Achyuth Jayadevan

First-order separation of pro-orderable groups from a bi-orderable matrix group over Z[F₂]

Let \(\mathcal O\) be the class of bi-orderable groups and let \(\mathcal O^*\) consist of the groups in which every bi-invariant partial order extends to a bi-invariant total order. We construct a countable upper triangular matrix group \(H\) over \(\mathbb Z[F_2]\) and a senten...

A non-concise formula in the integer Heisenberg group

A first-order formula in the language of groups is concise in a class of groups if, in every group in that class, finiteness of its solution set implies finiteness of the subgroup generated by that set. We give a parameter-free positive formula with one free variable that is not...

Free products of free-group automorphism groups: a finite-rank embedding by separating cocycles

Let \(I\) be finite and let \(G_i\leq\operatorname{Aut}(F_{n_i})\), where the ranks \(n_i\) may vary with \(i\). We construct an embedding \(\mathop{\ast}_{i\in I}G_i\hookrightarrow\operatorname{Aut}(F_{\sum_i n_i+|I|+1})\). For two factors this gives \(G*H\hookrightarrow\operato...

First-order separation of pro-orderable groups from a bi-orderable matrix group over Z[F₂]

Let \(\mathcal O\) be the class of bi-orderable groups and let \(\mathcal O^*\) consist of the groups in which every bi-invariant partial order extends to a bi-invariant total order. We construct a countable upper triangular matrix group \(H\) over \(\mathbb Z[F_2]\) and a senten...

Palindromic length in free groups through reflection length and noncrossing matchings

We express palindromic length in a finitely generated free group as the minimum of two reflection lengths in a universal Coxeter group. The identity converts optimal reflection factorizations into optimal palindromic factorizations. Combined with the classical cancellation-norm r...

Punctured point spectrum of a four-regular bipartite Cayley graph

Let \(G=\langle a,t\mid a^3=t^2=1\rangle\) and \(S=\{a^ita^j:1\le i,j\le2\}\). The Cayley graph \(\Gamma=\operatorname{Cay}(G,S)\) is infinite, connected, simple, four-regular, bipartite and vertex-transitive. Its adjacency operator on \(\mathbb C^G\) has point spectrum exactly \...

A non-concise formula in the integer Heisenberg group

A first-order formula in the language of groups is concise in a class of groups if, in every group in that class, finiteness of its solution set implies finiteness of the subgroup generated by that set. We give a parameter-free positive formula with one free variable that is not...

Finite first-order recognition of freeness from subsemigroup lattices

Let \(\mathcal L(S)\) be the lattice of all subsemigroups of a nonempty semigroup \(S\), including the empty subsemigroup. There are sentences \(\Phi_{\rm fr}\) and \(\Phi_{\rm ab}\) in the language \(\{\wedge,\vee\}\) such that \(\mathcal L(S)\models\Phi_{\rm fr}\) exactly when...