dbenbenn

Solved 50

• Finitely generated nilpotent groups are finitely presented (external)Proved

  Sep 2026

• Corollary 4.7: residually elementary amenable groups have property (P)Proved

  Sep 2026

• Proposition 4.2: every elementary amenable group has property (P)Proved

  Sep 2026

• Lemma 4.1 (a): directed unions preserve property (P)Proved

  Sep 2026

• Finitely generated abelian groups have property (P)Proved

  Sep 2026

• Milnor's Lemmas 1 to 3 in the form Chou uses them: in a finitely generated exponentially bounded group, a normal subgroup with virtually polycyclic quotient is finitely generatedProved

  Sep 2026

• A normal form is nontrivialProved

  Sep 2026

• Theorem 2.5: the word of a tree diagramProved

  Sep 2026

• Theorem 4.10: $F$ is not elementary amenableProved

  Sep 2026

• The abelianization of $F$ is $\mathbb{Z} \oplus \mathbb{Z}$Proved

  Sep 2026

• Every proper quotient of $F$ is AbelianProved

  Sep 2026

• Every tree diagram represents an element of $F$Proved

  Sep 2026

• The commutator subgroup of $F$Proved

  Sep 2026

• The commutator subgroup of $F$ is simpleProved

  Sep 2026

• A reduced tree diagram is uniqueProved

  Sep 2026

• Lemma 2.8: positive elements are closed under multiplicationProved

  Sep 2026

• Every nontrivial normal subgroup of $F$ contains a copy of $F$Proved

  Sep 2026

• Every element of $F$ is given by a pair of standard dyadic partitionsProved

  Sep 2026

• Elements supported in a dyadic interval form a copy of $F$Proved

  Sep 2026

• Every element of $F$ has a reduced tree diagramProved

  Sep 2026

• Normal-form data for a reduced tree diagramProved

  Sep 2026

• $F$ is a totally ordered groupProved

  Sep 2026

• Corollary-Definition 2.7: the unique normal form in $F$Proved

  Sep 2026

• $F$ is generated by $A$ and $B$Proved

  Sep 2026

• No translation-invariant normalised functional on all real functions on the integersProved

  Sep 2026

• Theorem 4.17: if conjugation on a free abelian normal subgroup has an eigenvalue off the unit circle then the group has a free subsemigroup of rank twoProved

  Sep 2026

• Corollary 2.5: two elements whose translates of a set are disjoint and stay inside it generate a free subsemigroupProved

  Sep 2026

• Theorem 4.7: a finitely generated solvable group with no free subsemigroup of rank two is polycyclicProved

  Sep 2026

• An exponent passes through a commutator with a lower central series member, modulo two steps further downProved

  Sep 2026

• The commutators of a generating set with a set generating one lower central factor generate the next factorProved

  Sep 2026

• Independent cyclic subgroups generated by elements of infinite order give linearly independent generators over the integersProved

  Sep 2026

• A module over the integers spanned by a family has rank at most the number of members that are not torsionProved

  Sep 2026

• Proposition 3.6: the growth function of a free abelian group of rank $n$Proved

  Sep 2026

• Proposition 3.6: the growth function of a free abelian group of rank $n$Disproved

  Sep 2026

• Lemma 3.5: polynomial growth bounds do not depend on the generating setProved

  Sep 2026

• Lemma 3.4: two rescalings of a polynomial lower bound on the growth functionProved

  Sep 2026

• Hirsch's theorem: a normal series with finitely generated abelian quotients, every such series, and the maximal conditionProved

  Sep 2026

• Proposition 4.1: seven equivalent characterisations of a polycyclic groupProved

  Sep 2026

• A subgroup of finite index has a generating set on which its elements are words no longer than in the whole groupProved

  Sep 2026

• A group with a finitely generated subgroup of finite index is finitely generatedProved

  Sep 2026

• In a group with no free subsemigroup on two generators, the conjugates of one element by the powers of another generate a finitely generated subgroupProved

  Sep 2026

• Rosenblatt's Lemmas 4.8 and 4.9 in the form Chou uses them: with no free subsemigroup on two generators, a normal subgroup with virtually polycyclic quotient is finitely generatedProved

  Sep 2026

• In a group without exponential growth, the conjugates of one element by the powers of another generate a finitely generated subgroupProved

  Sep 2026

• Free groups are residually finite (external)Proved

  Sep 2026

• An extension of a finitely presented group by a finite group is finitely presentedProved

  Sep 2026

• A finitely presented group is presented on any finite generating setProved

  Sep 2026

• An extension of a finitely presented group by a finitely presented group is finitely presentedProved

  Sep 2026

• An extension of a finitely generated group by a finitely generated group is finitely generatedProved

  Sep 2026

• Theorem: a solvable group which is not polycyclic has exponential growthProved

  Sep 2026

• Lemma 3: if $B/A$ is polycyclic and $B$ is not of exponential growth, then $B$ is polycyclicProved

  Sep 2026

Posted 50

• No translation-invariant normalised functional on all real functions on the integersProved

  Sep 2026

• Theorem 4.12, linear-algebra half: if every eigenvalue of every conjugation lies on the unit circle then the group is almost nilpotentOpen

  Sep 2026

• Theorem 4.17: if conjugation on a free abelian normal subgroup has an eigenvalue off the unit circle then the group has a free subsemigroup of rank twoProved

  Sep 2026

• Lemma 4.18 (Mal'cev): a solvable group of real matrices has a finite-index subgroup that can be simultaneously triangularized over the complex numbersOpen

  Sep 2026

• Corollary 2.5: two elements whose translates of a set are disjoint and stay inside it generate a free subsemigroupProved

  Sep 2026

• Theorem 4.12, core step: a polycyclic extension of a free abelian group by a nilpotent group, with no free subsemigroup of rank two, is almost nilpotentOpen

  Sep 2026

• Theorem 4.12: a polycyclic group is almost nilpotent or contains a free subsemigroup of rank twoOpen

  Sep 2026

• Theorem 4.7: a finitely generated solvable group with no free subsemigroup of rank two is polycyclicProved

  Sep 2026

• An exponent passes through a commutator with a lower central series member, modulo two steps further downProved

  Sep 2026

• The commutators of a generating set with a set generating one lower central factor generate the next factorProved

  Sep 2026

• Independent cyclic subgroups generated by elements of infinite order give linearly independent generators over the integersProved

  Sep 2026

• A module over the integers spanned by a family has rank at most the number of members that are not torsionProved

  Sep 2026

• Theorem 3.2, upper bound: a finitely generated nilpotent group grows at most like $m^{E_2}$Open

  Sep 2026

• Theorem 3.2, lower bound: a finitely generated nilpotent group grows at least like $m^{E_1}$Open

  Sep 2026

• Theorem 4.3 (2): a polycyclic group with no nilpotent subgroup of finite index grows at least exponentiallyOpen

  Sep 2026

• Proposition 3.6: the growth function of a free abelian group of rank $n$Proved

  Sep 2026

• A subgroup of finite index has a generating set on which its elements are words no longer than in the whole groupProved

  Sep 2026

• A group with a finitely generated subgroup of finite index is finitely generatedProved

  Sep 2026

• Theorem 4.8: the Milnor–Wolf theoremOpen

  Sep 2026

• Theorem 4.3: a polycyclic group has polynomial growth or exponential growthOpen

  Sep 2026

• Hirsch's theorem: a normal series with finitely generated abelian quotients, every such series, and the maximal conditionProved

  Sep 2026

• Proposition 4.1: seven equivalent characterisations of a polycyclic groupProved

  Sep 2026

• Theorem 3.11: a finite-index subgroup is finitely generated, and polynomial growth passes up to the groupOpen

  Sep 2026

• Theorem 3.2: a finitely generated nilpotent group has polynomial growth between $m^{E_1}$ and $m^{E_2}$Open

  Sep 2026

• Lemma 3.7: independent generating sets adapted to the lower central seriesOpen

  Sep 2026

• Proposition 3.6: the growth function of a free abelian group of rank $n$Disproved

  Sep 2026

• Lemma 3.5: polynomial growth bounds do not depend on the generating setProved

  Sep 2026

• Lemma 3.4: two rescalings of a polynomial lower bound on the growth functionProved

  Sep 2026

• Rosenblatt's theorem: a finitely generated solvable group is almost nilpotent or contains a free subsemigroup on two generatorsOpen

  Sep 2026

• In a group with no free subsemigroup on two generators, the conjugates of one element by the powers of another generate a finitely generated subgroupProved

  Sep 2026

• Rosenblatt's Lemmas 4.8 and 4.9 in the form Chou uses them: with no free subsemigroup on two generators, a normal subgroup with virtually polycyclic quotient is finitely generatedProved

  Sep 2026

• Milnor's Lemmas 1 to 3 in the form Chou uses them: in a finitely generated exponentially bounded group, a normal subgroup with virtually polycyclic quotient is finitely generatedProved

  Sep 2026

• In a group without exponential growth, the conjugates of one element by the powers of another generate a finitely generated subgroupProved

  Sep 2026

• A finitely presented group is presented on any finite generating setProved

  Sep 2026

• An extension of a finitely presented group by a finitely presented group is finitely presentedProved

  Sep 2026

• An extension of a finitely generated group by a finitely generated group is finitely generatedProved

  Sep 2026

• Theorem: a solvable group which is not polycyclic has exponential growthProved

  Sep 2026

• Lemma 3: if $B/A$ is polycyclic and $B$ is not of exponential growth, then $B$ is polycyclicProved

  Sep 2026

• Lemma 2: if $B/A$ is finitely presented, $A$ is the normal closure of finitely many elementsProved

  Sep 2026

• Lemma 1: the conjugates $\beta^k \alpha \beta^{-k}$ span a finitely generated subgroupProved

  Sep 2026

• Wolf's growth function, polynomial growth, polycyclic groups and the growth exponentsDefinition

  Sep 2026

• Superseded: Rosenblatt's Lemmas 4.8-4.9 without the abelian hypothesis — use Chou.fg_of_isVirtuallyPolycyclic_quotient_of_not_hasFreeSubsemigroupOfRankTwoOpen

  Sep 2026

• Free groups have property (P)Proved

  Sep 2026

• Free groups are residually finite (external)Proved

  Sep 2026

• Corollary 4.7: residually elementary amenable groups have property (P)Proved

  Sep 2026

• Lemma 4.6 (a): property (P) from quotients separating finite setsProved

  Sep 2026

• Proposition 4.2: every elementary amenable group has property (P)Proved

  Sep 2026

• Lemma 4.1 (b): extensions preserve property (P)Proved

  Sep 2026

• Lemma 4.1 (a): directed unions preserve property (P)Proved

  Sep 2026

• Finitely generated abelian groups have property (P)Proved

  Sep 2026