subgroups $\overrightarrow{F}_n$ and Jones' $3$-colorable subgroup $\mathcal F$. By a recent result of the first author, all maximal subgroups of $F$ of infinite index are closed. In this paper we prove that if $K\leq F$ is finitely g…
Finitely presented subgroups of direct products of graphs of groups with free abelian vertex groups
本文将自由群直积有限表现子群的相关结构定理推广到二维凝聚右角阿廷群（RAAGs）直积，刻画了其有限表现及$FP_n$型子群的结构，并证明这类子群的多重共轭问题与成员问题可判定。
Wreath products such as Z wr Z are not finitely-presentable yet can occur as subgroups of finitely presented groups. Here we compute the distortion of Z wr Z as a subgroup of Thompson's group F and as a subgroup of Baumslag's metabelian group G. We find that Z wr Z is undistorted in F but is at least exponentially distorted in G.
Fixed subgroups of generalised Baumslag-Solitar groups
本文研究广义Baumslag-Solitar群的自同构固定子群，刻画了在边稳定子严格包含于顶点稳定子条件下存在非有限生成固定子群的群，并证明有限阶自同构及树图情形下固定子群均有限生成。
On what finitely generated (left-orderable) simple groups can know about their subgroups
本文综述有限生成（左可序）单群嵌入的近期进展，探讨母群保留嵌入群算法等信息的嵌入方式，讨论新结果并扩展嵌入定理至可数类有限生成群。
In the first part of this note, we introduce Tietze transformations for $L$-presentations. These transformations enable us to generalize Tietze's theorem for finitely presented groups to invariantly finitely $L$-presented groups. Moreover, they allow us to prove that `being invariantly finitely $L$-presented' is an abstract property of a group which does not depend on the generating set. In the second part of this note, we consider finitely generated normal subgroups of finitely presented groups. Benli proved that a finitely generated normal subgroup of a finitely presented group is invariantl…
We prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type $\mathrm{FP}_\infty$ are virtual retracts. We describe a uniform solution to the membership problem for finitely presentable subgroups of residually free groups.
Recognition and constructive membership for discrete subgroups of $\mathrm{SL}_2(\mathbb{R})$
本文提出判定SL₂(ℝ)有限生成子群离散性、解决离散子群构造性成员资格问题并计算Fuchsian群基本域的算法，且已在Magma中实现。
We study the effects of subgroup distortion in the wreath products A wr Z, where A is finitely generated abelian. We show that every finitely generated subgroup of A wr Z has distortion function equivalent to some polynomial. Moreover, for A infinite, and for any polynomial l^k, there is a 2-generated subgroup of A wr Z having distortion function equivalent to the given polynomial. Also a formula for the length of elements in arbitrary wreath product H wr G easily shows that the group Z_2 wr Z^2 has distorted subgroups, while the lamplighter group Z_2 wr Z has no distorted (finitely generated)…
A result of Bridson, Howie, Miller and Short states that if $S$ is a subgroup of type $FP_{n}(\mathbb{Q})$ of the direct product of $n$ limit groups over free groups, then $S$ is virtually the direct product of limit groups over free groups. Furthermore, they characterise finitely presented residually free groups. In this paper these results are generalised to limit groups over Droms right-angled Artin groups. Droms RAAGs are the right-angled Artin groups with the property that all of their finitely generated subgroups are again RAAGs. In addition, we show that the generalised conjugacy proble…
We exhibit classes of groups in which the word problem is uniformly solvable but in which there is no algorithm that can compute finite presentations for finitely presentable subgroups. Direct products of hyperbolic groups, groups of integer matrices, and right-angled Coxeter groups form such classes. We discuss related classes of groups in which there does exist an algorithm to compute finite presentations for finitely presentable subgroups. We also construct a finitely presented group that has a polynomial Dehn function but in which there is no algorithm to compute the first Betti number of …
We generalize the concept of divergence of finitely generated groups by introducing the upper and lower relative divergence of a finitely generated group with respect to a subgroup. Upper relative divergence generalizes Gersten's notion of divergence, and lower relative divergence generalizes a definition of Cooper-Mihalik. While the lower divergence of Cooper-Mihalik can only be linear or exponential, relative lower divergence can be any polynomial or exponential function. In this paper, we examine the relative divergence (both upper and lower) of a group with respect to a normal subgroup or …
Embedding finitely presented self-similar groups into finitely presented simple groups
该研究证明有限表现自相似群可嵌入有限表现单群，确立这类群满足Boone-Higman猜想，还通过环上矩阵群例子证明$\n\b{GL}_n(\n\b{Q})$的有限生成子群也满足该猜想。
We characterize which permutational wreath products W^(X)\rtimes G are finitely presented. This occurs if and only if G and W are finitely presented, G acts on X with finitely generated stabilizers, and with finitely many orbits on the cartesian square X^2. On the one hand, this extends a result of G. Baumslag about standard wreath products; on the other hand, this provides nontrivial examples of finitely presented groups. For instance, we obtain two quasi-isometric finitely presented groups, one of which is torsion-free and the other has an infinite torsion subgroup. Motivated by the characte…
S. Gersten announced an algorithm that takes as input two finite sequences $\vec K=(K_1,\dots, K_N)$ and $\vec K'=(K_1',\dots, K_N')$ of conjugacy classes of finitely generated subgroups of $F_n$ and outputs: (1) $\mathsf{YES}$ or $\mathsf{NO}$ depending on whether or not there is an element $θ\in \mathsf{Out}(F_n)$ such that $θ(\vec K)=\vec K'$ together with one such $θ$ if it exists and (2) a finite presentation for the subgroup of $\mathsf{Out}(F_n)$ fixing $\vec K$. S. Kalajdžievski published a verification of this algorithm. We present a different algorithm from the point of view of Culle…
We prove the Girth Alternative for finitely generated subgroups of PL_o(I). We also prove that a finitely generated subgroup of Homeo(I) which is sufficiently rich with hyperbolic-like elements has infinite girth.
We prove that every countable group with solvable power problem embeds into a finitely presented 2-generated group with solvable power and conjugacy problems.
There exist right angled Artin groups $A$ such that the isomorphism problem for finitely presented subgroups of $A$ is unsolvable, and for certain finitely presented subgroups the conjugacy and membership problems are unsolvable. It follows that if $S$ is a surface of finite type and the genus of $S$ is sufficiently large, then the corresponding decision problems for the mapping class group $Mod(S)$ are unsolvable. Every virtually special group embeds in the mapping class group of infinitely many closed surfaces. Examples are given of finitely presented subgroups of mapping class groups that h…
Let G be a finitely generated relatively hyperbolic group. We show that if no peripheral subgroup of G is hyperbolic relative to a collection of proper subgroups, then the fixed subgroup of every automorphism of G is relatively quasiconvex. It follows that the fixed subgroup is itself relatively hyperbolic with respect to a natural family of peripheral subgroups. If all peripheral subgroups of G are slender (respectively, slender and coherent), our result implies that the fixed subgroup of every automorphism of G is finitely generated (respectively, finitely presented). In particular, this hap…
Malnormal Subgroups of Finitely Presented Groups
该研究改进了Higman嵌入定理，证明了有限生成群的递归呈示性与非正常嵌入的等距性质之间的关系，并改进了Clapham和Ol'shanskii的定理。
The set of finitely generated subgroups of the group $PL_+(I)$ of orientation-preserving piecewise-linear homeomorphisms of the unit interval includes many important groups, most notably R.~Thompson's group $F$. In this paper we show that every finitely generated subgroup $G<PL_+(I)$ is either soluble, or contains an embedded copy of Brin's group $B$, a finitely generated, non-soluble group, which verifies a conjecture of the first author from 2009. In the case that $G$ is soluble, we show that the derived length of $G$ is bounded above by the number of breakpoints of any finite set of generat…
We prove that every free metabelian non--cyclic group has a finitely generated isolated subgroup which is not separable in the class of nilpotent groups. As a corollary we prove that for every prime number $p$ an arbitrary free metabelian non--cyclic group has a finitely generated $p'$--isolated subgroup which is not $p$--separable.
In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties. We give the first examples of subgroups of hyperbolic groups which are of type $FP_2$ but not finitely presented. We give uncountably many groups of type $FP_2$ with similar properties to those subgroups of hyperbolic groups. Along the way we create more subgroups of hyperbolic groups which are finitely presented but not of type $FP_3$.
In this paper we characterize the finite permutation groups $F<S_d$ on $d$ letters such that every compact open subgroup of the associated universal group $U(F)<{\rm Aut} T_d$ is topologically finitely generated. Actually we show that in this case the groups are positively finitely generated.
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