# Questions tagged [geometric-group-theory]

Large scale properties of groups; growth functions; Dehn functions; small cancellation properties; hyperbolicity and CAT(0); actions and representations; combinatorial group theory; presentations

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### Solution of an equation over free group

Let $F_n$ be a free group on $n$ generators. Let $w \in F_n$ be a word such that there does not exist any solution in $F_n$ for the equation $w.w(t_1, \ldots, t_n) = 1$, where $t_1, \ldots, t_n$ are ...
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In his 1972 paper Sur la cohomologie des groupes topologiques II, Guichardet proved$^\ast$ that (non-abelian) free groups satisfy the following strong converse of property (T): The $1$-cohomology $H^1(... 6 votes 1 answer 191 views ### Does the inner automorphism group of the fundamental group of a closed aspherical manifold always have an element of infinite order? Let$\pi_1$be the fundamental group of a closed aspherical manifold of dimension$n$. In particular,$\pi_1$is finitely presented, torsion-free and its cohomology is finitely generated and satisfies ... 1 vote 0 answers 63 views ### Automorphic images of cones in free group Let$F_2$be the free group with basis$\{a,b\}$, with corresponding word metric$d$. For$x\in F_2$, the cone$C(x)$is$C(x):=\{y\in F_2\mid d(1,y)=d(1,x)+d(x,y)\}$, that is, the set of elements ... 1 vote 0 answers 63 views ### Basis of subgroup of free group Let$F_2$be a free group on$2$generators$a, b$. We know$b$and a conjugate of$b$, which is different from$b$, generate rank 2 free subgroup of$F_2$and they are free generating set of the ... -1 votes 0 answers 103 views ### Examples of a group with infinitely many ends which are not represented as a free product of groups Let$F_1$and$F_2$-non-trivial groups. Is it correct that the number of ends of the free product$F_1\ast F_2$is infinite? My thoughts about this: Since$e(G)=\infty$then$G=F_1\ast F_2$, a non-... 2 votes 1 answer 151 views ### Markov property for groups? My question again refers to the following article: Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.1007/s00222-023-01200-w, arXiv:... 1 vote 0 answers 75 views ### The free products of ﬁnitely many ﬁnitely generated groups are hyperbolic relative to the factors Are there any references how to show that:the free products of ﬁnitely many ﬁnitely generated groups are hyperbolic relative to the free factors. More precisely, how to show that$G = A \ast B $is ... 8 votes 2 answers 537 views ### Analogous results in geometric group theory and Riemannian geometry? As you can see from my other question I concern mmyself with the following article at the moment: Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–... 2 votes 0 answers 65 views ### Further questions to limit groups and an article of Fujiwara and Sela I already have asked a question to the following article: Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.1007/s00222-023-01200-w, ... 3 votes 1 answer 261 views ### Question to limit groups (over free groups) My question refers to the following article (to page 26: proof of Theorem 4.1): Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.... 5 votes 1 answer 284 views ### Does every sequence of group epimorphisms (between finitely generated groups) contain a stable subsequence? I have a question that is related to the topic of limit groups: Let$G$and$H$be finitely generated groups and let$(\varphi_n: G \to H)_{n \in \mathbb{N}}$be a sequence of group epimorphisms. Does ... 2 votes 1 answer 93 views ### Proof of the connection of the growth functions of a residually finite group and all of its finite quotients I was reading the research article entitled "Asymptotic growth of finite groups" by Sarah Black. Professor Black makes the following statement at the bottom of page 406: Indeed, given a f.g.... 1 vote 2 answers 130 views ### Is the canonical map from isometry group of a Gromov hyperbolic space to homeomorphisms of its Gromov boundary injective? Suppose X is a proper Gromov hyperbolic space and$\partial X$is its Gromov boundary. It is well-known that there is a canonical group homomorphism$\Phi$from the isometry group of X to the group ... 8 votes 1 answer 358 views ### Classes of groups with polynomial time isomorphism problem It is known that the isomorphism problem for finitely presented groups is in general undecidable. What are some classes of groups whose isomorphism problem is known to be solvable in polynomial time? (... 16 votes 1 answer 685 views ### A "simpler" description of the automorphism group of the lamplighter group I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts can point me to some relevant references. The lamplighter group is defined by the ... 9 votes 1 answer 354 views ### Morse theory on outer space via the lengths of finitely many conjugacy classes Let$F_n$be the free group on letters$\{x_1,\ldots,x_n\}$and let$X_n$be the (reduced) outer space of rank$n$. Points of$X_n$thus correspond to pairs$(G,\mu)$, where$G$is a finite connected ... 2 votes 0 answers 111 views ### Proof of Zimmer's cocycle super-rigidity theorem I was reading the proof of Zimmer's cocycle super-rigidity theorem from the book 'Ergodic theory and semi-simple groups' by Robert Zimmer (Theorem 5.2.5, page 98). But I am not able to understand it. ... 0 votes 0 answers 63 views ### Are Gromov-hyperbolic groups roughly starlike? [duplicate] Given a Cayley graph of a finitely generated Gromov-hyperbolic group$G$, does there exists$R>0$such that every element$g \in G$is at most distance$R$away from a geodesic ray starting at ... 2 votes 1 answer 54 views ### Can the stabiliser of a 'parabolic end' of a group stabilise an invariant line? Let$G$be a group acting freely and cocompactly on an infinite-ended graph$\Gamma$. In particular,$G$is finitely generated and acts as a convergence group on the Cantor set$\rm Ends(\Gamma)$. Let ... 12 votes 0 answers 177 views ### Are there free and discrete subgroups of$\mathrm{SL}_2(\mathbb{R}) \times \mathrm{SL}_2(\mathbb{R})$that are not Schottky on any factor?$\DeclareMathOperator\SL{SL}$Does there exist a free and discrete subgroup$\Gamma < \SL_2(\mathbb{R}) \times \SL_2(\mathbb{R})$such that neither$\pi_1(\Gamma)$nor$\pi_2(\Gamma)$is free and ... 5 votes 0 answers 69 views ### Integral over quotient of discrete group Let$Y$be a proper metric space. By a lattice we mean a discontinuous group of isometries$\Gamma$with compact quotient$Y/\Gamma$. You may also assume that$\Gamma$acts freely. Suppose we are ... 2 votes 0 answers 144 views ### Commuting conjugate elements in torsion-free groups I have come across the following question while studying projective modules over integral groups rings of torsion-free groups. Given a non-unit$x\in G$a torsion-free group, does there exist$g\in G$... 5 votes 0 answers 122 views ### Is fundamental group of a finite volume, negatively curved, cusped manifold a non-uniform lattice?$\DeclareMathOperator\Mob{Mob}$Some background: (1) A Riemannian manifold$M$is pinched negatively curved if there is a constant$\tau<\kappa<0$such that all the sectional curvatures are ... 4 votes 1 answer 82 views ###$K(\pi,1)$-conjecture ofr Artin groups behave well with respect to special subgroups. Reference-Request For a proof for an article I would need the following result: If$A_\Gamma$is an Artin group such that the$K(\pi,1)$-conjecture holds for it and$\Gamma'\subset\Gamma$is an induced subgraph, then ... 6 votes 1 answer 304 views ### Examples of groups that are unknown to be acylindrically hyperbolic Let$G$be a group. We say that$G$is acylindrically hyperbolic (for short, AH) if$G$admits an isometric, acylindrical, and non-elementary action on some Gromov hyperbolic space$X$. Here is the ... 5 votes 1 answer 200 views ### Extreme amenability of topological groups and invariant means Recently I'm reading the paper Ramsey–Milman phenomenon, Urysohn metric spaces, and extremely amenable groups by Pestov. When it comes to the definition of an extremely amenable topological group, it ... 5 votes 1 answer 206 views ### Cancellation of elements in the Gromov boundary of a free group Let$A$be a finite set of free generators and their inverses and$F$the free group generated by elements in$A$(some call$A$the alphabet of$F$). For each$g\in F$, use$\vert\,g\,\vert$to ... 3 votes 0 answers 156 views ### Write an Artin group as an HNN-extension Assume that$A_\Gamma$is an Artin group and$\chi:A_\Gamma\to(\mathbb{Z},+)$is a group homomorphism of the following form.$\Gamma=\Gamma_1\cup\Gamma_2$with$\Gamma_1\cap\Gamma_2=\emptyset,A_{\...
Let $F_2$ be free group of rank two with generators $a$ and $b$. If $H$ is a subgroup of $F_2$ generated by $d\geq 2$ elements with $$H=\langle a,b^{-k}ab^k, k=1,2,...,d-1\rangle,$$ I was trying to ...