This paper studies effective separability for subgroups of finitely generated nilpotent groups an... more 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 extend to virtually nilpotent groups and stating some open questions.
This paper initiates the study of effective twisted conjugacy separability for finitely generated... more This paper initiates the study of effective twisted conjugacy separability for finitely generated groups, which measures the complexity of separating distinct twisted conjugacy classes via finite quotients. The focus is on nilpotent groups, and our main result shows that there is a polynomial upper bound for twisted conjugacy separability. That allows us to study regular conjugacy separability in the case of virtually nilpotent groups, where we compute a polynomial upper bound as well. As another application, we improve the work of the second author by giving a possibly sharp upper bound for the conjugacy separability for finitely generated nilpotent groups of nilpotency class 2.
We study generalizations of conjugacy separability in restricted wreath products of groups. We pr... more We study generalizations of conjugacy separability in restricted wreath products of groups. We provide an effective upper bound for-conjugacy separability of a wreath product A ≀ B in terms of the-conjugacy separability of A and B, the growth of-cyclic subgroup separability of B and the-residual girth of B. As an application, we provide a characterization of when A ≀ B is p-conjugacy separable. We use this characterization to provide for each prime p an example of a wreath product with infinite base group that is p-conjugacy separable. We also provide asymptotic upper bounds for conjugacy separability for wreath products of nilpotent groups, which include the lamplighter groups and provide asymptotic upper bounds for conjugacy separability of the free metabelian groups.
In this article, we study the asymptotic behaviour of conjugacy separability for wreath products ... more In this article, we study the asymptotic behaviour of conjugacy separability for wreath products of abelian groups. We fully characterise the asymptotic class in the case of lamplighter groups and give exponential upper and lower bounds for generalised lamplighter groups. In the case where the base group is infinite, we give superexponential lower and upper bounds. We apply our results to obtain lower bounds for conjugacy depth functions of various wreath products of groups where the acting group is not abelian. 21 4.3. Finite base groups 4.4. Infinite base groups 5. Applications 6. Open questions 31 References
We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given t... more We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given two finitely generated torsion free nilpotent groups with the same degree of polynomial growth, but non-isomorphic Carnot completions (asymptotic cones). We show that there exists no injective Lipschitz function from one group to the other. It follows that neither group can act translation-like on the other. Examples of translation-like actions. We now give some examples of translation-like actions that do not arise from subgroups. • If there is a bilipschitz map Ψ : G→H, then H acts translation like on G by setting h ∈ H to act as the map g → ψ −1 (ψ(g)h −1). • Seward [20] showed that Z acts translation-like on every infinite group, or equivalently that having a translation-like action of Z is a complete obstruction to being finite. • Whyte [21] showed that Z * Z acts translation-like on every non-amenable group. • The first author [6] showed that Z × Z acts translation-like on the fundamental group of a closed hyperbolic 3-manifold. Known obstructions to translation-like actions. We briefly survey some known obstructions to the existence of a translation-like action. • Whyte [21] showed that Z * Z cannot act translation-like on an amenable group. Since it acts translation-like on every non-amenable group (as noted above), this shows that having a translation-like action by Z * Z is a complete obstruction to non-amenability. Note that many amenable groups (e.g., non-nilpotent elementary amenable groups [5]) admit Lipschitz injections from Z * Z, so admitting a translation-like action by some group is really a stronger notion than admitting a Lipschitz injection from that group. • If the growth function of H has a greater growth rate than G, then H cannot act translationlike on G. In particular, if G is nilpotent, then any group acting translation-like on G must also be nilpotent.
In previous work, the author claimed an effective characterization for residual finiteness and lo... more In previous work, the author claimed an effective characterization for residual finiteness and lower asymptotic bounds for conjugacy separability of all finitely generated nilpotent groups. However, a counterexample to the characterization of residual finiteness for finitely generated nilpotent groups was communicated to the author which also had consequences to the lower asymptotic bound provided for conjugacy separability for the class of finitely generated nilpotent groups. Subsequently, the statements of the asymptotic characterization of residual finiteness and the asymptotic lower bound for conjugacy separability for the same class of groups along with their proofs are incorrect. In this article, we provide correct asymptotic bounds for residual finiteness and asymptotic lower bounds for conjugacy separability for the class of finitely generated nilpotent groups. We additional provide upper asymptotic bounds for conjugacy separability for the class of finitely generated nilpotent groups
In this note, we complete the study of asymptotic behaviour of conjugacy separability of the gene... more In this note, we complete the study of asymptotic behaviour of conjugacy separability of the general case of wreath products of finitely abelian groups where the base group is possibly infinite. In particular, we provide super-exponential upper and lower bounds for conjugacy separability of wreath products where the base group contains $\mathbb{Z}$ and, combining with previous work of the authors, we provide asymptotic bounds for conjugacy separability depth functions of all wreath products of finitely generated abelian groups. As an application, we give exponential lower bounds for infinitely many wreath products where the acting group is not necessarily abelian.
The functions $F_{G}(n)$ measures the asymptotic behavior of residual finiteness for a finitely g... more The functions $F_{G}(n)$ measures the asymptotic behavior of residual finiteness for a finitely generated group $G$. In previous work \cite{Pengitore_1}, the author claimed a characterization for $F_{N}(n)$ when $N$ is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and subsequently, the statement of the asymptotic characterization of $F_{N}(n)$ is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for $F_{N}(n)$ when $N$ is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of \cite{Pengitore_1} can be improved. Finally, we construct of a class of finitely generated nilpotent groups $N$ for which the asymptotic behavior of $F_N(n)$ can be fully characterized.
In this article, we completely characterise the asymptotic behaviour of conjugacy separability fo... more In this article, we completely characterise the asymptotic behaviour of conjugacy separability for the lamplighter groups. More generally, we give exponential upper and lower bounds for all wreath products of finitely generated abelian groups where the acting group is infinite and base group is finite.
This paper initiates the study of effective twisted conjugacy separability for finitely generated... more This paper initiates the study of effective twisted conjugacy separability for finitely generated groups, which measures the complexity of separating distinct twisted conjugacy classes via finite quotients. The focus is on nilpotent groups, and our main result shows that there is a polynomial upper bound for twisted conjugacy separability. That allows us to study regular conjugacy separability in the case of virtually nilpotent groups, where we compute a polynomial upper bound as well. As another application, we improve the work of the second author by giving a possibly sharp upper bound for the conjugacy separability for finitely generated nilpotent groups of nilpotency class 2.
This paper studies effective separability for subgroups of finitely generated nilpotent groups an... more 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 extend to virtually nilpotent groups and stating some open questions.
We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given t... more We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given two finitely generated torsion-free nilpotent groups with the same degree of polynomial growth, but non-isomorphic Carnot completions (asymptotic cones). We show that there exists no injective Lipschitz function from one group to the other. It follows that neither group can act translation-like on the other. As Lipschitz injections need not be bi-Lipschitz embeddings, this is a strengthening of a classical result of Pansu in the context of groups of the same homogeneous dimension.
We give effective proofs of residual finiteness and conjugacy separability for finitely generated... more We give effective proofs of residual finiteness and conjugacy separability for finitely generated nilpotent groups. In particular, we give precise asymptotic bounds for a function introduced by Bou-Rabee that measures how large the quotients that are needed to separate non-identity elements of bounded length from the identity which improves the work of Bou-Rabee. Similarly, we give polynomial upper and lower bounds for an analogous function introduced by Lawton, Louder, and McReynolds that measures how large the quotients that are needed to separate pairs of distinct conjugacy classes of bounded word length using work of Blackburn and Mal'tsev.
2009 2nd International Symposium on Resillient Control Systems, 2009
This paper explores the overlaps between the Control community's work on System IdentiÀcation (Sy... more This paper explores the overlaps between the Control community's work on System IdentiÀcation (SysID) and the Physics, Mathematics, Chaos, and Complexity communities' work on phase-space reconstruction via time-delay embedding. There are numerous overlaps between the goals of each community. Nevertheless, the Controls community can gain new insight as well as some new very powerful tools for SysID from the latest developments within the Physics, Mathematics, Chaos, and Complexity communities. These insights are gained via the work on phase-space reconstruction of non-linear dynamics. New methods for discovering non-linear differential based equations that evolved from embedding operations can shed new light on hybrid-systems theory, Nyquest-Shannon's Theories, and network based control theory. This paper strives to guide the Controls community towards a closer inspection of the tools and additional insights being developed within the Physics, Mathematics, Chaos, and Complexity communities for discovery of system dynamics, the Àrst step in control system development. The paper introduces the concepts of phase-space reconstruction via time-delay embedding (made famous by Whitney, Takens, and Sauer's Thoreoms), intergrate-and-Àre embedding, and non-linear differential equation discovery based on Perona's method.
Separability for groups refers to the question which subsets of a group can be detected in its fi... more Separability for groups refers to the question which subsets of a group can be detected in its finite quotients. Classically, separability is studied in terms of which classes have a certain separability property, and this question is related to algorithmic problems in groups such as the word problem. A more recent perspective tries to study the order of the smallest finite quotient in which one detects the subset under consideration depending on its complexity, measured using the word norm on a finitely generated group. In this survey, we present what is currently known in the field of effective separability and give an overview of the open questions for several classes of groups.
In this note, we study non-transitive graphs and prove a number of results when they satisfy a co... more In this note, we study non-transitive graphs and prove a number of results when they satisfy a coarse version of transitivity. Also, for each finitely generated group G, we produce continuum many pairwise non-quasi-isometric regular graphs that have the same growth rate, number of ends, and asymptotic dimension as G.
A group is said to hae a rational growth with respect to the generating set if the growth series ... more A group is said to hae a rational growth with respect to the generating set if the growth series is a rational polynomial. It was shown by Parry that a subset of torus bundle groups exhibits rational growth. We generalize this result to other torus bundle groups.
We study generalisations of conjugacy separability in restricted wreath products of groups. We pr... more We study generalisations of conjugacy separability in restricted wreath products of groups. We provide an effective upper bound for $\mathcal{C}$-conjugacy separability of a wreath product $A \wr B$ in terms of the $\mathcal{C}$-conjugacy separability of $A$ and $B$, the growth of $\mathcal{C}$-cyclic subgroup separability of $B$, and the $\mathcal{C}$-residual girth of $B.$ As an application, we provide a characterisation of when $A \wr B$ is $p$-conjugacy separable. We use this characterisation to the provide for each prime $p$ an example of wreath products with infinite base group that are $p$-conjugacy separable. We also provide asymptotic upper bounds for conjugacy separability for wreath products of nilpotent groups which include the lamplighter groups and provide asymptotic upper bounds for conjugacy separability of the free metabelian groups. Along the way, we provide a polynomial upper bound for the shortest conjugator between two elements of length at most $n$ in a finitel...
The function FG(n){\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distin... more The function FG(n){\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distinguish a nontrivial element of G from the identity with a surjective group morphism as one varies over nontrivial elements of word length at most n. In previous work [M. Pengitore, Effective separability of finitely generated nilpotent groups, New York J. Math. 24 2018, 83–145], the author claimed a characterization for FN(n){\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and consequently, the statement of the asymptotic characterization of FN(n){\mathrm{F}_{N}(n)} is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for FN(n){\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of the above article can be improved. Finally, we construct a class o...
This paper studies effective separability for subgroups of finitely generated nilpotent groups an... more 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 extend to virtually nilpotent groups and stating some open questions.
This paper initiates the study of effective twisted conjugacy separability for finitely generated... more This paper initiates the study of effective twisted conjugacy separability for finitely generated groups, which measures the complexity of separating distinct twisted conjugacy classes via finite quotients. The focus is on nilpotent groups, and our main result shows that there is a polynomial upper bound for twisted conjugacy separability. That allows us to study regular conjugacy separability in the case of virtually nilpotent groups, where we compute a polynomial upper bound as well. As another application, we improve the work of the second author by giving a possibly sharp upper bound for the conjugacy separability for finitely generated nilpotent groups of nilpotency class 2.
We study generalizations of conjugacy separability in restricted wreath products of groups. We pr... more We study generalizations of conjugacy separability in restricted wreath products of groups. We provide an effective upper bound for-conjugacy separability of a wreath product A ≀ B in terms of the-conjugacy separability of A and B, the growth of-cyclic subgroup separability of B and the-residual girth of B. As an application, we provide a characterization of when A ≀ B is p-conjugacy separable. We use this characterization to provide for each prime p an example of a wreath product with infinite base group that is p-conjugacy separable. We also provide asymptotic upper bounds for conjugacy separability for wreath products of nilpotent groups, which include the lamplighter groups and provide asymptotic upper bounds for conjugacy separability of the free metabelian groups.
In this article, we study the asymptotic behaviour of conjugacy separability for wreath products ... more In this article, we study the asymptotic behaviour of conjugacy separability for wreath products of abelian groups. We fully characterise the asymptotic class in the case of lamplighter groups and give exponential upper and lower bounds for generalised lamplighter groups. In the case where the base group is infinite, we give superexponential lower and upper bounds. We apply our results to obtain lower bounds for conjugacy depth functions of various wreath products of groups where the acting group is not abelian. 21 4.3. Finite base groups 4.4. Infinite base groups 5. Applications 6. Open questions 31 References
We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given t... more We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given two finitely generated torsion free nilpotent groups with the same degree of polynomial growth, but non-isomorphic Carnot completions (asymptotic cones). We show that there exists no injective Lipschitz function from one group to the other. It follows that neither group can act translation-like on the other. Examples of translation-like actions. We now give some examples of translation-like actions that do not arise from subgroups. • If there is a bilipschitz map Ψ : G→H, then H acts translation like on G by setting h ∈ H to act as the map g → ψ −1 (ψ(g)h −1). • Seward [20] showed that Z acts translation-like on every infinite group, or equivalently that having a translation-like action of Z is a complete obstruction to being finite. • Whyte [21] showed that Z * Z acts translation-like on every non-amenable group. • The first author [6] showed that Z × Z acts translation-like on the fundamental group of a closed hyperbolic 3-manifold. Known obstructions to translation-like actions. We briefly survey some known obstructions to the existence of a translation-like action. • Whyte [21] showed that Z * Z cannot act translation-like on an amenable group. Since it acts translation-like on every non-amenable group (as noted above), this shows that having a translation-like action by Z * Z is a complete obstruction to non-amenability. Note that many amenable groups (e.g., non-nilpotent elementary amenable groups [5]) admit Lipschitz injections from Z * Z, so admitting a translation-like action by some group is really a stronger notion than admitting a Lipschitz injection from that group. • If the growth function of H has a greater growth rate than G, then H cannot act translationlike on G. In particular, if G is nilpotent, then any group acting translation-like on G must also be nilpotent.
In previous work, the author claimed an effective characterization for residual finiteness and lo... more In previous work, the author claimed an effective characterization for residual finiteness and lower asymptotic bounds for conjugacy separability of all finitely generated nilpotent groups. However, a counterexample to the characterization of residual finiteness for finitely generated nilpotent groups was communicated to the author which also had consequences to the lower asymptotic bound provided for conjugacy separability for the class of finitely generated nilpotent groups. Subsequently, the statements of the asymptotic characterization of residual finiteness and the asymptotic lower bound for conjugacy separability for the same class of groups along with their proofs are incorrect. In this article, we provide correct asymptotic bounds for residual finiteness and asymptotic lower bounds for conjugacy separability for the class of finitely generated nilpotent groups. We additional provide upper asymptotic bounds for conjugacy separability for the class of finitely generated nilpotent groups
In this note, we complete the study of asymptotic behaviour of conjugacy separability of the gene... more In this note, we complete the study of asymptotic behaviour of conjugacy separability of the general case of wreath products of finitely abelian groups where the base group is possibly infinite. In particular, we provide super-exponential upper and lower bounds for conjugacy separability of wreath products where the base group contains $\mathbb{Z}$ and, combining with previous work of the authors, we provide asymptotic bounds for conjugacy separability depth functions of all wreath products of finitely generated abelian groups. As an application, we give exponential lower bounds for infinitely many wreath products where the acting group is not necessarily abelian.
The functions $F_{G}(n)$ measures the asymptotic behavior of residual finiteness for a finitely g... more The functions $F_{G}(n)$ measures the asymptotic behavior of residual finiteness for a finitely generated group $G$. In previous work \cite{Pengitore_1}, the author claimed a characterization for $F_{N}(n)$ when $N$ is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and subsequently, the statement of the asymptotic characterization of $F_{N}(n)$ is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for $F_{N}(n)$ when $N$ is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of \cite{Pengitore_1} can be improved. Finally, we construct of a class of finitely generated nilpotent groups $N$ for which the asymptotic behavior of $F_N(n)$ can be fully characterized.
In this article, we completely characterise the asymptotic behaviour of conjugacy separability fo... more In this article, we completely characterise the asymptotic behaviour of conjugacy separability for the lamplighter groups. More generally, we give exponential upper and lower bounds for all wreath products of finitely generated abelian groups where the acting group is infinite and base group is finite.
This paper initiates the study of effective twisted conjugacy separability for finitely generated... more This paper initiates the study of effective twisted conjugacy separability for finitely generated groups, which measures the complexity of separating distinct twisted conjugacy classes via finite quotients. The focus is on nilpotent groups, and our main result shows that there is a polynomial upper bound for twisted conjugacy separability. That allows us to study regular conjugacy separability in the case of virtually nilpotent groups, where we compute a polynomial upper bound as well. As another application, we improve the work of the second author by giving a possibly sharp upper bound for the conjugacy separability for finitely generated nilpotent groups of nilpotency class 2.
This paper studies effective separability for subgroups of finitely generated nilpotent groups an... more 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 extend to virtually nilpotent groups and stating some open questions.
We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given t... more We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given two finitely generated torsion-free nilpotent groups with the same degree of polynomial growth, but non-isomorphic Carnot completions (asymptotic cones). We show that there exists no injective Lipschitz function from one group to the other. It follows that neither group can act translation-like on the other. As Lipschitz injections need not be bi-Lipschitz embeddings, this is a strengthening of a classical result of Pansu in the context of groups of the same homogeneous dimension.
We give effective proofs of residual finiteness and conjugacy separability for finitely generated... more We give effective proofs of residual finiteness and conjugacy separability for finitely generated nilpotent groups. In particular, we give precise asymptotic bounds for a function introduced by Bou-Rabee that measures how large the quotients that are needed to separate non-identity elements of bounded length from the identity which improves the work of Bou-Rabee. Similarly, we give polynomial upper and lower bounds for an analogous function introduced by Lawton, Louder, and McReynolds that measures how large the quotients that are needed to separate pairs of distinct conjugacy classes of bounded word length using work of Blackburn and Mal'tsev.
2009 2nd International Symposium on Resillient Control Systems, 2009
This paper explores the overlaps between the Control community's work on System IdentiÀcation (Sy... more This paper explores the overlaps between the Control community's work on System IdentiÀcation (SysID) and the Physics, Mathematics, Chaos, and Complexity communities' work on phase-space reconstruction via time-delay embedding. There are numerous overlaps between the goals of each community. Nevertheless, the Controls community can gain new insight as well as some new very powerful tools for SysID from the latest developments within the Physics, Mathematics, Chaos, and Complexity communities. These insights are gained via the work on phase-space reconstruction of non-linear dynamics. New methods for discovering non-linear differential based equations that evolved from embedding operations can shed new light on hybrid-systems theory, Nyquest-Shannon's Theories, and network based control theory. This paper strives to guide the Controls community towards a closer inspection of the tools and additional insights being developed within the Physics, Mathematics, Chaos, and Complexity communities for discovery of system dynamics, the Àrst step in control system development. The paper introduces the concepts of phase-space reconstruction via time-delay embedding (made famous by Whitney, Takens, and Sauer's Thoreoms), intergrate-and-Àre embedding, and non-linear differential equation discovery based on Perona's method.
Separability for groups refers to the question which subsets of a group can be detected in its fi... more Separability for groups refers to the question which subsets of a group can be detected in its finite quotients. Classically, separability is studied in terms of which classes have a certain separability property, and this question is related to algorithmic problems in groups such as the word problem. A more recent perspective tries to study the order of the smallest finite quotient in which one detects the subset under consideration depending on its complexity, measured using the word norm on a finitely generated group. In this survey, we present what is currently known in the field of effective separability and give an overview of the open questions for several classes of groups.
In this note, we study non-transitive graphs and prove a number of results when they satisfy a co... more In this note, we study non-transitive graphs and prove a number of results when they satisfy a coarse version of transitivity. Also, for each finitely generated group G, we produce continuum many pairwise non-quasi-isometric regular graphs that have the same growth rate, number of ends, and asymptotic dimension as G.
A group is said to hae a rational growth with respect to the generating set if the growth series ... more A group is said to hae a rational growth with respect to the generating set if the growth series is a rational polynomial. It was shown by Parry that a subset of torus bundle groups exhibits rational growth. We generalize this result to other torus bundle groups.
We study generalisations of conjugacy separability in restricted wreath products of groups. We pr... more We study generalisations of conjugacy separability in restricted wreath products of groups. We provide an effective upper bound for $\mathcal{C}$-conjugacy separability of a wreath product $A \wr B$ in terms of the $\mathcal{C}$-conjugacy separability of $A$ and $B$, the growth of $\mathcal{C}$-cyclic subgroup separability of $B$, and the $\mathcal{C}$-residual girth of $B.$ As an application, we provide a characterisation of when $A \wr B$ is $p$-conjugacy separable. We use this characterisation to the provide for each prime $p$ an example of wreath products with infinite base group that are $p$-conjugacy separable. We also provide asymptotic upper bounds for conjugacy separability for wreath products of nilpotent groups which include the lamplighter groups and provide asymptotic upper bounds for conjugacy separability of the free metabelian groups. Along the way, we provide a polynomial upper bound for the shortest conjugator between two elements of length at most $n$ in a finitel...
The function FG(n){\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distin... more The function FG(n){\mathrm{F}_{G}(n)} gives the maximum order of a finite group needed to distinguish a nontrivial element of G from the identity with a surjective group morphism as one varies over nontrivial elements of word length at most n. In previous work [M. Pengitore, Effective separability of finitely generated nilpotent groups, New York J. Math. 24 2018, 83–145], the author claimed a characterization for FN(n){\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. However, a counterexample to the above claim was communicated to the author, and consequently, the statement of the asymptotic characterization of FN(n){\mathrm{F}_{N}(n)} is incorrect. In this article, we introduce new tools to provide lower asymptotic bounds for FN(n){\mathrm{F}_{N}(n)} when N is a finitely generated nilpotent group. Moreover, we introduce a class of finitely generated nilpotent groups for which the upper bound of the above article can be improved. Finally, we construct a class o...
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