# Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

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### Is there a higher categorical structure which models the (higher) conjugation actions of a group acting on itself?

Let $G$ be a group, and consider the action of $G$ on itself by conjugation. If we think of $G$ as a one object category, then the conjugation action can be realised as automorphisms of this category, ...
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### Large subgroups of infinite-dimensional vector spaces

Let $V$ be an infinite-dimensional vector space over $\mathbb{Q}$. Consider a proper subgroup $W$ of $V, +$ with the following property: each vector line $L$ (which we see as a subgroup of $V, +$) has ...
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### Ping pong with parabolic isometries on Gromov hyperbolic spaces

For a group $G$ with a non-elementary general type action by isometries on a Gromov hyperbolic geodesic space $(X,d)$, it is well known that you can construct free subgroups of $G$ via the ping pong ...
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### Subgroup rank of finite simple groups

Definition: The subgroup rank of a finite group $G$ is the minimal natural number $n$ such that every subgroup of $G$ can be generated by $n$ elements (or fewer). This invariant has been studied ...
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### Empty preimage under homomorphism of finitely presented groups with decidable word problems

Let $f:G\to H$ be a homomorphism of finitely presented groups with decidable word problems. Assume you are given explicit finite presentations for both $G$ and $H$ and you are given the words to which ...
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### Is the diagonal of finitely presented groups computable?

Let $f:G\to H$ be a surjective homomorphism of finitely presented groups. If the kernel of $f$ is finitely generated then is $G\times_H G$ is a finitely presented group? Can one compute an explicit ...
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### Element being trivial in a finitely presented group independent of ZFC

Is there an explicit finitely presented group $G$ and an element $g\in G$ such that the statement "$g$ is equal to the identity" is independent of ZFC?
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### Empty preimage under homomorphism of finitely presented groups independent of ZFC

Is there a homomorphism of finitely presented groups $f:G\to H$ and an element $h\in H$ such that the statement "$f^{-1}(h)$ is empty" is independent of ZFC?
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### Empty preimage under homomorphism of finitely presented groups with decidable word problems

Let $G, H$ be finitely presented groups with decidable word problems. Can there be a homomorphism $f:G\to H$ such that there is no algorithm deciding given $w\in H$ whether $f^{-1}(w)$ is empty or not?...
Let $G$ be a countable finitely generated group, with word metric $d_S$ induced by some generating set $S$. Let $B_r$ be the ball of radius $r$ around the identity under $d_S$. We say that $A \subset ... 0answers 45 views ### Quasi-isometry of solvable minimax groups [Edits in brackets] Consider two finitely generated solvable minimax groups$G_i$($i = 1,2$) so that$1 \to N_i \to G_i \to Z_i \to 1$[splits] with$N_i$nilpotent,$Z_i$infinite cyclic and$G_i$... 0answers 58 views ### When is the submonoid preserving a subspace finitely generated? Let$T$be a topological space with at least one open set whose closure is not open. Let$G$be a finitely generated group acting by homeomorphisms on$T$. Let$S\subset T$be a subspace. Under what ... 2answers 196 views ### Groups acting on products of hyperbolic spaces I am interested in groups acting properly and cocompactly by isometries on finite products of Gromov-hyperbolic metric spaces. I am mostly interested in the case where the group itself is not ... 1answer 462 views ### The sum (with multiplicity) of the cubes of irreducible character degrees of a finite group Throughout$G$is a finite, non-abelian group.$\DeclareMathOperator\Irr{Irr}\DeclareMathOperator\AD{AD}\DeclareMathOperator\cp{cp}\newcommand\card[1]{\lvert#1\rvert}$Let$\Irr(G)$be the set of ... 0answers 137 views ### Random walk on non-abelian free group Let$F_2$be the free non-abelian group with generators$a, b\in F_2$. Has the "random walk" where we start with the identity and then multiply it by$a$or$b$or$a^{-1}$or$b^{-1}$... 2answers 192 views ### Is the automorphism group of free group of rank two relatively hyperbolic? By Behrstock, Drutu and Mosher [BDM], we know that the (outer) automorphism groups$\mathrm{Aut}(F_n)$and$\mathrm{Out}(F_n)$of free group of rank$n$are not relatively hyperbolic if$n \geq 3$(... 0answers 91 views ### Calculating the polynomials which are invariant under the action of a simple finite group Let$G$be a simple, finite group. In general,$G$is not abelian. Let$\rho$be a representation of this group, where each$\rho(g)$for$g\in G$is a unitary, complex,$d$-dimensional matrix,$\rho(...
Let $X$ be a rack and $A$ be an $X$-module. By this paper, p. 33, we can associate a cochain complex $C^\bullet(X,A)$ to the pair $(X,A)$. This complex is explicitly defined by a differential $d$. I ...